††thanks: Present address: Department of Physics and Astronomy, University of
Tennessee, Knoxville TN 37996, USA††thanks: Present address: Neutron
Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA††thanks: Present address: Photon Science Division, Swiss Light
Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

# Nematic fluctuations in the non-superconducting iron pnictide
BaFe1.9-xNi0.1CrxAs2

Dongliang Gong  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China  Ming Yi
Department of Physics and Astronomy, Rice University, Houston, Texas 77005,
USA  Department of Physics, University of California, Berkeley, California
94720, USA  Meng Wang  School of Physics, Sun Yat-Sen University, Guangzhou
510275, China  Tao Xie  Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  University of Chinese Academy of Sciences, Beijing 100049, China
Wenliang Zhang  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China  Sergey
Danilkin  Australian Centre for Neutron Scattering, Australian Nuclear Science
and Technology Organisation, Lucas Heights NSW-2234, Australia  Guochu Deng
Australian Centre for Neutron Scattering, Australian Nuclear Science and
Technology Organisation, Lucas Heights NSW-2234, Australia  Xinzhi Liu  School
of Physics, Sun Yat-Sen University, Guangzhou 510275, China  Jitae T. Park
Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, D-85748
Garching, Germany  Kazuhiko Ikeuchi  Neutron Science and Technology Center,
Comprehensive Research Organization for Science and Society, Tokai, Ibaraki
319-1106, Japan  Kazuya Kamazawa  Neutron Science and Technology Center,
Comprehensive Research Organization for Science and Society, Tokai, Ibaraki
319-1106, Japan  Sung-Kwan Mo  Lawrence Berkeley National Laboratory,
Berkeley, 94720 California, USA  Makoto Hashimoto  Stanford Synchrotron
Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park,
California 94025, USA  Donghui Lu  Stanford Synchrotron Radiation Lightsource,
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA  Rui
Zhang  Department of Physics and Astronomy, Rice Center for Quantum Materials,
Rice University, Houston, Texas 77005, USA  Pengcheng Dai  Department of
Physics and Astronomy, Rice Center for Quantum Materials, Rice University,
Houston, Texas 77005, USA  Robert J. Birgeneau  Department of Physics,
University of California, Berkeley, California 94720, USA  Materials Science
Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA  Department of Materials Science and Engineering, University of
California, Berkeley, California 94720, USA  Shiliang Li  Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing 100190, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Songshan Lake Materials Laboratory, Dongguan, Guangdong
523808, China  Huiqian Luo<EMAIL_ADDRESS>Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  Songshan Lake Materials Laboratory, Dongguan, Guangdong
523808, China

###### Abstract

The main driven force of the electronic nematic phase in iron-based
superconductors is still under debate. Here, we report a comprehensive study
on the nematic fluctuations in a non-superconducting iron pnictide system
BaFe1.9-xNi0.1CrxAs2 by electronic transport, angle-resolved photoemission
spectroscopy (ARPES) and inelastic neutron scattering (INS) measurements.
Previous neutron diffraction and transport measurements suggested that the
collinear antiferromagnetism persists to $x=0.8$, with similar Néel
temperature $T_{N}$ and structural transition temperature $T_{s}$ around 32 K,
but the charge carriers change from electron type to hole type around $x=$
0.5. In this study, we have found that the in-plane resistivity anisotropy
also highly depends on the Cr dopings and the type of charge carriers. While
ARPES measurements suggest possibly weak orbital anisotropy onset near $T_{s}$
for both $x=0.05$ and $x=0.5$ compounds, INS experiments reveal clearly
different onset temperatures of low-energy spin excitation anisotropy, which
is likely related to the energy scale of spin nematicity. These results
suggest that the interplay between the local spins on Fe atoms and the
itinerant electrons on Fermi surfaces is crucial to the nematic fluctuations
of iron pnictides, where the orbital degree of freedom may behave differently
from the spin degree of freedom, and the transport properties are intimately
related to the spin dynamics.
Keywords: Iron-based superconductors, Electronic nematic phase, Nematic
fluctuations, Resistivity, Spin excitations, Orbital ordering, Neutron
scattering

###### pacs:

74.70.Xa, 75.50.Ee, 75.25.-j, 74.25.F-

##  I Introduction

Electronic nematic phase breaks the rotational symmetry but preserves the
translational symmetry of the underlying lattice in correlated materials
oganesyan2001 ; fradkin2010a ; fradkin2010b ; wwang2021 . In iron-based
superconductors, the nematic order associated with a tetragonal-to-
orthorhombic structural transition at temperature $T_{s}$ acts as a precursor
of the magnetic order below $T_{N}$ and the superconducting state below
$T_{c}$ fernandes2012 ; fernandes2017 ; chen2014 ; pdai2015 ; si2016 ;
gong2018 . The nematic fluctuations can be described by the electronic nematic
susceptibility, which is defined as the susceptibility of electronic
anisotropy to the uniaxial in-plane strain bohmer2016 . Divergent nematic
susceptibility upon approaching $T_{s}$ from high temperature are revealed by
the elastoresistance and elastic moduli measurements, suggesting nematic
fluctuations well above $T_{s}$ jhchu2012 ; hhkuo2014 ; hhkuo2013 ; bohmer2014
; dgong2017 . The nematic fluctuations commonly exist in iron-based
superconductors, and are even present in compounds with tetragonal crystal
symmetry without any static nematic order bohmer2020 . Accumulating evidence
suggests that the optimal superconductivity with maximum $T_{c}$ usually
occurs near a nematic quantum critical point where the nematic fluctuations
are the strongesthhkuo2016 ; yshizawa2012 ; jdai2009 ; skasahara2012 ;
tshibauchi2013 ; slderer2015 ; hluo2012 ; xlu2013 ; dhu2015 ; wzhang2019 ;
zliu2016 ; ygu2017 . However, the charge, spin and orbital degrees of freedom
are always intertwined in the presence of nematic fluctuations chandra1990 ;
jphu2012 ; fernandes2014 ; fernandes2012b ; fwang2015 ; cma2014 ;
cthorsmolle2014 ; qswang2015 ; chubukov2015 ; yamakawa2016 , giving a twofold
rotational ($C_{2}$) symmetry in many physical properties fernandes2012 ;
fernandes2017 ; chen2014 ; pdai2015 ; si2016 ; gong2018 ; bohmer2016 ;
cclee2009 ; kruger2009 ; wclv2009 ; cchen2010 ; valenzuela2010  including
anisotropic in-plane electronic resistivity and optical conductivity jhchu2010
; matanatar2010 ; jjying2011 ; hman2015 ; xluo2015 ; cmirri2014 ; fisher2011 ,
lifting of degeneracy between $d_{xz}$/$d_{yz}$ orbitals myi2011 ; myi2014 ;
myi2012 ; myi2019 ; yzhang2012 ; myi2017 ; watson2019 , anisotropic spin
excitations at low energies hluo2013 ; xlu2014 ; wzhang2016 ; ysong2015 ;
xlu2018 , phonon-energy split in lattice dynamics xren2015 ; ywhu2016 , and
splitting of the Knight shift baek2014 ; tiye2015 . In addition, it has been
proposed that the local anisotropic impurity scattering of chemical dopants
likely induces the twofold symmetry in the transport properties rosenthal2014
; sishida2013 ; allan2013 . Such complex cases make it is difficult to clarify
the main driven force of nematic phase by a single experimental probe.

Our previous works suggest that the Cr substitution is an effective way both
to suppress the superconductivity and to tune the magnetism in iron-based
superconductors wzhang2019 ; rzhang2014 ; rzhang2015 ; dgong2018 .
Specifically in the BaFe1.9-xNi0.1CrxAs2 system, by continuously doping Cr to
the optimally superconducting compound BaFe1.9Ni0.1As2 with $T_{c}=20$ K, the
superconductivity is quickly suppressed above $x=0.05$, but the magnetic
transition temperature $T_{N}$ and the structural transition temperature
$T_{s}$ remain between 30 K and 35 K as shown by neutron diffraction results
on naturally twinned samples [Fig. 1(a)]. Moreover, the effective moment $m$
is significantly enhanced first then suppressed for dopings higher than
$x=0.5$, where the charge carriers change from electron type to hole type as
shown by the sign of Hall and Seebeck coefficients dgong2018 . These make
BaFe1.9-xNi0.1CrxAs2 a rare example to separately tune the magnetically
ordered temperature $T_{N}$ by the local spin interactions and the
magnetically ordered strength by the scattering of itinerant electrons on
Fermi surfaces, respectively. The extra holes introduced by Cr substitutions
compensate the electron doping thus may drive those non-superconducting
compounds to a half-filled Mott insulator similar to the parent compounds of
cuprate and nickelate superconductors jmpizarro2017 ; medelmann2017 ;
medici2014 ; paLee2014 ; qgu2022 ; ysong2016b . It would be interesting to
monitor the evolution of the nematic fluctuations starting from a metallic
state toward to a localized insulating state zpyin2011 ; georges2013 ;
ysong2016b , especially on the detwinned samples [Fig. 1(b)].

In this paper, we further report a multi-probe study on the nematic
fluctuations in the non-superconducting compounds BaFe1.9-xNi0.1CrxAs2
($x=0.05\sim 0.8$) by electronic transport, angle resolved photoemission
spectroscopy (ARPES) and inelastic neutron scattering (INS) measurements. The
in-plane resistivity anisotropy measured in the detwinned samples under
uniaxial pressure shows a strong dependence on the Cr content with a clear
sign change above $x=0.6$. By focusing on two compounds with $x=0.05$ and 0.5,
ARPES measurements suggest possible band shifts induced by orbital anisotropy
near $T_{s}$/$T_{N}$ for both dopings, but INS experiments reveal clearly
different behaviors on the spin nematicity. The onset temperature of low-
energy spin excitation anisotropy between $Q=(1,0,1)$ and $Q=(0,1,1)$ for
$x=0.05$ is about 110 K, but for $x=0.5$ is much lower, only about 35 K near
the magnetic transition. Such temperature dependence of spin nematicity is
consistent with the results of in-plane resistivity anisotropy. At high
energies, the spin nematicity for $x=0.05$ extends to about 120 meV, much
larger than the case for $x=0.5$ (about 40 meV), suggesting a possible linear
correlation between the highest energy scale and the onset temperature of spin
nematicity. Therefore, the nematic behaviors in iron pnictides is highly
related to the interplay between local moments and itinerant electrons. While
the $C_{2}$-type anisotropies in spin excitations and in-plane resistivity are
strongly correlated to each other xlu2014 , the orbital anisotropy induced
band splitting may behave differently as affected by the complex Fermi surface
topology myi2009 ; richard2011 ; ysong2017 ; ysong2016 ; txie2018a ; txie2018b
; txie2020 ; twang2020a ; jguo2019 ; cliu2022 .

Figure 1: (Color online) Phase diagram and in-plane resistivity anisotropy of
BaFe1.9-xNi0.1CrxAs2. (a) The PM, AF and SC mark the region of paramagnetic,
antiferromagnetic, superconducting phases defined by $T_{s}$, $T_{N}$ and
$T_{c}$, respectively. Here $T_{s}$ and $T_{N}$ were measured by neutron
diffraction in our previous work on the naturally twinned samples dgong2018 .
(b) The gradient color maps the in-plane resistivity anisotropy
$\delta_{\rho}$ measured on detwinned samples. The vertical dashed line
divides the regions for electron-type and hole-type charge carriers as
determined by the sign of Hall and Seebeck coefficients dgong2018 .

##  II Experiment details

Figure 2: (Color online) In-plane resistivity anisotropy of
BaFe1.9-xNi0.1CrxAs2 under uniaxial pressure. Here $\rho_{b}$ is the in-plane
resistivity along the direction of uniaxial pressure, and $\rho_{a}$ is the
in-plane resistivity perpendicular to the direction of uniaxial pressure. For
easy comparison, each curve is normalized by its resistivity at 150 K, there
is no resistivity anisotropy above this temperature.  Figure 3: (Color online)
Temperature dependence of the in-plane resistivity anisotropy $\delta_{\rho}$
from $x=0.05$ to 0.8. (a) In electron-type compounds, $\delta_{\rho}$ gets
weaker but keeps positive when increasing Cr doping. (b) In hole-type
compounds, $\delta_{\rho}$ is very weak and becomes negative when $x\geq 0.7$.

High-quality single crystals of BaFe1.9-xNi0.1CrxAs2 were grown by the self-
flux method rzhang2014 ; rzhang2015 ; dgong2018 ; hqluo2008 ; ychen2011 ;
txie2017 ; twang2020b , the characterization results of our sample can be
found in previous reports rzhang2014 ; dgong2018 . The crystalline directions
of our sample were determined by a X-ray Laue camera (Photonic Sciences) in
backscattering mode with incident beam along $c-$axis. After that, the
crystals were cut into rectangle shapes (typical sizes: 1 mm $\times$ 2 mm) by
a wire saw under the directions [1, 0, 0] $\times$ [0, 1, 0] in orthorhombic
lattice notation ($a=b=5.6$ Å). By applying a uniaxial pressure around 10 MPa,
the crystal can be fully detwinned at low temperature, where the direction of
pressure was defined as $b$ direction, and the pressure-free direction was
defined as $a$ direction xlu2014 ; wzhang2016 ; ysong2015 ; xlu2018 ; xlu2016
; dwtam2019 ; dwtam2020 ; pliu2020 . The in-plane resistivity ($\rho_{a,b}$)
was measured by the standard four-probe method with the Physical Property
Measurement System (PPMS) from Quantum Design. To compare the temperature
dependence of resistivity at different directions, we normalized the
resistivity $\rho_{a,b}(T)$ data at 150 K for each sample. The in-plane
resistivity anisotropy was defined by
$\delta_{\rho}=(\rho_{b}-\rho_{a})/(\rho_{b}+\rho_{a})$ same as other
literatures jhchu2010 ; matanatar2010 ; jjying2011 .

