# Stability and chaotic behaviors of Bose-Einstein condensates in optical
lattices with two- and three-body interactions

Yan Chen  Institute of Theoretical Physics, Lanzhou University, Lanzhou
$730000$, China  Ke-Zhi Zhang  Physics and Electronics Engineering College,
Northwest Normal University, Lanzhou 730070, China  Yong Chen  Corresponding
author. Email<EMAIL_ADDRESS>Institute of Theoretical Physics, Lanzhou
University, Lanzhou $730000$, China  Key Laboratory for Magnetism and Magnetic
Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000,
China

###### Abstract

The stability and chaotic behaviors of Bose-Einstein condensates with two- and
three-atom interactions in optical lattices are discussed with analytical and
numerical methods. It is found that the steady-state relative population
appears tuning-fork bifurcation when the system parameters are changed to
certain critical values. In particular, the existence of three-body
interaction not only transforms the bifurcation point of the system but also
affects greatly on the macroscopic quantum self-trapping behaviors of the
system associated with the critically stable steady-state solution. In
addition, we also investigated the influence of the initial conditions, three-
body interaction and the energy bias on the macroscopic quantum self-trapping.
Finally, by applying the periodic modulation on the energy bias, we find that
the relative population oscillation exhibits a process from order to chaos,
via a series of period-doubling bifurcations.

###### pacs:

03.75.Kk, 67.85.Jk, 03.65.Ge,

##  I Introduction

In Recent years, Bose-Einstein condensates (BECs) in optical lattices have
attracted enormous attention both experimentally and theoretically ce1 ; ce2 .
This is mainly because the lattice parameters and interaction strength can be
manipulated using a modern experimental technique. Making use of this,
researchers have discovered many long-predicted phenomena, for example non-
linear Landau-Zener tunneling, energetic and dynamical instability and the
strongly inhibited transport of one-dimensional BEC in optical lattices ce3 ;
ce4 ; ce5 ; ce6 ; ce7 ; ce8 ; ce9 ; ce10 . More attracting phenomena, namely,
self-trapping, was recently observed experimentally in this system ce11 . In
such an experiment, a BEC cloud with repulsive interaction initially loaded in
optical lattices was self-trapped. Many theoretical analysis was also
presented about self-trapping ce12 ; ce13 ; ce14 ; ce15 . It is well know the
macroscopic quantum self-trapping (MQST) means self-maintained population
imbalance with non-zero average value of the fractional population imbalance
which was detailed discussed ce16 ; ce17 . Marino et. al. considered that the
damping decays all different oscillations to the zero-phase mode ce18 .
Besides, macroscopic quantum fluctuations have also been discussed by taking
advantage of second-quantization approaches ce19 . However, when the trapping
potential is time dependent and the damping and finite-temperature effect can
not be neglected, chaos emerges. Abdullaev and Kraenkel studied the nonlinear
resonances and chaotic oscillation of the fractional imbalance between two
coupled BEC’s in a double-well trap with a time-dependent tunneling amplitude
for different damping ce20 . When the asymmetry of the trap potential is time-
dependent and its amplitude is so small that can be took as a perturbation,
Lee et al. studied the chaotic and frequency-locked atomic population
oscillation between two coupled BECs with a weak damping, and discovered that
the system comes to an stationary frequency-locked atomic population
oscillations from transient chaos ce21 .

It is important to note that theoretical studies of stability are mainly
focused on the effect of two-body interactions. It is clear that in low
temperature and density, where interatomic distance is much greater than the
distance scale of atom-atom interactions, two-body s-wave scattering should be
important and three-body interactions can be neglected. But, if the atom
density is higher, for example, in the case of BEC in optical lattices, three-
body interactions will play an important role ce22 . As reported in Ref. ce23
, even for a small strength of the three-body force, the region of stability
for the condensate can be extended considerably.