ARPES experiments were performed at beamline 10.0.1 of the Advanced Light
Source and beamline 5-4 of the Stanford Synchrotron Radiation Light source
with R4000 electron analyzers. The angular resolution was 0.3 degree, and the
total energy resolution was 15 meV. All samples were cleaved _in-situ_ at 10 K
and measured in ultra-high vacuum with a base pressure lower than 4 $\times
10^{-11}$ Torr. We note that we used twinned samples without uniaxial pressure
for the ARPES experiments. INS experiments were carried out at two thermal
triple-axis spectrometers: PUMA at Heinz Maier-Leibnitz Zentrum (MLZ)
sobolev2015 , Germany, and TAIPAN at Australian Centre for Neutron Scattering
(ACNS) danilkin2009 , ANSTO, Australia. The wave vector $\bf Q$ at ($q_{x}$,
$q_{y}$, $q_{z}$) was defined as
$(H,K,L)=(q_{x}a/2\pi,q_{y}b/2\pi,q_{z}c/2\pi)$ in reciprocal lattice units
(r.l.u.) using the orthorhombic lattice parameters $a\approx b=5.6$ Å, and
$c\approx 13$ Å. All measurements were done with a fixed final energy
$E_{f}=14.8$ meV, and double focusing monochromator and analyzer using
pyrolytic graphite crystals. To gain better signal-noise ratio, 8 pieces of
rectangularly cut crystals (typical sizes: 7 mm $\times$ 8 mm $\times$ 0.5 mm)
were assembled in a detwinned device made by aluminum and springy gaskets
xlu2014 ; wzhang2016 ; ysong2015 ; xlu2018 . To reach both $Q=(1,0,1)$ and
$Q=(0,1,1)$, the sample holder was designed to easily rotate by 90 degrees,
thus the scattering plane can switch from $[H,0,0]\times[0,0,L]$ to
$[0,K,0]\times[0,0,L]$. The total mass of the crystals used in INS experiments
was about 2 grams from each sample set of $x=0.05$ and $x=$0.5. Time-of-flight
neutron scattering experiments were carried out on the same sample sets at
4SEASONS spectrometer (BL-01) at J-PARC nakamura2009 ; kajimoto2011 , Tokai,
Japan, with multiple incident energies $E_{i}=250,73,34,20$ meV, $k_{i}$
parallel to the $c$ axis, and chopper frequency $f=250$ Hz. The data were only
corrected by the efficiency of detectors from the incoherent scattering of
vanadium with white beam. As we were comparing two samples with similar mass
under the same measured conditions at the same spectrometer, it was not
necessary to do the vanadium normalization with mono beam. The data was
analyzed by the Utsusemi and MSlice software packages inamura2013 ; mslice .

##  III Results and discussions

We first present the resistivity results in Fig. 1, Fig. 2 and Fig. 3.
Apparently, the in-plane resistivity anisotropy show a strong dependence on
the Cr doping level. In the Cr free sample BaFe1.9Ni0.1As2, difference between
$\rho_{a}$ and $\rho_{b}$ presents above the superconducting transition
temperature $T_{c}=20$ K, where $\rho_{a}$ is metallic and $\rho_{b}$ is
semiconducting-like with a upturn at low temperature (namely
$\rho_{a}<\rho_{b}$) [Fig. 2(a)]. The superconductivity is completely
suppressed at $x=0.05$, there is a dramatic difference between $\rho_{a}$ and
$\rho_{b}$ with a anisotropy $\delta_{\rho}$ persisting to about $T=110$ K
[Fig. 2(b)]. By further increasing Cr doping, both $\rho_{a}$ and $\rho_{b}$
become semiconducting-like even insulating-like above $x=0.1$, and the
resistivity anisotropy gets weaker and weaker, until it nearly disappears at
$x=0.5$ and 0.6 compounds [Fig. 2(c) - (h)]. For those high doping compounds
$x=0.7$ and 0.8, it seems that $\delta_{\rho}$ changes sign with
$\rho_{a}>\rho_{b}$ at low temperatures [Fig. 2(i) and (j)]. To clearly
compare the resistivity anisotropy upon Cr doping, we plot $\delta_{\rho}$ as
gradient color mapping in Fig. 1(b) and show its detailed temperature
dependence in Fig. 3. Interestingly, the sign of $\delta_{\rho}$ is also
related to the type of charge carriers. $\delta_{\rho}$ keeps strong and
positive in the electron-type compounds but changes to negative and weak
($<1\%$) in the hole-type compounds [Fig. 1(b) and Fig. 3(b)]. This is
consistent with the results in the electron doped BaFe2-x(Ni, Co)xAs2 and the
hole doped Ba1-xKxFe2As2, Ca1-xNaxFe2As2, BaFe2-xCrxAs2 systems jhchu2010 ;
matanatar2010 ; jjying2011 ; hman2015 ; blomberg2013 ; jqma2013 ;
kobayashi2015 ; ishida2020 . However, in those cases, the onset temperature of
$\delta_{\rho}$ decreases with the structural transition temperature $T_{s}$
when increasing the doping level from the non-superconducting parent compounds
to optimally doped superconducting compounds. Here in BaFe1.9-xNi0.1CrxAs2
system, both $T_{N}$ and $T_{s}$ are actually within the range $32\sim 35$ K
for all probed dopings dgong2018 , but the onset temperature of
$\delta_{\rho}$ still extends to high temperatures, it is then strongly
suppressed by Cr doping [Fig. 3(a)]. In those hole-type compounds,
$\delta_{\rho}$ shows a peak feature (for $x=0.5$ and 0.6) or a kink (for
$x=0.7$ and 0.8) responding to the magnetic and structural transitions [Fig.
3(b)]. The non-monotonic behavior of $\delta_{\rho}$ may come from the
competition between the scattering from hole bands and electron bands, similar
behaviors were observed in the nematic susceptiblity of the Cr doped
BaFe2(As1-xPx)2 system wzhang2019 .

Figure 4: (Color online) ARPES results on $x=0.05$ (left) and $x=0.5$ (right)
compounds. (a)-(b) The measured Fermi surfaces around the $\Gamma$ and $X$
points. (c)-(d) Band dispersions along the high symmetry direction
$\Gamma$-$X$ obtained from the second derivatives in the energy direction.
(e)-(f) Temperature dependence of the fitted band position from the $X$ point.
All dashed lines are guides for eyes.  Figure 5: (Color online) Inelastic
neutron scattering results on the spin excitations of uniaxially detwinned
samples for $x=0.05$ (left) and $x=0.5$ (right) compounds measured by two
triple-axis spectrometers TAIPAN and PUMA. We compared the constant-energy
scans ($Q-$scans along $[H,0,1]$ or $[0,K,1]$, $S1$ rocking scans at
$Q=(1,0,1)$ or (0, 1, 1)) for $E=$ 3, 6, 9, 12 meV, respectively. All data are
corrected by a linearly $Q-$dependent background, and the solid lines are
gaussian fittings. The spurious signals in 6 meV data are ignored.  Figure 6:
(Color online) The order parameter of antiferromagnetism and spin nematicity
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ for $x=0.05$ and $x=0.5$
compounds.(a) and (b) The magnetic order parameters measured at $Q=(1,0,3)$ on
twinned samples, $Q=(1,0,1)$ and $Q=(0,1,1)$ on detwinned samples by elastic
neutron scattering. All data are subtracted by the normal state background and
normalized by the intensity at base temperature for $Q=(1,0,3)$ or
$Q=(1,0,1)$. (c) and (d) Spin nematicity measured by inelastic neutron
scattering. The solid symbols are the differences of local susceptibility
$\chi^{\prime\prime}$ between $Q=(1,0,1)$ and $Q=(0,1,1)$ (left y-axis), and
the open symbols are similar but obtained by integrating the constant-energy
scans in Fig. 5 corrected by the Bose population factor (right y-axis). The
vertical dash lines mark the magnetic transition temperature $T_{N}$ on
twinned samples and $T^{\prime}_{N}$ on detwinned samples. All solid lines are
guides to eyes.

Next, we focus on the electronic structure and the spin excitations in two
typical dopings $x=0.05$ with $T_{N}=32$ K and $x=0.5$ with $T_{N}=35$ K. The
Fermi surface topology and band structure measured by ARPES on naturally
twinned samples are shown in Fig. 4. From the Fermi surface mapping in Fig. 4
(a) and (b), we can find typical hole pockets around the zone center $\Gamma$
point. Near the $X$ point, an electron pocket is observed for $x=0.05$. For
$x=0.5$, however, the Fermi surface resembles that of the hole-doped
(Ba,K)Fe2As2myi2014 . This is due to the hole doping introduced by the Cr
substitution, which also introduces disorder directly in the Fe-planes, thus
resulting in spectral features that appear broad myi2009 . Figures 4 (c) and
(d) show the second energy derivatives of the spectral images along the high
symmetry direction ($\Gamma$-$X$). Larger hole pockets can indeed be seen for
$x=0.5$ compared to $x=0.05$. As has been demonstrated previously on BaFe2As2,
NaFeAs, and FeSe, the onset of $T_{s}$ is associated with the onset of an
observed anisotropic shift of the $d_{xz}$ and $d_{yz}$ orbital-dominated
bands where the $d_{xz}$ band shifts down and the $d_{yz}$ band shifts
upmyi2011 ; myi2014 ; myi2012 ; myi2019 ; yzhang2012 . This shift is most
prominently observed near the X point of the Brillouin zone. Moreover, such
band splitting as measured on uniaxially-strained crystals can be observed
above $T_{s}$ in the presence of this symmetry-breaking field. On a
structurally twinned crystal, the anisotropic band shifts would appear in the
form of a band splitting due to domain mixing. While we do not observe clearly
the band splitting as shown in Fig. 4(c)-(d), we can clearly observe the lower
branch with dominant intensity that shifts with temperature. This can be
understood as the lower $d_{xz}$ band. We can fit the energy position of the
band extracted from the X point and plot as a function of temperature. The
temperature evolution clearly identifies a temperature scale associated with
an onset of the band shift myi2014 ; myi2012 ; myi2019 . As shown in Fig. 4
(e) and (f), the $X$ band shifts at low temperature $T\approx 25$ K for
$x=0.05$ and $T\approx 45$ K for $x=0.5$, respectively, closing to their
structural or magnetic transition temperatures. We do note that while we
cannot conclusively state that this represents the orbital anisotropy, the
behavior we observe here on these twinned crystals is consistent with the
expectation of the onset of orbital anisotropymyi2011 ; myi2017 ; ysong2015 .
We note here that the observed onset temperature of band splitting is close to
the $T_{s}$ (or $T_{N}$), in contrast to the much higher onset in the
resistivity anisotropy shown in Fig. 3 measured on a strained crystal.

We then turn to search the connection between the resistivity anisotropy and
the spin excitation anisotropy. The first evidence of spin nematicity was
observed in BaFe2-xNixAs2 ($x=0$, 0.065, 0.085, 0.10, 0.12) xlu2014 ;
wzhang2016 ; ysong2015 ; xlu2018 , where BaFe1.9Ni0.1As2 is the starting
compound of this study. Low-energy spin excitations are measured on the
detwinned BaFe1.9-xNi0.1CrxAs2 ($x=0.05$ and 0.5) samples by INS experiments
using two triple-axis spectrometers. The results of constant-energy scans at
$E=$ 3, 6, 9, and 12 meV are summarized in Fig. 5. With convenient design of
the detwinned device and sample holder, we can easily perform constant-energy
scans ($Q-$scans) either along $[H,0,1]$ or $[0,K,1]$ direction after rotating
the whole sample set by 90 degree. For $x=0.5$ sample, we instead do the $S1$
rocking scans at $Q=(1,0,1)$ and (0, 1, 1). It should be noticed that the Néel
temperature $T_{N}$ is slightly enhanced by the applied uniaxial pressure in
$x=0.05$ sample from 32 K to 40 K (so does $T_{s}$), but does not change for
$x=0.5$ sample ($T_{N}\approx T^{\prime}_{N}=35$ K) [Fig. 6(a) and (b)]. Such
effect has been detected in the BaFe2-x(Ni, Co)xAs2 system dwtam2017 . The
detwinned ratio can be estimated by comparing the integrated intensities of
magnetic Bragg peak between $Q=(1,0,1)$ and $Q=(0,1,1)$ positions, which is
about 10 : 1 for the $x=0.05$ samples, and 4 : 1 for the $x=0.5$ samples,
respectively. Such large ratio means successful detwin for both sample sets.
At the first glance, it is very clear for the difference of the spin
excitations between $Q=(1,0,1)$ and $Q=(0,1,1)$ especially at low
temperatures, which could be attributed to the spin Ising-nematic correlations
(so called spin nematicity). After warming up to high temperatures, the spin
excitations at $Q=(1,0,1)$ decrease and become nearly identical to those at
$Q=(0,1,1)$. The nematic order parameter for the spin system can be
approximately represented by
$\chi^{\prime\prime}_{nematic}=\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$,
in which $\chi^{\prime\prime}_{10}$ (or $\chi^{\prime\prime}_{01}$) is the
local spin susceptibility at $Q=(1,0,1)$ (or $Q=(0,1,1)$). Figure 6 (c) and
(d) show the temperature dependence of
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ for both compounds, where
the Bose population factor is already corrected. We also plot the data (open
symbols) obtained from the integrated intensity of those $Q-$scans in Fig. 5.
For $x=0.05$ compound, the spin nematicity decreases slightly upon increasing
energy and terminates well above $T^{\prime}_{N}=40$ K [Fig. 6(c)] dgong2018 .
For the lowest energy we measured (3 meV), the onset temperature of spin
nematicity is about 110 K, similar to the in-plane resistivity anisotropy in
Fig. 3(a). The results for $x=0.5$ compound show markedly differences, where
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ quickly decreases both
with energy and temperature, and the onset temperature is around
$T^{\prime}_{N}=35$ K [Fig. 6(d)]. No spin anisotropy can be detected above 40
K both for $Q-$scans and energy scans, this is also consistent with the very
weak in-plane resistivity anisotropy for $x=0.5$ [Fig. 3(b)]. The spin nematic
theory predicts that the nematic fluctuations enhance both the intensity and
the correlation length of spin excitations at ($\pi$, 0) but suppress those at
(0, $\pi$) even above $T_{s}$. This was firstly testified in the detwinned
BaFe1.935Ni0.065As2 and can be also seen here in Fig. 5 wzhang2016 . Although
the peak intensities at $Q=(1,0,1)$ seem stronger than those at $Q=(0,1,1)$ in
Fig. 5(g) and (h), the peak width is smaller, and the integrated intensity of
the $Q$-scans are closed to each other. The above results of spin nemacticity
in BaFe1.9-xNi0.1CrxAs2 ($x=0.05$ and 0.5) resemble to those in BaFe2-xNixAs2,
where spin excitations at low energies change from $C_{4}$ to $C_{2}$ symmetry
in the tetragonal phase at temperatures approximately corresponding to the
onset of the in-plane resistivity anisotropy.

Figure 7: (Color online) Inelastic neutron scattering results on the spin
excitations of the uniaxially detwinned samples for $x=0.05$ (left) and
$x=0.5$ (right) compounds measured at $T=5$ K by time-of-flight spectrometer
4SEASONS. All data are presented both in 2D slices for $[H,K]$ plane and 1D
cuts along $[H,0]$ or $[0,K]$ at typical energy windows $E=3\pm 1,15\pm
2,42\pm 4,110\pm 10$ meV. The solid lines are gaussian fittings guiding for
eyes, which are not shown in panel (p) due to poor data quality. The dashed
diamonds in panel (d) and (l) illustrate the integrated Brillouin zone for
spin excitations around $Q=(1,0)$ and $Q=(0,1)$, respectively.  Figure 8:
(Color online) Energy dependence of the total spin fluctuations
$\chi^{\prime\prime}_{10}+\chi^{\prime\prime}_{01}$ and the spin nematicity
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ of uniaxially detwinned
samples for $x=0.05$ (left) and $x=0.5$ (right) compounds. Different symbols
correspond to different incident energy in the measurements. Both of
$\chi^{\prime\prime}_{10}+\chi^{\prime\prime}_{01}$ and
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ can be fitted with a
power-law dependence on the energy, $\sim A/E^{\alpha}$, where the amplitude
$A$ and exponent $\alpha$ are listed in each panel. Inset of panel (d) shows
the correlation between the highest energy and the onset temperature at low
energy of $\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$.

Moreover, INS experiments on detwinned BaFe2As2 and BaFe1.9Ni0.1As2 suggest
the spin anisotropy can persist to very high energy ysong2015 ; xlu2018 , even
in the later case the splitting of the $d_{xz}$ and $d_{yz}$ bands nearly
vanishes myi2017 . To quantitatively determine the energy dependence of spin
excitation anisotropy, we have performed time-of-flight INS experiments on the
uniaxially detwinned BaFe1.9-xNi0.1CrxAs2 ($x=0.05$ and 0.5), the results are
shown in Fig. 7 and Fig. 8. It should be noted that for such experiments, the
energy transfer is always in coupled with $L$ due to $k_{i}\parallel c$
nakamura2009 ; kajimoto2011 . The two-dimensional (2D) energy slices and one-
dimensional (1D) cuts along $[H,0]$ and $[0,K]$ at various energies are
presented in Fig. 7. Indeed the spin excitations are twofold symmetric below
100 meV for both compounds. The spin excitations at $E=3$ meV, $Q=(0,\pm 1)$
are very weak in $x=0.05$ compound, then continuously increase upon energy and
become nearly the same as $Q=(\pm 1,0)$ around 110 meV. For $x=0.5$ compound,
although the spin excitations at $Q=(0,\pm 1)$ can be initially observed at
$E=3$ meV, the spin anisotropy still exists at 15 meV, then disappears above
42 meV. To further compare the spin excitations in both compounds, we have
calculated the total spin fluctuations
$\chi^{\prime\prime}_{10}+\chi^{\prime\prime}_{01}$ and the spin nematicity
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ from the integrated
intensity marked by the dashed diamonds in Fig. 7 (d) and (l). In principle,
the local dynamic susceptibility $\chi^{\prime\prime}$ can be estimated from
the integration outcome of the spin excitations within one Brillouin zone,
here $\chi^{\prime\prime}$ can simply be calculated through dividing the
integration signal in the $Q=$ (0, 0), (1, 1), (2, 0), (1, -1) boxes, giving
the diamond shape integration zone pdai2015 . The total spin susceptibility
$\chi^{\prime\prime}_{10}+\chi^{\prime\prime}_{01}$ in $x=0.5$ compound is
stronger than that in $x=0.05$, but decays much quickly with energy [Fig. 8
(a) and (c)]. The spin nematicity
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ apparently has different
energy scale for two compounds, where it is about 120 meV for $x=0.05$ but
only 40 meV for $x=0.5$, respectively. The energy scale of
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ in the superconducting
compound BaFe1.9Ni0.1As2 is 60 meV ysong2015 , and for the parent compound
BaFe2As2 is about 200 meV up to the band top of the spin waves xlu2018 . These
facts lead to a possible linear correlation between the highest energy and the
onset temperature of spin nematicity at low energy [inset of Fig. 8(d)].
Within the measured energy range, both
$\chi^{\prime\prime}_{10}+\chi^{\prime\prime}_{01}$ and
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ can be fit with a power-
law dependence on the energy, $\sim A/E^{\alpha}$, where the amplitude $A$ and
exponent $\alpha$ are listed in each panel of Fig. 8. Indeed, the larger value
of $\alpha$ for $x=0.5$ in comparison to that for $x=0.05$ suggests faster
decay with energy both for the spin fluctuations and the spin nematicity.
Similar fitting on the results of BaFe1.9Ni0.1As2 gives parameters in between
them ysong2015 . Although the low energy data below 10 meV may be affected by
the $L$-modulation of spin excitations, and by the superconductivity in
BaFe1.9Ni0.1As2, the similar quantum critical behavior both for
$\chi^{\prime\prime}_{10}+\chi^{\prime\prime}_{01}$ and
$\chi^{\prime\prime}_{10}-\chi^{\prime\prime}_{01}$ in these three compounds
is expected by the Ising-nematic scenario ysong2015 ; xlu2014 ; xlu2018 ;
wzhang2016 .

In our previous neutron diffraction results on the BaFe1.9-xNi0.1CrxAs2
system, the Cr dopings have limited effects on the magnetically ordered
temperature $T_{N}$ but significantly enhance the effective ordered moment $m$
by reaching a maximum value at $x=0.5$ dgong2018 . The Néel temperature
$T_{N}$ is mostly determined by the local magnetic coupling related to the
local FeAs4 tetrahedron structure. The evolution of ordered moment probably
induced by the changes of the density of states and the orbital angular
momentum from itinerant electrons on the Fermi surfaces. The Cr doping
introduces both local distortion on the lattices and hole doping on the Fermi
pockets, yielding a non-monotonic change of the conductivity of charge
carriers. As shown in Fig. 2, the low-temperature upturn of resistivity is
enhanced by Cr doping first but then weakens in those hole-type compounds.
Among these dopings, $x=0.5$ has the most insulating-like behavior, and thus
strongly localized charge carriers and maximum ordered moment, but its spin
nematicity quickly drops down both for the temperature and energy dependence.
In contrast to the magnetically ordered strength, both the structural
transition temperature $T_{s}$ and the lattice orthorhombicity
$\delta=(a-b)/(a+b)$ are nearly Cr doping independent dgong2018 . This means
the static nematic order is also nearly Cr independent in this system, as
opposed to the case for dynamic nematic fluctuations.

The nature of iron-based superconductor can be theoretically described as a
magnetic Hund’s metal, in which the strong interplay between the local spins
on Fe atoms and the itinerant electrons on Fermi surfaces gives correlated
electronic states zpyin2011 ; georges2013 . Indeed, time-of-flight INS
experiments on the detwinned BaFe2As2 suggest the spin waves in parent
compound are preferably described by a multi-orbital Hubbard-Hund model based
on the itinerant picture with moderate electronic correlation effects, instead
of a Heisenberg model with effective exchange couplings from local spins. Upon
warming up to high temperatures, the intensities of spin excitation anisotropy
decrease gradually with increasing energy and finally cut off at a energy away
from the band top of spin waves xlu2018 . Therefore, the energy scale of spin
nematicity sets an upper limit for the characteristic temperature for the
nematic spin correlations, as well as the onset temperature of resistivity
anisotropy. Here by adding up the results on the in-plane anisotropies of
resistivity, orbital energy and spin excitations in BaFe1.9-xNi0.1CrxAs2, they
clearly suggest that the electronic nematicity is intimately related to the
spin dynamics, which seems consistent with the Hund’s metal picture.
Specifically, by doping Cr to suppress the superconductivity in
BaFe1.9Ni0.1As2 makes the charge carriers initially localized with enhanced
electron correlations dgong2018 , which may enhance the electronic
correlations by increasing the intra- and inter-orbital onsite repulsion $U$
as well as the Hund’s coupling $J_{H}$ xlu2018 , and thus gives rise to
stronger spin excitations and larger spin anisotropy in Cr doping $x=0.05$
compound. Another effect is the lifting up of $d_{yz}$ and $d_{xy}$ along
$\Gamma$-X direction to the Fermi level, which primarily contributes to the
effective moments zpyin2011 . The orbital-weight redistribution triggered by
the spin order suggests that the orbital degree of freedom are coupled to the
spin degree of freedom mdaghofer2010 . By further increasing Cr doping to
$x=0.5$, the localization effect is so strong that electron system becomes
insulating at low temperature. In this case, the itinerant picture based on
Hund’s metal may not be applicable anymore. The low density of itinerant
electrons weakens the nematic fluctuations and probably limits them inside the
magnetically ordered state. In either case for $x=0.05$ or $x=0.5$, the band
splitting does not directly correspond to the spin nematic correlations but
only present below the nematic ordered temperature. This may attribute to the
weak spin-orbit coupling in this system, as the spin anisotropy in spin space
can only present at very low energies hluo2013 . In addition, our results can
rule out the picture of local impurity scattering driven nematicity, since the
impurity scattering from Cr substitutions is certainly stronger in the $x=0.5$
compound but it does not promote the nematic fluctuations.

##  IV Conclusion

In conclusion, we have extensively studied the in-plane resistivity
anisotropy, orbital ordering and spin nematicity in a non-superconducting
BaFe1.9-xNi0.1CrxAs2 system. We have found the Cr doping strongly affect on
the anisotropy of resistivity and spin excitations along with the itinerancy
of charge carriers. While the onset temperatures of resistivity anisotropy and
spin nematicity are similar and correlated to the energy scale of spin
anisotropy, the orbital anisotropy shows a onset temperature irrelevant to
them. These results suggest that the electronic correlations from the
interplay between local moments and itinerant electrons is crucial to
understand the nematic fluctuations, thus inspire the quest for the driven
force of the electronic nematic phase in iron-pnictide superconductors.

Additional Requirements For additional requirements for specific article types
and further information please directly refer to Huiqian Luo
(hqluo@iphy.ac.cn).

Conflict of Interest Statement The authors declare that the research was
conducted in the absence of any commercial or financial relationships that
could be construed as a potential conflict of interest.

Author Contributions H. L. and D. G. proposed and designed the research. D.
G., T. X., W. Z. and R. Z. contributed in sample growth and resistivity
measurements. M. Y., M. W., S. M., M. H., D. L. and R. J. B. contributed to
the ARPES measurements. D. G. and H. L. carried out the neutron scattering
experiments with S. D., G. D., X. L., J. T. P., K. I., and K. K.. D. G., H.
L., S. L. and P. D. analyzed the data. H. L., D. G. and M. Y. wrote the paper.
All authors participated in discussion and comment on the paper.

Funding This work is supported by the National Key Research and Development
Program of China (Grants No. 2018YFA0704200, No. 2017YFA0303100 and No.
2017YFA0302900), the National Natural Science Foundation of China (Grants No.
11822411, No. 11961160699, and No. 12061130200), the Strategic Priority
Research Program (B) of the CAS (Grants No. XDB25000000 and No. XDB07020300)
and K. C. Wong Education Foundation (GJTD-2020-01). H. Luo is grateful for the
support from the Youth Innovation Promotion Association of CAS (Grant No.
Y202001) and Beijing Natural Science Foundation (Grants No. JQ19002). M. Wang
is supported by the National Natural Science Foundation of China (Grants No.
11904414, No.12174454), the Guangdong Basic and Applied Basic Research
Foundation (No. 2021B1515120015), National Key Research and Development
Program of China (No. 2019YFA0705702). Work at University of California,
Berkeley and Lawrence Berkeley National Laboratory was funded by the U.S.
Department of Energy (DOE), Office of Science, Office of Basic Energy
Sciences, Materials Sciences and Engineering Division under Contract No. DE-
AC02-05-CH11231 within the Quantum Materials Program (KC2202) and the Office
of Basic Energy Sciences. The ARPES work at Rice University was supported by
the Robert A. Welch Foundation Grant No. C-2024 (M. Y.).