Therefore, the purpose of this paper is to investigate the steady-state
solution of BEC in an one-dimensional periodic optical lattice when both the
two-body and three-body interactions are taken into account. By using the
mean-field approximation and linear stability theorem, one interesting result
is found that the tuning-fork bifurcation of steady-state relative population
appears when the system parameters are changed to certain critical values. The
existence of three-body interaction not only transforms the bifurcation point
of the system but also affects greatly on the self-trapping behaviors of the
system associated with the critically stable steady-state solutions.
Additionally, we also study the effects of the initial conditions, three-body
interaction and the energy bias on the MQST. Besides, we discuss the chaos
behaviors of the system by applying the periodic modulation on the energy
bias. The result shows the relative population oscillation can undergo a
process from order to chaos, via a series of period-doubling bifurcations.

This paper is organized as follows. In Sec. II, we introduce the mean-field
description of BEC in optical lattices with two- and three-atom interactions.
In Sec. III, with linear stability theorem, we analysis the stability of
steady-state solutions. Then the influences of three-body interaction on the
macroscopic quantum self-trapping of the system are displayed In Sec. IV. In
Sec. V, by applying the periodic modulation in the energy bias, we discuss
chaotic behaviors of the system using the numerical simulation method. In the
last section, summary and conclusion of our work are presented.

##  II Mean-field description of BEC in optical lattices with two- and three-
atom interactions

We focus our attention on a BEC with both two- and three-body interactions is
subjected to one dimensional (1D) optical lattices where the motion in the
perpendicular directions is confined. In the mean-field approximation , the
dynamics of BEC can be modeled by the 1D Gross-Pitaevskii (GP) equation in the
comoving frame of the lattice ce3 ; ce6 ; ce24 ; ce25 ,

$i\hbar\frac{\partial\Phi}{\partial
t}=-\frac{1}{2m}\left(\hbar\frac{\partial}{\partial
t}-ima_{l}t\right)^{2}\Phi+\upsilon_{0}\cos(2K_{l}x)\Phi+\frac{2\hbar^{2}a_{s}}{a_{\bot}^{2}m}|\Phi|^{2}\Phi+\frac{g_{2}}{3\pi^{2}a_{\bot}^{4}}|\Phi|^{4}\Phi,$
(1)

where $\Phi$ is the wave function of the condensate, $m$ is the mass of atoms,
$a_{s}$ is the two-body s-wave scattering length, $\upsilon_{0}$ is the
strength of the periodic potential, $K_{l}$ is the wave number of the laser
light which is used to generate the optical lattice, $ma_{l}$ stands for
either the inertial force in the comoving frame of an accelerating lattice or
the gravity force, $a_{\bot}=\sqrt{\hbar/(m\omega_{\bot})}$, where
$\omega_{\bot}$ is the radial frequencies of the anisotropic harmonic trap,
$g_{2}|\Phi|^{4}\Phi/(3\pi^{2}a_{\bot}^{4})$ is three-body interactions
related to the GP equation. Among Eq. (1), all the variables can be rescaled
to be dimensionless by the following system’s basic parameter $x\sim
2K_{l}x,\Phi\sim\frac{\Phi}{\sqrt{2K_{l}N}},t\sim\frac{4\hbar}{m}K_{l}^{2}t$.
we obtain the normalized 1D-GP equation in optical lattices with cubic and
quintic nonlinearities,

$i\frac{\partial\Phi}{\partial t}=-{{1}\over{2}}\left(\frac{\partial}{\partial
t}-i\alpha
t\right)^{2}\Phi+\upsilon\cos(x)\Phi+c|\Phi|^{2}\Phi+\lambda|\Phi|^{4}\Phi,$
(2)

where
$\upsilon=\frac{m\upsilon_{0}}{4\hbar^{2}K_{l}^{2}},\alpha=\frac{m^{2}}{8\hbar^{2}K_{l}^{3}}a_{l},c=\frac{Na_{s}}{K_{l}a\bot^{2}}$
is the effective two-body interaction, $N$ is the total numbers of atoms,
$\lambda=\frac{mg_{2}N^{2}}{3\pi^{2}\hbar^{2}a_{\bot}^{4}}$ is the effective
interaction among three atoms, here the three-body interaction is expected to
be positive with a value of $0<\lambda<1$.