Acknowledgements The authors thank the helpful discussion with Xingye Lu at
Beijing Normal University and Yu Song at Zhejiang University. The neutron
scattering experiments in this work are performed at thermal triple-axis
spectrometer PUMA at Heinz Maier-Leibnitz Zentrum (MLZ), Germany, thermal
triple-axis spectrometer TAIPAN at Australian Centre for Neutron Scattering
(ACNS), Australian Nuclear Science and Technology Organisation (ANSTO),
Australia (Proposal No. P4263), and time-of-flight Fermi-chopper spectrometer
4SEASONS (BL-01) at the Materials and Life Science Experimental Facility of
J-PARC (Proposal Nos. 2015A0005, 2016A0169). ARPES measurements were performed
at the Advanced Light Source and the Stanford Radiation Lightsource, which are
both operated by the Office of Basic Energy Sciences, U.S. DOE.

## References

  * (1) Oganesyan V, Kivelson SA, Fradkin E. Quantum theory of a nematic Fermi fluid. Phys. Rev. B (2001) 64: 195109.
  * (2) Fradkin E, Kivelson SA. Electron Nematic Phases Proliferate. Science (2010) 327: 155-156.
  * (3) Fradkin E, Kivelson SA, Lawler MJ, Eisenstein JP, Mackenzie AP. Nematic Fermi fluids in condensed matter physics. Annu. Rev. Condens. Matter Phys. (2010) 1: 153.
  * (4) Wang W, Luo J, Wang C, Yang J, Kodama Y, Zhou R, Zheng G-Q. Microscopic evidence for the intra-unit-cell electronic nematicity inside the pseudogap phase in YBa2Cu4O8. Sci. China Phys. Mech. Astron. (2021) 64: 237413.
  * (5) Fernandes RM, Schmalian J. Manifestations of nematic degrees of freedom in the magnetic, elastic, and superconducting properties of the iron pnictides. Supercond. Sci. Technol. (2012) 25: 084005.
  * (6) Fernandes RM, Chubukov AV. Low-energy microscopic models for iron-based superconductors: a review. Rep. Prog. Phys. (2017) 80: 014503.
  * (7) Chen X, Dai P, Feng D, Xiang T, Zhang F-C. Iron-based high transition temperature superconductors. Nat. Sci. Rev.(2014) 1: 371-395.
  * (8) Dai P. Antiferromagnetic order and spin dynamics in iron-based superconductors. Rev. Mod. Phys. (2015) 87: 855.
  * (9) Si Q, Yu R, Abrahams E. High-temperature superconductivity in iron pnictides and chalcogenides. Nat. Rev. Mater. (2016) 1: 16017.
  * (10) Gong D, Luo H. Antiferromagnetic order and spin dynamics in iron-based superconductors. Acta Phys. Sin. (2018) 67: 207407.
  * (11) Böhmer AE, Meingast C. Electronic nematic susceptibility of iron-based superconductors. C. R.Physique (2016) 17: 90-112.
  * (12) Chu J-H, Kuo H-H, Analytis JG, Fisher IR. Divergent Nematic Susceptibility in an Iron Arsenide Superconductor. Science (2012) 337: 710-712.
  * (13) Kuo H-H, Fisher IR. Effect of Disorder on the Resistivity Anisotropy Near the Electronic Nematic Phase Transition in Pure and Electron-Doped BaFe2As2. Phys. Rev. Lett. (2014) 112: 227001.
  * (14) Kuo H-H, Shapiro MC, Riggs SC, Fisher IR. Measurement of the elastoresistivity coefficients of the underdoped iron arsenide Ba(Fe0.975Co0.025)2As2, Phys. Rev. B (2013) 88: 085113.
  * (15) Böhmer AE, Burger P, Hardy F, Wolf T, Schweiss P, Fromknecht R, Reinecker M, Schranz W, Meingast C. Nematic Susceptibility of Hole-Doped and Electron-Doped BaFe2As2 Iron-Based Superconductors from Shear Modulus Measurements. Phys. Rev. Lett. (2014) 112: 047001.
  * (16) Gong D, Liu Z, Gu Y, Xie T, Ma X, Luo H, Yang Y, Li S. Nature of the antiferromagnetic and nematic transitions in Sr1-xBaxFe1.97Ni0.03As2. Phys. Rev. B (2017) 96: 104514.
  * (17) Böhmer AE, Chen F, Meier WR, Xu M, Drachuck G, Merz M, Wiecki PW, Bud’ko SL, Borisov V, Valenti R, Christensen MH, Fernandes RM, Meingast C, Canfield PC. Evolution of nematic fluctuations in CaK(Fe1-xNix)4As4 with spin-vortex crystal magnetic order [Preprint] (2020). Available at: https://arxiv.org/abs/2011.13207.
  * (18) Kuo H-H, Chu J-H, Kivelson SA, Fisher IR. Ubiquitous signatures of nematic quantum criticality in optimally doped Fe-based superconductors. Science (2016) 352: 958-962.
  * (19) Yoshizawa M, Kimura D, Chiba T, Simayi S, Nakanishi Y, Kihou K, Lee C-H, Iyo A, Eisaki H, Nakajima M, Uchida S. Structural Quantum Criticality and Superconductivity in Iron-Based Superconductor Ba(Fe1-xCox)2As2, J. Phys. Soc. Jpn. (2012) 81: 024604.
  * (20) Dai J, Si Q, Zhu JX, Abrahams E. Iron pnictides as a new setting for quantum criticality. Proc. Natl. Acad. Sci. USA (2009) 106: 4118-4121.
  * (21) Kasahara S, Shi HJ, Hashimoto K, Tonegawa S, Mizukami Y, Shibauchi T, Sugimoto K, Fukuda T, Terashima T, Nevidomskyy AH, Matsuda Y. Electronic nematicity above the structural and superconducting transition in BaFe2(As1-xPx)2. Nature (2012) 486: 382-385.
  * (22) Shibauchi T, Carrington A, Matsuda Y. Quantum critical point lying beneath the superconducting dome in iron-pnictides. Annu. Rev. Condens. Matter Phys. (2014) 5: 113-135.
  * (23) Lederer S, Schattner Y, Berg E, Kivelson SA. Enhancement of Superconductivity near a Nematic Quantum Critical Point. Phys. Rev. Lett. (2015) 114: 097001.
  * (24) Luo H, Zhang R, Laver M, Yamani Z, Wang M, Lu X, Wang M, Chen Y, Li S, Chang S, Lynn JW, Dai P. Coexistence and Competition of the Short-Range Incommensurate Antiferromagnetic Order with the Superconducting State of BaFe2-xNixAs2. Phys. Rev. Lett. (2012) 108: 247002.
  * (25) Lu X, Gretarsson H, Zhang R, Liu X, Luo H, Tian W, Laver M, Yamani Z, Kim Y-J, Nevidomskyy AH, Si Q, Dai P. Avoided Quantum Criticality and Magnetoelastic Coupling in BaFe2-xNixAs2. Phys. Rev. Lett. (2013) 110: 257001.
  * (26) Hu D, Lu X, Zhang W, Luo H, Li S, Wang P, Chen G, Han F, Banjara SR, Sapkota A, Kreyssig A, Goldman AI, Yamani Z, Niedermayer C, Skoulatos M, Georgii R, Keller T, Wang P, Yu W, Dai P. Structural and Magnetic Phase Transitions near Optimal Superconductivity in BaFe2(As1-xPx)2. Phys. Rev. Lett. (2015) 114: 157002.
  * (27) Zhang W, Wei Y, Xie T, Liu Z, Gong D, Ma X, Hu D, Čermák P, Schneidewind A, Tucker G, Meng S, Huesges Z, Lu Z, Song J, Luo W, Xu L, Zhu Z, Yin X, Li H, Yang Y, Luo H, Li S. Unconventional Antiferromagnetic Quantum Critical Point in Ba(Fe0.97Cr0.03)2(As1-xPx)2. Phys. Rev. Lett. (2019) 122: 037001.
  * (28) Liu Z, Gu Y, Zhang W, Gong D, Zhang W, Xie T, Lu X, Ma X, Zhang X, Zhang R, Zhu J, Ren C, Shan L, Qiu X, Dai P, Yang Y, Luo H, Li S. Nematic Quantum Critical Fluctuations in BaFe2-xNixAs2. Phys. Rev. Lett. (2016) 117: 157002.
  * (29) Gu Y, Liu Z, Xie T, Zhang W, Gong D, Hu D, Ma X, Li C, Zhao L, Lin L, Xu Z, Tan G, Chen G, Meng ZY, Yang Y, Luo H, Li S. Unified Phase Diagram for Iron-Based Superconductors. Phys. Rev. Lett. (2017) 119: 157001.
  * (30) Chandra P, Coleman P, Larkin AI. Ising transition in frustrated Heisenberg models. Phys. Rev. Lett. (1990) 64: 88.
  * (31) Hu JP, Xu CK. Nematic orders in iron-based superconductors. Physica C (2012) 481: 215.
  * (32) Fernandes RM, Chubukov AV, Schmalian J. What drives nematic order in iron-based superconductors? Nat. Phys. (2014) 10: 97-104.
  * (33) Fernandes RM, Chubukov AV, Knolle J, Eremin I, Schmalian J. Preemptive nematic order, pseudogap, and orbital order in the iron pnictides. Phys. Rev. B (2012) 85: 024534.
  * (34) Wang F, Kivelson SA, Lee D-H. Nematicity and quantum paramagnetism in FeSe. Nat. Phys. (2015) 11: 959-963.
  * (35) Ma C, Wu L, Yin W-G, Yang H, Shi H, Wang Z, Li J, Homes CC, Zhu Y. Strong Coupling of the Iron-Quadrupole and Anion-Dipole Polarizations in Ba(Fe1-xCox)2As2. Phys. Rev. Lett. (2014) 112: 077001.
  * (36) Thorsmølle VK, Khodas M, Yin ZP, Zhang C, Carr SV, Dai P, Blumberg G. Critical quadrupole fluctuations and collective modes in iron pnictide superconductors. Phys. Rev. B (2016) 93: 054515.
  * (37) Wang Q, Shen Y, Pan B, Hao Y, Ma M, Zhou F, Steffens P, Schmalzl K, Forrest TR, Abdel-Hafiez M, Chen X, Chareev DA, Vasiliev AN, Bourges P, Sidis Y, Cao H, Zhao J. Strong interplay between stripe spin fluctuations, nematicity and superconductivity in FeSe. Nat. Mater. (2015) 15: 159.
  * (38) Chubukov AV, Fernandes RM, Schmalian J. Origin of nematic order in FeSe. Phys. Rev. B (2015) 91: 201105(R).
  * (39) Yamakawa Y, Onari S, Kontani H. Nematicity and Magnetism in FeSe and Other Families of F-Based Superconductors. Phys. Rev. X (2016) 6: 021032.
  * (40) Lee CC, Yin WG, Ku W. Ferro-Orbital Order and Strong Magnetic Anisotropy in the Parent Compounds of Iron-Pnictide Superconductors. Phys. Rev. Lett. (2009) 103: 267001.
  * (41) Krüger F, Kumar S, Zaanen J, van den Brink J. Spin-orbital frustrations and anomalous metallic state in iron-pnictide superconductors. Phys. Rev. B (2009) 79: 054504.
  * (42) Lv WC, Wu JS, Phillips P. Orbital ordering induces structural phase transition and the resistivity anomaly in iron pnictides. Phys. Rev. B (2009) 80: 224506.
  * (43) Chen C-C, Maciejko J, Sorini AP, Moritz B, Singh RRP, Devereaux TP. Orbital order and spontaneous orthorhombicity in iron pnictides. Phys. Rev. B (2010) 82: 100504(R).
  * (44) Valenzuela B, Bascones E, Calderón MJ. Conductivity Anisotropy in the Antiferromagnetic State of Iron Pnictides. Phys. Rev. Lett. (2010) 105: 207202.
  * (45) Chu J-H, Analytis JG, De Greve K, McMahon PL, Islam Z, Yamamoto Y, Fisher IR. In-Plane Resistivity Anisotropy in an Underdoped Iron Arsenide Superconductor. Science (2010) 329: 824.
  * (46) Tanatar MA, Blomberg EC, Kreyssig A, Kim MG, Ni N, Thaler A, Bud’ko SL, Canfield PC, Goldman AI, Mazin II, Prozorov R. Uniaxial-strain mechanical detwinning of CaFe2As2 and BaFe2As2 crystals: Optical and transport study. Phys. Rev. B (2010) 81: 184508.
  * (47) Ying JJ, Wang XF, Wu T, Xiang ZJ, Liu RH, Yan YJ, Wang AF, Zhang M, Ye GJ, Cheng P, Hu JP, Chen XH. Measurements of the Anisotropic In-Plane Resistivity of Underdoped FeAs-Based Pnictide Superconductors. Phys. Rev. Lett. (2011) 107: 067001.
  * (48) Man H, Lu X, Chen JS, Zhang R, Zhang W, Luo H, Kulda J, Ivanov A, Keller T, Morosan E, Si Q, Dai P. Electronic nematic correlations in the stress-free tetragonal state of BaFe2-xNixAs2. Phys. Rev. B (2015) 92: 134521.
  * (49) Luo X, Stanev V, Shen B, Fang L, Ling XS, Osborn R, Rosenkranz S, Benseman TM, Divan R, Kwok W-K, Welp U. Antiferromagnetic and nematic phase transitions in BaFe2(As1-xPx)2 studied by ac microcalorimetry and SQUID magnetometry. Phys. Rev. B (2015) 91: 094512.
  * (50) Mirri C, Dusza A, Bastelberger S, Chu J-H, Kuo H-H, Fisher IR, Degiorgi L. Hysteretic behavior in the optical response of the underdoped Fe arsenide Ba(Fe1-xCox)2As2 in the electronic nematic phase0. Phys. Rev. B (2014) 89, 060501(R).
  * (51) Fisher IR, Degiorgi L, Shen Z-X. In-plane electronic anisotropy of underdoped ’122’ Fe-arsenide superconductors revealed by measurements of detwinned single crystals. Rep. Prog. Phys. (2011) 74: 124506.
  * (52) Yi M, Lu DH, Chu JH, Analytis JG, Sorini AP, Kemper AF, Moritz B, Moore RG, Hashimoto M, Lee WS, Hussain Z, Devereaux TP, Fisher IR, Shen ZX. Symmetry-breaking orbital anisotropy observed for detwinned Ba(Fe1-xCox)2As2 above the spin densitywave transition. Proc. Natl. Acad. Sci. USA (2011) 108: 6878.
  * (53) Yi M, Zhang Y, Liu Z-K, Ding X, Chu J-H, Kemper AF, Plonka N, Moritz B, Hashimoto M, Mo S-K, Hussain Z, Devereaux TP, Fisher IR, Wen H-H, Shen Z-X, Lu DH. Dynamic competition between spin-density wave order and superconductivity in underdoped Ba1-xKxFe2As2. Nat. Comm. (2014) 5: 3711.
  * (54) Yi M, Lu DH, Moore RG, Kihou K, Lee C-H, Iyo A, Eisaki H, Yoshida T, Fujimori A, Shen Z-X. Electronic reconstruction through the structural and magnetic transitions in detwinned NaFeAs. New J. Phys. (2012) 14: 073019.
  * (55) Yi M, Pfau H, Zhang Y, He Y, Wu H, Chen T, Ye Z-R, Hashimoto M, Yu R, Si Q, Lee D-H, Dai P, Shen Z-X, Lu DH, Birgeneau RJ. Nematic Energy Scale and the Missing Electron Pocket in FeSe. Phys. Rev. X (2019) 9: 041049.
  * (56) Zhang Y, He C, Ye ZR, Jiang J, Chen F, Xu M, Ge QQ, Xie BP, Wei J, Aeschlimann M, Cui XY, Shi M, Hu JP, Feng DL. Symmetry breaking via orbital-dependent reconstruction of electronic structure in detwinned NaFeAs. Phys. Rev. B (2012) 85: 085121.
  * (57) Yi M, Zhang Y, Shen Z-X, Lu D. Role of the orbital degree of freedom in iron-based superconductors. npj Quantum Materials (2017) 2: 57.
  * (58) Watson MD, Dudin P, Rhodes LC, Evtushinsky DV, Iwasawa H, Aswartham S, Wurmehl S, Büchner B, Hoesch M, Kim TK. Probing the reconstructed Fermi surface of antiferromagnetic BaFe2As2 in one domain. npj Quantum Mater. (2019) 4: 36.
  * (59) Luo H, Wang M, Zhang C, Lu X, Regnault L-P, Zhang R, Li S, Hu J, Dai P. Spin Excitation Anisotropy as a Probe of Orbital Ordering in the Paramagnetic Tetragonal Phase of Superconducting BaFe1.904Ni0.096As2. Phys. Rev. Lett. (2013) 111: 107006.
  * (60) Lu X, Park JT, Zhang R, Luo H, Nevidomskyy AH, Si Q, Dai P. Nematic spin correlations in the tetragonal state of uniaxial-strained BaFe2-xNixAs2. Science (2014) 345: 657.
  * (61) Zhang W, Park JT, Lu X, Wei Y, Ma X, Hao L, Dai P, Meng ZY, Yang Y, Luo H, Li S. Effect of Nematic Order on the Low-Energy Spin Fluctuations in Detwinned BaFe1.935Ni0.065As2. Phys. Rev. Lett. (2016) 117: 227003.
  * (62) Song Y, Lu X, Abernathy DL, Tam DW, Niedziela JL, Tian W, Luo H, Si Q, Dai P. Energy dependence of the spin excitation anisotropy in uniaxial-strained BaFe1.9Ni0.1As2. Phys. Rev. B (2015) 92: 180504(R).
  * (63) Lu X, Scherer DD, Tam DW, Zhang W, Zhang R, Luo H, Harriger LW, Walker HC, Adroja DT, Andersen BM, Dai P. Spin Waves in Detwinned BaFe2As2. Phys. Rev. Lett. (2018) 121: 067002.
  * (64) Ren X, Duan L, Hu Y, Li J, Zhang R, Luo H, Dai P, Li Y. Nematic Crossover in BaFe2As2 under Uniaxial Stress. Phys. Rev. Lett. (2015) 115: 197002.
  * (65) Hu Y, Ren X, Zhang R, Luo H, Kasahara S, Watashige T, Shibauchi T, Dai P, Zhang Y, Matsuda Y, Li Y. Nematic magnetoelastic effect contrasted between Ba(Fe1-xCox)2As2 and FeSe. Phys. Rev. B (2016) 93: 060504(R).
  * (66) Baek S-H, Efremov DV, Ok JM, Kim JS, van den Brink J, Büchner B. Orbital-driven nematicity in FeSe. Nat. Mater. (2015) 14: 210-214.
  * (67) Iye T, Julien M-H, Mayaffre H, Horvatić M, Berthier C, Ishida K, Ikeda H, Kasahara S, Shibauchi T, Matsuda Y. Emergence of Orbital Nematicity in the Tetragonal Phase of BaFe2(As1-xPx)2. J. Phys. Soc. Jpn. (2015) 84: 043705.
  * (68) Rosenthal EP, Andrade EF, Arguello CJ, Fernandes RM, Xing LY, Wang XC, Jin CQ, Millis AJ, Pasupathy A. N. Visualization of electron nematicity and unidirectional antiferroic fluctuations at high temperatures in NaFeAs. Nat. Phys. (2014) 10: 225-232.
  * (69) Ishida S, Nakajima M, Liang T, Kihou K, Lee CH, Iyo A, Eisaki H, Kakeshita T, Tomioka Y, Ito T, Uchida S. Anisotropy of the In-Plane Resistivity of Underdoped Ba(Fe1-xCox)2As2 Superconductors Induced by Impurity Scattering in the Antiferromagnetic Orthorhombic Phase. Phys. Rev. Lett. (2013) 110: 207001.
  * (70) Allan MP, Chuang T-M, Massee F, Xie Y, Ni N, Bud’ko SL, Boebinger GS, Wang Q, Dessau DS, Canfield PC, Golden MS, Davis JC. Anisotropic impurity states, quasiparticle scattering and nematic transport in underdoped Ca(Fe1-xCox)2As2. Nat. Phys. (2013) 9: 220-224.
  * (71) Zhang R, Gong D, Lu X, Li S, Dai P, Luo H. The effect of Cr impurity to superconductivity in electron-doped BaFe2-xNixAs2. Supercond. Sci. Technol. (2014) 27: 115003.
  * (72) Zhang R, Gong D, Lu X, Li S, Laver M, Niedermayer C, Danilkin S, Deng G, Dai P, Luo H. Doping evolution of antiferromagnetism and transport properties in nonsuperconducting BaFe2-2xNixCrxAs2. Phys. Rev. B (2015) 91: 094506.
  * (73) Gong D, Xie T, Zhang R, Birk J, Niedermayer C, Han F, Lapidus SH, Dai P, Li S, Luo H. Doping effects of Cr on the physical properties of BaFe1.9-xNi0.1CrxAs2. Phys. Rev. B (2018) 98: 014512.
  * (74) Pizarro JM, Calderón MJ, Liu J, Muñoz MC, Bascones E. Strong correlations and the search for high-$T_{c}$ superconductivity in chromium pnictides and chalcogenides, Phys. Rev. B (2017) 95: 075115.
  * (75) Edelmann M, Sangiovanni G, Capone M, de’Medici L. Chromium analogs of iron-based superconductors, Phys. Rev. B (2017) 95: 205118.
  * (76) de’Medici L, Giovannetti G, Capone M. Selective Mott physics as a key to iron superconductors. Phys. Rev. Lett. (2014) 112: 177001.
  * (77) Lee PA, Nagaosa N, Wen X-G. Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys. (2006) 78: 17.
  * (78) Gu Q, Wen H-H. Superconductivity in nickel-based 112 systems. The Innovation (2022) 3: 100202.
  * (79) Song Y _et al._ , A Mott insulator continuously connected to iron pnictide superconductors, Nat. Commun. (2016) 7: 13879.
  * (80) Yin ZP, Haule K, Kotliar G. Kinetic frustration and the nature of the magnetic and paramagnetic states in iron pnictides and iron chalcogenides. Nat. Mater. (2011) 10: 932-935.
  * (81) Georges A, de’Medici L, Mravlje J. Strong Correlations from Hund’s Coupling. Annu. Rev. Condens. Matter Phys. (2013) 4: 137.
  * (82) Yi M, Lu DH, Analytis JG, Chu J-H, Mo S-K, He R-H, Moore RG, Zhou XJ, Chen GF, Luo JL, Wang NL, Hussain Z, Singh DJ, Fisher IR, Shen Z-X. Electronic structure of the BaFe2As2 family of iron-pnictide superconductors. Phys. Rev. B (2009) 80: 024515.
  * (83) Richard P, Sato T, Nakayama K, Takahashi T, Ding H. Fe-based superconductors: an angle-resolved photoemission spectroscopy perspective. Rep. Prog. Phys. (2011) 74: 124512.
  * (84) Song Y, Wang W, Zhang C, Gu Y, Lu X, Tan G, Su Y, Bourdarot F, Christianson AD, Li S, Dai P. Temperature and polarization dependence of low-energy magnetic fluctuations in nearly optimally doped NaFe0.9785Co0.0215As. Phys. Rev. B (2017) 96: 184512.
  * (85) Song Y, Man H, Zhang R, Lu X, Zhang C, Wang M, Tan G, Regnault L-P, Su Y, Kang J, Fernandes RM, Dai P. Spin anisotropy due to spin-orbit coupling in optimally hole-doped Ba0.67K0.33Fe2As2. Phys. Rev. B (2016) 94: 214516.
  * (86) Xie T, Wei Y, Gong D, Fennell T, Stuhr U, Kajimoto R, Ikeuchi K, Li S, Hu J, Luo H. Odd and Even Modes of Neutron Spin Resonance in the Bilayer Iron-Based Superconductor CaKFe4As4. Phys. Rev. Lett. (2018) 120: 267003.
  * (87) Xie T, Gong D, Ghosh H, Ghosh A, Soda M, Masuda T, Itoh S, Bourdarot F, Regnault L-P, Danilkin S, Li S, Luo H. Neutron Spin Resonance in the 112-Type Iron-Based Superconductor. Phys. Rev. Lett. (2018) 120: 137001.
  * (88) Xie T, Liu C, Bourdarot F, Regnault L-P, Li S, Luo H. Spin-excitation anisotropy in the bilayer iron-based superconductor CaKFe4As4. Phys. Rev. Research (2020) 2: 022018(R).
  * (89) Wang T, Zhang C, Xu LC, Wang JH, Jiang S, Zhu ZW, Wang ZS, Chu JN, Feng JX, Wang LL, Li W, Hu T, Liu XS, Mu G. Strong Pauli paramagnetic effect in the upper critical field of KCa2Fe4As4F2. Sci. China Phys. Mech. Astron. (2020) 63: 227412.
  * (90) Guo J, Yue L, Iida K, Kamazawa K, Chen L, Han T, Zhang Y, Li Y. Preferred Magnetic Excitations in the Iron-Based Sr1-xNaxFe2As2 Superconductor. Phys. Rev. Lett. (2019) 122: 017001.
  * (91) Liu C, Bourges P, Sidis Y, Xie T, He G, Bourdarot F, Danilkin S, Ghosh H, Ghosh S, Ma X, Li S, Li Y, Luo H. Preferred Spin Excitations in the Bilayer Iron-Based Superconductor CaK(Fe0.96Ni0.04)4As4 with Spin-Vortex Crystal Order. Phys. Rev. Lett. (2022) 128: 137003.
  * (92) Luo H, Wang Z, Yang H, Cheng P, Zhu X, Wen H-H. Growth and characterization of A1-xKxFe2As2 (A = Ba, Sr) single crystals with $x=$ 0 - 0.4. Supercond. Sci. Technol. (2008) 21: 125014.
  * (93) Chen Y, Lu X, Wang M, Luo H, Li S. Systematic growth of BaFe2-xNixAs2 large crystals, Supercond. Sci. Technol. (2011) 24: 065004.
  * (94) Xie T, Gong D, Zhang W, Gu Y, Huesges Z, Chen D, Liu Y, Hao L, Meng S, Lu Z, Li S, Luo H. Crystal growth and phase diagram of 112-type iron pnictide superconductor Ca1-yLayFe1-xNixAs2. Supercond. Sci. Technol. (2017) 30: 095002.
  * (95) Wang T, Chu JN, Feng JX, Wang LL, Xu XG, Li W, Wen HH, Liu XS, Mu G. Low temperature specific heat of 12442-type KCa2Fe4As4F2 single crystals. Sci. China Phys. Mech. Astron. (2020) 63: 297412.
  * (96) Lu X, Tseng K-F, Keller T, Zhang W, Hu D, Song Y, Man H, Park JT, Luo H, Li S, Nevidomskyy AH, Dai P. Impact of uniaxial pressure on structural and magnetic phase transitions in electron-doped iron pnictides. Phys. Rev. B (2016) 93: 134519.
  * (97) Tam DW, Wang W, Zhang L, Song Y, Zhang R, Carr SV, Walker HC, Perring TG, Adroja DT, Dai P. Weaker nematic phase connected to the first order antiferromagnetic phase transition in SrFe2As2 compared to BaFe2As2. Phys. Rev. B (2019) 99: 134519.
  * (98) Tam DW, Yin ZP, Xie Y, Wang W, Stone MB, Adroja DT, Walker HC, Yi M, Dai P. Orbital selective spin waves in detwinned NaFeAs. Phys. Rev. B (2020) 102: 054430.
  * (99) Liu P, Klemm ML, Tian L, Lu X, Song Y, Tam DW, Schmalzl K, Park JT, Li Y, Tan G, Su Y, Bourdarot F, Zhao Y, Lynn JW, Birgeneau RJ, Dai P. In-plane uniaxial pressure-induced out-of-plane antiferromagnetic moment and critical fluctuations in BaFe2As2. Nat. Commun. (2020) 11: 5728.
  * (100) Sobolev O, Park JT. PUMA: Thermal three axes spectrometer. J. Large Scale Res. Facil. (2015) 1: A13.
  * (101) Danilkin SA, Yethiraj M. TAIPAN: Thermal Triple-Axis Spectrometer, Neutron News (2009) 20: 37-39.
  * (102) Nakamura M, Kajimoto R, Inamura Y, Mizuno F, Fujita M, Yokoo T, Arai M. First Demonstration of Novel Method for Inelastic Neutron Scattering Measurement Utilizing Multiple Incident Energies. J. Phys. Soc. Jpn. (2009) 78: 093002.
  * (103) Kajimoto R _et al._ The Fermi Chopper Spectrometer 4SEASONS at J-PARC. J. Phys. Soc. Jpn. (2011) 80: SB025.
  * (104) Inamura Y, Nakatani T, Suzuki J, Otomo T. Development Status of Software ”Utsusemi” for Chopper Spectrometers at MLF, J-PARC. J. Phys. Soc. Jpn. (2013) 82: SA031.
  * (105) ISIS Facility, Rutherford Appleton Laboratory, UK (2000) https://www.isis.stfc.ac.uk/Pages/Excitations-Software.aspx.
  * (106) Blomberg EC, Tanatar MA, Fernandes RM, Mazin II, Shen B, Wen H-H, Johannes MD, Schmalian J, Prozorov R. Sign-reversal of the in-plane resistivity anisotropy in hole-doped iron pnictides. Nat. Commun. (2013) 4: 1914.
  * (107) Ma JQ, Luo XG, Cheng P, Zhu N, Liu DY, Chen F, Ying JJ, Wang AF, Lu XF, Lei B, Chen XH. Evolution of anisotropic in-plane resistivity with doping level in Ca1-xNaxFe2As2 single crystals. Phys. Rev. B (2014) 89: 174512.
  * (108) Kobayashi T, Tanaka K, Miyasaka S, Tajima S. Importance of Fermi Surface Topology for In-Plane Resistivity Anisotropy in Hole- and Electron-Doped Ba(Fe1-xTMx)2As2 (TM = Cr, Mn, and Co). J. Phys. Soc. Jpn. (2015) 84: 094707.
  * (109) Ishida K, Tsujii M, Hosoi S, Mizukami Y, Ishida S, Iyo A, Eisaki H, Wolf T, Grube K, Löhneysen Hv, Fernandes RM, Shibauchi T. Novel electronic nematicity in heavily hole-doped iron pnictide superconductors. Proc. Natl. Acad. Sci. USA (2020) 117: 6424-6429.
  * (110) Tam DW _et al._ Uniaxial pressure effect on the magnetic ordered moment and transition temperatures in BaFe2-xTxAs2 (T = Co, Ni). Phys. Rev. B (2017) 95: 060505(R).
  * (111) Daghofer M, Luo Q-L, Yu R, Yao DX, Moreo A, Dagotto E. Orbital-weight redistribution triggered by spin order in the pnictides. Phys. Rev. B (2010) 81: 180514(R).


# Bifurcations and transition to chaos in generalized fractional maps of the
orders $0<\alpha<1$

Mark Edelman<EMAIL_ADDRESS>Avigayil B. Helman
<EMAIL_ADDRESS>Rasa Smidtaite<EMAIL_ADDRESS>Stern College
for Women, Yeshiva University, 245 Lexington Ave., New York, NY 10016, USA
Courant Institute of Mathematical Sciences at NYU, 251 Mercer Street, New
York, NY 10012, USA  School of Engineering and Applied Sciences, Columbia
University, 500 W. 120th Street, New York, NY 10027, USA  Department of
Applied Mathematics, Kaunas University of Technology, Studentu 50-429, 51368,
Kaunas, Lithuania

###### Abstract

Generalized fractional maps of the orders $0<\alpha<1$ are Volterra difference
equations of convolution type with kernels, which differences are absolutely
summable, but the series of kernels are diverging. Commonly used in
applications fractional (with the power-law kernels) and fractional difference
(with the falling factorial kernels) maps of the orders $0<\alpha<1$ belong to
this class of maps. We derived the algebraic equations which allow the
calculation of bifurcation points of generalized fractional maps. We
calculated the bifurcation points and drew bifurcation diagrams for the
fractional and fractional difference logistic maps. Although the transition to
chaos on individual trajectories in fractional maps may be characterized by
the cascade of bifurcations type behavior and zero Lyapunov exponents, the
results of our numerical simulations allow us to make a conjecture that the
cascade of bifurcations scenarios of transition to chaos in generalized
fractional maps and regular maps are similar, and the value of the generalized
fractional Feigenbaum constant $\delta_{f}$ is the same as the value of the
regular Feigenbaum constant $\delta=4.669...$.

###### keywords:

Fractional maps , Periodic points , Bifurcations , Fractional difference

††journal: Chaos, Solitons & Fractals

##  1 Introduction

Introduced in [1] for orders $\alpha>1$ and extended in [2] to $\alpha>0$
fractional and proposed in [3, 4] fractional difference maps are Volterra
difference equations of convolution type with the power-law (in fractional
maps) and falling factorial (in fractional difference maps) kernels. These
maps were recently used in cryptography [5] and biological [6], economic [7],
communication [8], and control (multiple papers) applications. It is the
researchers’ task to find which maps to use in a particular application or
which one is a better fit for a natural phenomenon. Fractional and fractional
difference maps share many features like cascade of bifurcations type
trajectories and power-law convergence to the fixed points (see, e.g. reviews
[9, 10]). Their properties to a high degree are defined by the common
properties of their kernels which belong to a class of discrete functions
whose series are diverging; however, the series of their differences belong to
the space of absolutely converging series $l_{1}$. For this reason, the
generalized fractional maps, which include fractional and fractional
difference maps, were introduced in [11, 12].

The universal cascade of bifurcations scenario of transition to chaos is well-
known (the most popular reference is Robert May’s “Nature” article [13]). The
foundational papers on universality in non-linear dynamics are collected in
the reprinted selection of articles [14]. Mathematically, the universality in
regular nonlinear dynamics is expressed through the Feigenbaum function and
constants (see, e.g., [15, 16]), and is based on the analysis of periodic
points and their bifurcations. Fractional systems have no bifurcation points
[17, 18, 19, 20, 21, 22, 23], and the corresponding analysis cannot be
directly extended to the maps with the asymptotically power-law memory.

Although the fractional maps do not have periodic points, they have
asymptotically periodic points. Asymptotically periodic points and stability
of the fixed points in generalized fractional maps were investigated in the
recent papers [11, 12, 24, 25]. In this paper, we use the results of [11] to
draw the bifurcation diagrams for the fractional difference logistic map. We
also derive the equations which define bifurcation points in generalized
fractional maps. We use them to calculate the bifurcation points in fractional
and fractional difference logistic maps. The problem that we are trying to
address is whether the universal behavior persists in generalized fractional
maps. Our numerical results suggest that the answer is “yes” and that the
fractional Feigenbaum constant $\delta$ has the same value as the regular
Feigenbaum constant.

We performed numerical simulations of the fractional and fractional difference
logistic maps for various values of the order $\alpha\in(0,1)$. In this paper,
we present only selected numerical results for the order $\alpha=0.5$ (the
value which is exactly in the middle of the interval between two integer
values) to demonstrate our findings, which are valid for all considered values
of $\alpha$.

##  2 Preliminaries

The generalized universal $\alpha$-family of maps of the orders $0<\alpha<1$
is defined as (see [11, 12]):

$\displaystyle x_{n}=x_{0}-\sum^{n-1}_{k=0}G^{0}(x_{k})U_{\alpha}(n-k),$ (1)

where $G^{0}(x)=h^{\alpha}G_{K}(x)/\Gamma(\alpha)$, $x_{0}$ is the initial
condition, $h$ is the time step of the map, $\alpha$ is the order of the map,
$G_{K}(x)$ is a nonlinear function depending on the parameter $K$,
$U_{\alpha}(n)=0$ for $n\leq 0$, and
$U_{\alpha}(n)\in\mathbb{D}^{0}(\mathbb{N}_{1})$. The space
$\mathbb{D}^{i}(\mathbb{N}_{1})$ is defined as (see [12])

$\displaystyle\mathbb{D}^{i}(\mathbb{N}_{1})\ \ =\ \
\\{f:|\sum^{\infty}_{k=1}\Delta^{i}f(k)|>N,$ $\displaystyle\forall N,\ \
N\in\mathbb{N},\sum^{\infty}_{k=1}|\Delta^{i+1}f(k)|=C,\ \
C\in\mathbb{R}_{+}\\},$ (2)

where the forward difference operator $\Delta$ is defined as

$\Delta f(n)=f(n+1)-f(n).$ (3)

For any real number $a$, $\mathbb{N}_{a}=\\{a,a+1,a+2,a+3,...\\}$, and
$\mathbb{N}=\mathbb{N}_{0}$. To generalize a one-dimensional map

$x_{n+1}=F_{K}(x_{n})$ (4)

function $G_{K}(x)$ is defined as

$G_{K}(x_{n})=x_{n}-F_{K}(x_{n})$ (5)

to converge to the map Eq. (4) when $\alpha=1$, $h=1$, and $U_{1}(n)=1$.

In the case of Caputo fractional maps [1, 2], $U_{\alpha}(n)$ is a power
function

$U_{\alpha}(n)=n^{\alpha-1}.$ (6)

In the case Caputo fractional difference maps, which are defined as solutions
of the Caputo h-difference equation [3, 4, 26]

$(_{0}\Delta^{\alpha}_{h,*}x)(t)=-G_{K}(x(t+(\alpha-1)h)),$ (7)

where $t\in(h\mathbb{N})_{m}$, with the initial conditions

$(_{0}\Delta^{k}_{h}x)(0)=c_{k},\ \ \ k=0,1,...,m-1,\ \ \
m=\lceil\alpha\rceil,$ (8)

the kernel $U_{\alpha}(n)$ is the falling factorial function:

$\displaystyle U_{\alpha}(n)=(n+\alpha-2)^{(\alpha-1)},\ \
U_{\alpha}(1)=(\alpha-1)^{(\alpha-1)}=\Gamma(\alpha).$ (9)

The falling factorial $t^{(\alpha)}$ is defined as

$t^{(\alpha)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)},\ \ t\neq-1,-2,-3....$
(10)

The falling factorial is asymptotically a power function (See Eq. (32) and
Fig. 4 from [4]):

$\lim_{t\rightarrow\infty}\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)t^{\alpha}}=1,\
\ \ \alpha\in\mathbb{R}.$ (11)

The $h$-falling factorial $t^{(\alpha)}_{h}$ is defined as [27]

$\displaystyle
t^{(\alpha)}_{h}=h^{\alpha}\frac{\Gamma(\frac{t}{h}+1)}{\Gamma(\frac{t}{h}+1-\alpha)}=h^{\alpha}\Bigl{(}\frac{t}{h}\Bigr{)}^{(\alpha)},\
\ \frac{t}{h}\neq-1,-2,-3,....$ (12)

The asymptotically period-$l$ points in generalized fractional maps are
defined by the following equations [11]

$\displaystyle
x_{l,m+1}-x_{l,m}=S_{1,l}G^{0}(x_{l,m})+\sum^{m-1}_{j=1}S_{j+1,l}G^{0}(x_{l,m-j})$
$\displaystyle+\sum^{l-1}_{j=m}S_{j+1,l}G^{0}(x_{l,m-j+l}),\ \ 0<m<l,$ (13)
$\displaystyle\sum^{l}_{j=1}G^{0}(x_{l,j})=0,$ (14)

where

$\displaystyle
S_{j+1,l}=\sum^{\infty}_{k=0}\Bigl{[}U_{\alpha}(lk+j)-U_{\alpha}(lk+j+1)\Bigr{]},\
\ 0\leq j<l.$ (15)

In this paper we will use the following easily verifiable identity:

$\sum^{l}_{j=1}S_{j,l}=0.$ (16)

##  3 Equations defining $T=2^{n-1}$ – $T=2^{n}$ bifurcation points

Let us consider a cycle with the period $l=2^{n}$. We expect that near the
$T=2^{n-1}$ – $T=2^{n}$ bifurcation point the difference
$x_{2^{n},i+2^{n-1}}-x_{2^{n},i}$ will be small. To obtain the value of this
difference, we perform the summation of Eq. (13) from $m=i$ to $m=i+2^{n-1}-1$
and use the results of the following transformations:

$\displaystyle\sum^{i+2^{n-1}-1}_{m=i}(x_{2^{n},m+1}-x_{2^{n},m})=x_{2^{n},i+2^{n-1}}-x_{2^{n},i};$
(17)
$\displaystyle\sum^{i+2^{n-1}-1}_{m=i}\sum^{m-1}_{j=1}S_{j+1,2^{n}}G^{0}(x_{2^{n},m-j})=\sum^{i+2^{n-1}-1}_{m=i}\sum^{m-1}_{j=1}S_{m-j+1,2^{n}}G^{0}(x_{2^{n},j})$
$\displaystyle=\sum^{i-1}_{j=1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}+\sum^{i+2^{n-1}-2}_{j=i}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=j+1}S_{m-j+1,2^{n}};$
(18)
$\displaystyle\sum^{i+2^{n-1}-1}_{m=i}\sum^{2^{n}-1}_{j=m}S_{j+1,2^{n}}G^{0}(x_{2^{n},m-j+2^{n}})=\sum^{i+2^{n-1}-1}_{m=i}\sum^{2^{n}}_{j=m+1}S_{m-j+2^{n}+1,2^{n}}G^{0}(x_{2^{n},j})$
$\displaystyle=\sum^{i+2^{n-1}}_{j=i+1}G^{0}(x_{2^{n},j})\sum^{j-1}_{m=i}S_{m-j+2^{n}+1,2^{n}}$
$\displaystyle+\sum^{2^{n}}_{j=i+2^{n-1}+1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+2^{n}+1,2^{n}};$
(19) $\displaystyle 0<i\leq 2^{n-1}.$

The resulting set of equations can be written as

$\displaystyle
x_{2^{n},i+2^{n-1}}-x_{2^{n},i}=\sum^{i+2^{n-1}-1}_{j=i}G^{0}(x_{2^{n},j})S_{1,2^{n}}+\sum^{i-1}_{j=1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}$
$\displaystyle+\sum^{i+2^{n-1}-2}_{j=i}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=j+1}S_{m-j+1,2^{n}}+\sum^{i+2^{n-1}}_{j=i+1}G^{0}(x_{2^{n},j})\sum^{j-1}_{m=i}S_{m-j+2^{n}+1,2^{n}}$
$\displaystyle+\sum^{2^{n}}_{j=i+2^{n-1}+1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+2^{n}+1,2^{n}};$
(20) $\displaystyle 0<i\leq 2^{n-1}.$

It is easy to see that this formula can be written as

$\displaystyle
x_{2^{n},i+2^{n-1}}-x_{2^{n},i}=\sum^{2^{n-1}}_{j=1}[G^{0}(x_{2^{n},j})-G^{0}(x_{2^{n},2^{n-1}+j})]\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}};$
(21) $\displaystyle 0<i\leq 2^{n-1},$

where coefficients $S_{k,2^{n}}$, defined by Eq. (15) for $0<k\leq 2^{n}$, are
continued periodically for $\forall k,\ \ k\in\mathbb{Z}$:

$\displaystyle S_{k,2^{n}}=S_{k+2^{n},2^{n}}.$ (22)

Indeed, let us consider the terms on RHS of Eq. (20) containing the same
factor $G^{0}(x_{2^{n},p})$. Only the second sum contains the terms with
$0<p<i$. They all can be written as

$\displaystyle G^{0}(x_{2^{n},p})\sum^{i+2^{n-1}-1}_{m=i}S_{m-p+1,2^{n}}.$
(23)

Only the first and the third sums contain the terms with $p=i$:

$\displaystyle
G^{0}(x_{2^{n},p})\Bigl{[}S_{1}+\sum^{i+2^{n-1}-1}_{m=i+1}S_{m-p+1,2^{n}}\Bigr{]}=G^{0}(x_{2^{n},p})\sum^{i+2^{n-1}-1}_{m=i}S_{m-p+1,2^{n}}.$
(24)

The first, the third, and the fourth sums contain the terms with $i<p\leq
i+2^{n-1}-2$:

$\displaystyle
G^{0}(x_{2^{n},p})\Bigl{[}S_{1}+\sum^{i+2^{n-1}-1}_{m=p+1}S_{m-p+1,2^{n}}+\sum^{p-1}_{m=i}S_{m-p+1,2^{n}}\Bigr{]}$
$\displaystyle=G^{0}(x_{2^{n},p})\sum^{i+2^{n-1}-1}_{m=i}S_{m-p+1,2^{n}}.$
(25)

Only the first and the fourth sums contain the terms with $p=i+2^{n-1}-1$:

$\displaystyle
G^{0}(x_{2^{n},p})\Bigl{[}S_{1}+\sum^{i+2^{n-1}-2}_{m=i}S_{m-p+1,2^{n}}\Bigr{]}=G^{0}(x_{2^{n},p})\sum^{i+2^{n-1}-1}_{m=i}S_{m-p+1,2^{n}}.$
(26)

It is obvious that for $p=i+2^{n-1}$ the fourth sum and for $i+2^{n-1}<p\leq
2^{n}$ the fifth sum produce the same formula Eq. (23). Now we may write

$\displaystyle
x_{2^{n},i+2^{n-1}}-x_{2^{n},i}=\sum^{2^{n-1}}_{j=1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}$
$\displaystyle+\sum^{2^{n}}_{j=2^{n-1}+1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}};\
\ 0<i\leq 2^{n-1}.$ (27)

After substitutions $p=j-2^{n-1}$ and $q=m-2^{n-1}$, the second sum in the
last expression can be written as

$\displaystyle\sum^{2^{n}}_{j=2^{n-1}+1}G^{0}(x_{2^{n},j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}=\sum^{2^{n-1}}_{p=1}G^{0}(x_{2^{n},2^{n-1}+p})\sum^{i-1}_{q=i-2^{n-1}}S_{q-p+1,2^{n}}$
$\displaystyle=\sum^{2^{n-1}}_{p=1}G^{0}(x_{2^{n},2^{n-1}+p})\Bigl{[}\sum^{i+2^{n-1}-1}_{q=i-2^{n-1}}S_{q-p+1,2^{n}}-\sum^{i+2^{n-1}-1}_{q=i}S_{q-p+1,2^{n}}\Bigr{]}$
$\displaystyle=-\sum^{2^{n-1}}_{j=1}G^{0}(x_{2^{n},2^{n-1}+j})\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}};\
\ 0<i\leq 2^{n-1}.$ (28)

Validity of Eq. (21) follows from Eqs. (27) and (28).

Near the $T=2^{n-1}$ – $T=2^{n}$ bifurcation point,
$x_{2^{n},j+2^{n-1}}\approx x_{2^{n},j}\approx x_{2^{n-1}bif,j}$ and the
differences $x_{2^{n},j+2^{n-1}}-x_{2^{n},j}$ are small. The set
$\\{x_{2^{n-1}bif,1},x_{2^{n-1}bif,2},...,x_{2^{n-1}bif,2^{n-1}}\\}$ is the
$2^{n-1}$-cycle at the bifurcation point. In the linear approximation, we may
write

$\displaystyle
G^{0}(x_{2^{n},j+2^{n-1}})-G^{0}(x_{2^{n},j})=\frac{dG^{0}(x)}{dx}\Bigl{|}_{x_{2^{n-1}bif,j}}(x_{2^{n},j+2^{n-1}}-x_{2^{n},j}).$
(29)

Near the bifurcation point, Eq. (21) can be written as

$\displaystyle\sum^{2^{n-1}}_{j=1}\Bigl{[}\frac{dG^{0}(x)}{dx}\Bigl{|}_{x_{2^{n-1}bif,j}}\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}+\delta_{i,j}\Bigr{]}(x_{2^{n},j+2^{n-1}}-x_{2^{n},j})=0;$
(30) $\displaystyle 0<i\leq 2^{n-1},$

where $\delta_{i,j}$ is the Kronecker symbol. System of $2^{n-1}$ linear
equations Eq. (30) has a nonzero solution only if the determinant of $2^{n-1}$
by $2^{n-1}$ matrix $A$ of its coefficients

$\displaystyle
A_{i,j}=\frac{dG^{0}(x)}{dx}\Bigl{|}_{x_{2^{n-1}bif,j}}\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}+\delta_{i,j}$
(31)

is equal to zero:

$\displaystyle\det(A)=0.$ (32)

The results of this section may be summarized as a theorem:

###### Theorem 3.1

The $T=2^{n-1}$ – $T=2^{n}$ bifurcation points, $2^{n-1}$ values of
$x_{2^{n-1}bif,i}$ with $0<i\leq 2^{n-1}$ and the value of the nonlinear
parameter $K_{2^{n-1}bif}$, of a fractional generalization of a nonlinear one-
dimensional map $x_{n+1}=F_{K}(x_{n})$ written as the Volterra difference
equations of convolution type

$\displaystyle x_{n}=x_{0}-\sum^{n-1}_{k=0}G^{0}(x_{k})U_{\alpha}(n-k),$ (33)

where $G^{0}(x)=h^{\alpha}G_{K}(x)/\Gamma(\alpha)$, $x_{0}$ is the initial
condition, $h$ is the time step of the map, $\alpha$ is the order of the map,
$G_{K}(x)=x-F_{K}(x)$, $U_{\alpha}(n)=0$ for $n\leq 0$,
$U_{\alpha}(n)\in\mathbb{D}^{0}(\mathbb{N}_{1})$, and

$\displaystyle\mathbb{D}^{i}(\mathbb{N}_{1})\ \ =\ \
\\{f:|\sum^{\infty}_{k=1}\Delta^{i}f(k)|>N,$ $\displaystyle\forall N,\ \
N\in\mathbb{N},\sum^{\infty}_{k=1}|\Delta^{i+1}f(k)|=C,\ \
C\in\mathbb{R}_{+}\\},$ (34)

are defined by the system of $2^{n-1}+1$ equations

$\displaystyle
x_{2^{n-1}bif,m+1}-x_{2^{n-1}bif,m}=S_{1,2^{n-1}}G^{0}(x_{2^{n-1}bif,m})+\sum^{m-1}_{j=1}S_{j+1,2^{n-1}}G^{0}(x_{2^{n-1}bif,m-j})$
$\displaystyle+\sum^{2^{n-1}-1}_{j=m}S_{j+1,2^{n-1}}G^{0}(x_{2^{n-1}bif,m-j+2^{n-1}}),\
\ 0<m<2^{n-1},$ (35)
$\displaystyle\sum^{2^{n-1}}_{j=1}G^{0}(x_{2^{n-1}bif,j})=0,$ (36)
$\displaystyle\det(A)=0,$ (37)

where

$\displaystyle
S_{j+1,l}=\sum^{\infty}_{k=0}\Bigl{[}U_{\alpha}(lk+j)-U_{\alpha}(lk+j+1)\Bigr{]},\
\ 0\leq j<l,$ $\displaystyle S_{i,l}=S_{i+l,l},\ \ i\in\mathbb{Z},$ (38)

and the elements of the $2^{n-1}$-dimensional matrix $A$ are

$\displaystyle
A_{i,j}=\frac{dG^{0}(x)}{dx}\Bigl{|}_{x_{2^{n-1}bif,j}}\sum^{i+2^{n-1}-1}_{m=i}S_{m-j+1,2^{n}}+\delta_{i,j}\
\ 0<i,j\leq 2^{n-1}.$ (39)

##  4 Bifurcation diagrams and transition to chaos in the fractional and
fractional difference logistic maps with $0<\alpha<1$

In the generalized fractional logistic map

$\displaystyle G_{K}(x)=x-Kx(1-x).$ (40)

We solved Eqs. (13) and (14) numerically using the Mathematica’s Newton’s
method (the exact solution) to draw the bifurcation diagrams for the
fractional difference and fractional logistic maps with various values of
$\alpha\in(0,1)$ (we assume $h=1$). Fig. 1 shows the results of our
calculations for the fractional difference logistic map with $\alpha=0.5$ (the
value which is far from the integer values one and zero).

Figure 1: A part of the bifurcation diagram for the Caputo fractional
difference logistic map of the order $\alpha=0.5$ from $K=2$ (fixed point) to
approximately $K=3.16$ (T=128 periodic point). In figures a-c the steady line
represents the solution of Eqs. (13) and (14) (the exact solution) and the
dots represent numerical calculations on a single trajectory with $x_{0}=0.3$
after $10^{5}$ iterations. Figure d represents the exact solution from $T=32$
on the left to $T=128$ on the right. Figures e and f represent the
calculations on a single trajectory.

From Fig. 1b one may see that already for the period four ($T=4$) points there
is a noticeable difference between the exact solutions and the results
obtained after $10^{5}$ iterations on a single trajectory. For $T>16$ (Figs. 1
b, e, f) the results obtained after $10^{5}$ iterations on a single trajectory
seem to be noticeably inaccurate. Multiple papers investigating various
fractional difference maps contain bifurcation diagrams obtained by iterations
on a single trajectory and the number of iterations in all these papers is
much less than $10^{5}$.

Figure 2: Transition to chaos in the Caputo fractional difference logistic map
of the order $\alpha=0.2$ with $K=3.3$. a. The trajectory $x=x_{n}(n)$ with
the initial condition $x_{0}=10^{-6}$; b. log-log plot of the deviation of two
trajectories $x=x_{n,2}-x_{n,1}$ with the initial conditions $x_{0,1}=10^{-6}$
and $x_{0,2}=10^{-6}+10^{-9}$; c. $\log_{10}(x)$ vs. $n$ plot for the case in
b; d. logarithm of deviation vs. $n$ plot for the trajectories with
$x_{0,1}=0.3$ and $x_{0,2}=0.3+10^{-9}$.

Bifurcation diagrams for the fractional and fractional difference logistic
maps with various values of $0<\alpha<1$ are qualitatively the same.
Bifurcation diagrams obtained with the increasing number of iterations on a
single trajectory (and the exact bifurcation diagrams when the number of
iterations goes to infinity), in many cases, demonstrate the increasing shift
to the left (see Figs. 1 b and 1 c). This phenomenon was first noticed and
explained on the examples of the fractional and fractional difference standard
maps in [4] (see Fig. 8 there). The explanation of the left shift is based on
another phenomenon typical for fractional and fractional difference maps,
cascade of bifurcations type trajectories (CBTT), first noticed in [28] on the
example of the order $\alpha=1.65$ standard map (Fig. 5 in that article). CBTT
appear in higher periodicity asymptotically stable trajectories which first
converge to a fixed point and then evolve following the cascade of
bifurcations type scenario converging to an asymptotically periodic cycle
(see, e.g., Figs. 6.6a and 6.7a from [6]). Even chaotic trajectories may first
converge to a fixed point or a periodic cycle (see Fig. 7 from [4] or Fig. 3
from [10]). In fractional difference maps, convergence of trajectories to
fixed points follows the power law [29] and the corresponding Lyapunov
exponents are equal to zero. In the case of chaotic trajectories, the rate of
convergence depends on the initial conditions. Two initially close
trajectories in the asymptotically chaotic fractional difference logistic map
($\alpha=0.2$ and $K=3.3$) Fig. 2a, depending on the initial conditions, may
diverge exponentially Fig. 2d and the corresponding Lyapunov exponent is
positive. But the graph of the deviation of two initially close to zero
trajectories Fig. 2b and Fig. 2c shows the transition to chaos through a
series of power-law convergencies followed by jumps.

Fractional and fractional difference maps may also demonstrate inverse CBTT,
when an asymptotically periodic trajectory starts as a chaotic or higher
periodicity trajectory. Examples of the inverse CBTT in the fractional and
fractional difference logistic maps are Figs. 7 and 8 from [2] and Figs. 7.6b
and 6.7c from [6]. The inverse CBTT will cause the right shift in bifurcation
diagrams. Whether the asymptotic bifurcation diagram of a
fractional/fractional difference map is shifted to the left or to the right
with respect to the bifurcation diagram obtained after a finite number of
iterations on single trajectories depends on the type of the map and the
initial conditions used.

$n$ | $K_{1}(n)$ | $K_{.5}(n)$ | $\frac{\Delta K_{1}(n-2)}{\Delta K_{1}(n-1)}$ | $\frac{\Delta K_{.5}(n-2)}{\Delta K_{.5}(n-1)}$ | $\delta_{1}-\delta_{F}$ | $\delta_{.5}-\delta_{F}$
---|---|---|---|---|---|---
1 | 3 | 2.41421356 |  |  |  |
2 | 3.4494897 | 3.03150807 |  |  |  |
3 | 3.5440904 | 3.12880294 | 4.751446 | 6.3446 | $8.2\times 10^{-2}$ | 1.6754
4 | 3.5644073 | 3.15225457 | 4.656251 | 4.1487 | $-1.3\times 10^{-2}$ | -.5205
5 | 3.5687594 | 3.15698186 | 4.668242 | 4.9609 | $-9.6\times 10^{-4}$ | .2917
6 | 3.5696916 | 3.15802086 | 4.668739 | 4.5499 | $-4.6\times 10^{-4}$ | -.1193
7 | 3.5698913 | 3.15824198 | 4.669132 | 4.7199 | $-6.9\times 10^{-5}$ | .0507
8 | 3.5699340 | 3.15828833 | 4.669183 | 4.6496 | $-1.9\times 10^{-5}$ | -.0196
9 | 3.5699432 | 3.15829845 | 4.669195 | 4.6766 | $-7\times 10^{-6}$ | .0074
10 | 3.5699451 | 3.15830062 |  | 4.6665 |  | -.0027
11 |  | 3.158301084 |  | 4.6702 |  | .0010
12 |  | 3.158301183 |  | 4.6688 |  | -.0004
Table 1: Approaching the Feigenbaum constant $\delta$ in the regular and
fractional difference logistic maps. $K_{1}(n)$ are the values of the map
parameter for the period $2^{n-1}$ – period $2^{n}$ bifurcation points in the
regular logistic map ($\alpha=1$); $K_{.5}(n)$ are the same points in the case
$\alpha=.5$; $\delta_{1}=\Delta K_{1}(n-2)/\Delta
K_{1}(n-1)=[K_{1}(n-1)-K_{1}(n-2)]/[K_{1}(n)-K_{1}(n-1)]$; $\delta_{.5}$ is
the same value in the case $\alpha=.5$; $\delta_{F}=\delta=4.6692016$.

Table 1 shows the values of the map parameter for the period $2^{n-1}$ –
period $2^{n}$ bifurcation points $K_{1}(n)$ (for the regular logistic map)
and $K_{.5}(n)$ (for the order $\alpha=0.5$ fractional difference logistic
map). The results for $n=1,2,...,10$ and $\alpha=0.5$ were obtained using Eqs.
(13) and (14). The table also shows the corresponding values of the ratios
$\delta_{1}=\Delta K_{1}(n-2)/\Delta
K_{1}(n-1)=[K_{1}(n-1)-K_{1}(n-2)]/[K_{1}(n)-K_{1}(n-1)]$ and
$\delta_{.5}=\Delta K_{.5}(n-2)/\Delta
K_{.5}(n-1)=[K_{.5}(n-1)-K_{.5}(n-2)]/[K_{.5}(n)-K_{.5}(n-1)]$. All results
for $\alpha=0.5$ in Table 1 were later calculated using Theorem 3.1 (Eqs.
(35)-(37)), and for the first ten values of $n$ the results are identical.
Numerical calculations were performed using Matlab and the obtained results
had higher accuracy than the corresponding solutions of Eqs. (13) and (14)
obtained using Mathematica. The values for the regular logistic map are
available from many sources (see, e.g., [30, 31]. The number of significant
digits in the tables is dictated by the requirement that the table should fit
the standard journal’s page. Higher accuracy results for the case $\alpha=1$
are available at multiple sites on the Internet. Corresponding results for the
order $\alpha=0.5$ fractional logistic map are presented in Table 2.

$n(T=2^{n})$ | $K_{.5}(n)$ | $\frac{\Delta K_{.5}(n-2)}{\Delta K_{.5}(n-1)}$ | $\delta_{.5}-\delta_{F}$
---|---|---|---
1 | 3.93016668194129 |  |
2 | 4.88614847910753 |  |
3 | 5.05712582113625 | 5.591278 | 0.922076
4 | 5.09634066404542 | 4.360016 | -.309186
5 | 5.10446168317469 | 4.828808 | 0.159606
6 | 5.10622604348767 | 4.602812 | -.066389
7 | 5.10660170543380 | 4.696670 | 0.027469
8 | 5.10668234310063 | 4.658641 | -.010561
9 | 5.10669959857451 | 4.673164 | 0.003963
10 | 5.10670329532456 | 4.667742 | -.001460
Table 2: The order $\alpha=0.5$ fractional logistic map: the values of the map
parameter for the bifurcation points $K_{.5}$ and the ratios
$\delta_{.5}=\Delta K_{.5}(n-2)/\Delta
K_{.5}(n-1)=[K_{.5}(n-1)-K_{.5}(n-2)]/[K_{.5}(n)-K_{.5}(n-1)]$ converging to
the Feigenbaum number $\delta_{F}=\delta=4.6692016$.

As in the case of the regular logistic map, the values of $\delta_{.5}$
oscillate around the Feigenbaum number but converge significantly slower. The
slow convergence is expected because, in many cases, the convergence in
fractional maps follows the power law while the convergence in regular maps is
exponential. From the authors’ point of view, the data present sufficient
evidence to make a conjecture that the Feigenbaum number $\delta$ exists in
fractional and fractional difference maps and has the same value as in regular
maps.

##  5 Conclusion

In this paper, we derived the analytic expressions for the coefficients of the
equations that define bifurcation points in generalized fractional maps of the
orders $0<\alpha<1$ (Eqs. (35)–(39)). They allow calculations of asymptotic
bifurcation points in various cases which include fractional and fractional
difference maps and maps. We showed that derived in [11] equations, which
define asymptotically periodic points, Eqs. (13) and (14), allow drawing of
bifurcation diagrams which are much more accurate than bifurcation diagrams
obtained by iterations on single trajectories.

Based on the data from Tables 1 and 2, we made a conjecture that the
Feigenbaum number $\delta$ exists in fractional and fractional difference maps
and has the same value as in regular maps. This should be a result of the
symmetry and self-similarity in the system of equations Eqs. (35)–(39). We
hope that the following mathematical analysis of the equations will lead to
the explanation of this symmetry.

## Data availability

Data will be made available on request.

## Acknowledgements

The first author acknowledges support from Yeshiva University’s 2021–2022
Faculty Research Fund, expresses his gratitude to the administration of
Courant Institute of Mathematical Sciences at NYU for the opportunity to
perform some of the computations at Courant, and expresses his gratitude to
Virginia Donnelly for technical help.

The last author acknowledges support from the European Social Fund under the
No 09.3.3–LMT–K–712 ”Development of Competences of Scientists, other
Researchers and Students through Practical Research Activities” measure
(Project No. 09.3.3–LMT–K–712–23–0235).

## References

  * [1] Tarasov VE, Zaslavsky GM. Fractional equations of kicked systems and discrete maps. J Phys A 2008;41:435101.
  * [2] Edelman M. Universal Fractional Map and Cascade of Bifurcations Type Attractors. Chaos 2013;23:033127.
  * [3] Wu G-C, Baleanu D, Zeng S-D. Discrete chaos in fractional sine and standard maps. Phys Lett A 2014;378:484–7.
  * [4] Edelman M. Caputo standard $\alpha$-family of maps: Fractional difference vs. fractional. Chaos 2014;24:023137.
  * [5] Huang L-L, Baleanu D, Wu G-C, Zeng S. A new application of the fractional logistic map. Romanian Journal of Physics 2016;61(7):1172–1179.
  * [6] Edelman M. Evolution of Systems with Power-Law Memory: Do We Have to Die? (Dedicated to the Memory of Valentin Afraimovich). In: Skiadas CH, Skiadas C, editors. Demography of Population Health, Aging and Health Expenditures. 65–85, Springer, eBook; 2020.
  * [7] Tarasova VV, Tarasov VE. Logistic map with memory from economic model. Chaos, Solitons and Fractals 2017;95:84–91.
  * [8] Petkeviciute-Gerlach D, Timofejeva I, Ragulskis M. Clocking convergence of the fractional difference logistic map. Nonlinear Dynamics 2020;100:3925–35.
  * [9] Edelman M. Maps with power-law memory: direct introduction and Eulerian numbers, fractional maps, and fractional difference maps. In: A. Kochubei and Yu. Luchko (eds.), Handbook of Fractional Calculus With Applications, Volume 2, Theory. 47–64. Berlin: De Gruyter; 2019.
  * [10] Edelman M. Dynamics of nonlinear systems with power-law memory. In: V. E. Tarasov (editor), Handbook of Fractional Calculus with Applications, Volume 4, Applications in Physics. 103–132. Berlin: De Gruyter; 2019.
  * [11] Edelman M. Cycles in asymptotically stable and chaotic fractional maps. Nonlinear Dynamics 2021;104:2829–41.
  * [12] Edelman M, Helman AB. Asymptotic cycles in fractional maps of arbitrary positive orders. Fract. Calc. Appl. Anal. 2022;https://doi.org/10.1007/s13540-021-00008-w.
  * [13] May RM. Simple mathematical models with very complicated dynamics. Nature 1976;261:459–67.
  * [14] Cvitanovic P. Universality in Chaos. Bristol and New York: Adam Hilger; 1989.
  * [15] Feigenbaum MJ. Quantitative universality for a class of nonlinear transformations. J of Stat Phys 1978;19:25–52.
  * [16] Feigenbaum MJ. Universal behavior in nonlinear systems. Physica D: Nonlinear Phenomena 1983;7:16–39.
  * [17] Jagan Mohan J. Periodic solutions of fractional nabla difference equations. Communications in Applied Analysis 2016;20:585–609.
  * [18] Jagan Mohan J. Quasi-periodic solutions of fractional nabla difference systems. Fractional Differential Calculus 2017;7:339–55.
  * [19] Area I, Losada J, and Nieto JJ. On fractional derivatives and primitives of periodic functions Abstract and Applied Analysis 2014;2014:392598.
  * [20] Kaslik E, Sivasundaram S. Nonexistence of periodic solutions in fractional order dynamical systems and a remarkable difference bet ween integer and fractional order derivatives of periodic functions. Nonlinear Analysis. Real World Applications 2012;13:1489–97.
  * [21] Tavazoei MS, Haeri M. A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 2009;45:1886–90.
  * [22] Wang J, Feckan M, Zhou Y. Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations. Commun in Nonlin. Sci. Numer. Simul. 2013;18:246–56.
  * [23] Yazdani M, Salarieh H. On the existence of periodic solutions in time-invariant fractional order Systems. Automatica 2011;47:1834–37.
  * [24] Edelman M. Stability of fixed points in generalized fractional maps of the orders $0<\alpha<1$. 2022 http://arxiv.org/abs/2209.01719.
  * [25] Edelman M, Helman AB. Asymptotically periodic points, bifurcations, and transition to chaos in fractional difference maps. 2022 http://arxiv.org/abs/2209.15462.
  * [26] Chen F, Luo X, Zhou Y. Existence Results for Nonlinear Fractional Difference Equation. Adv Differ Eq 2011;2011:713201.
  * [27] Ferreira RAC, Torres DFM. Fractional h-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 2011;5:110–21.
  * [28] Edelman M, Tarasov VE. Fractional standard map. Phys Lett A 2009;374:279–285.
  * [29] Anh PT, Babiarz A, Czornik A, Niezabitowski M, Siegmund S.: Asymptotic properties of discrete linear fractional equations. Bulletin of the Polish Academy of Sciences, Technical Sciences 2019;67:749–59.
  * [30] Alligood KT, Sauer TD, Yorke JA. Chaos An Introduction to Dynamical Systems. New York: Springer; 1996.
  * [31] Sarmah HK, Baishya TK. Period doubling route in the periodic and chaotic region of the logistic map. Int J Appl Math Stat Sci 2013;2:49–62.