In the neighborhood of the Brillouin Zone edge $k=1/2$, the wave function can
be approximated by ce3

$\Phi(x,t)=a(t)e^{ikx}+b(t)e^{i(k-1)x},$ (3)

where $a(t)$, $b(t)$ are the probability amplitudes of atoms in each of the
two wells respectively and $|a|^{2}+|b|^{2}=1$. By inserting such wave
functions into Eq. (2) and performing some spatial integrals, we obtain the
dynamical equations with two- and three-body interactions.

$\displaystyle i\frac{\partial a}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}a+\frac{c}{2}\left(|b|^{2}-|a|^{2}\right)a+\lambda\left(1+2|a|^{2}|b|^{2}+2|b|^{2}\right)a+\frac{\upsilon}{2}b,$
(4) $\displaystyle i\frac{\partial b}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\gamma}{2}b-\frac{c}{2}\left(|b|^{2}-|a|^{2}\right)b+\lambda\left(1+2|a|^{2}|b|^{2}+2|b|^{2}\right)b+\frac{\upsilon}{2}a.$
(5)

Here, the level bias $\gamma(t)=\alpha t$, and $\alpha$ is the sweeping rate,
$c$ and $\lambda$ represent the nonlinear parameters, $\upsilon$ is the
coupling constant between the two condensates. We introduce the relative
population variance

$s=|b|^{2}-|a|^{2},$ (6)

with the parameters $a=|a|e^{i}\theta_{a}$, $b=|b|e^{i}\theta_{b}$,

$\theta=\theta_{b}-\theta_{a}.$ (7)

Combining Eqs. (4-7), one yields the equations of the relative population and
relative phase,

$\displaystyle\dot{s}$ $\displaystyle=$
$\displaystyle-\upsilon\sqrt{1-s^{2}}\sin\theta,$ (8)
$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\gamma+(c+2\lambda)s+\frac{\upsilon
s}{\sqrt{1-s^{2}}}\cos\theta.$ (9)

$\dot{s}$ and $\dot{\theta}$ denote the time derivative of the relative
population and the relative phase. If we regard $s$ and $\theta$ as the
canonically conjugate variables Eqs. (8) and (9), become a pair of Hamilton’s
canonical equations with the conserved effective Hamiltonian

$H=\gamma s+\frac{1}{2}(c+2\lambda)s^{2}+\upsilon\sqrt{1-s^{2}}\cos\theta.$
(10)

In the following section, we will discuss the stability of steady-state in the
symmetric condition ($\gamma=0$) with linear stability theorem.

##  III Stability analysis of the steady-state solutions

In Sec. II, we have given the dynamical equations of the system with three-
body interaction. In this section, we will discuss the stability of steady-
state in the symmetric condition. Generally, there are two ways to study the
stability of nonlinear system, the linear stability theorem and the Lyapunov
direct method. We will investigate the stability of the system with the first
method.

The steady-state solution of this system can be obtained by setting Eqs. (8)
and (9) to zero. The forms of steady-state solutions are very complicated when
the level bias $\gamma\neq 0$. For simplicity, we set $\gamma=0$, leading to

$\displaystyle\dot{s}$ $\displaystyle=$ $\displaystyle
f_{1}(s,\theta)=-\upsilon\sqrt{1-s^{2}}\sin\theta,$ (11)
$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle
f_{2}(s,\theta)=(c+2\lambda)s+\frac{\upsilon s}{\sqrt{1-s^{2}}}\cos\theta.$
(12)

and the conserved energy

$H=\frac{1}{2}(c+2\lambda)s^{2}+\upsilon\sqrt{1-s^{2}}\cos\theta.$ (14)

Taking $\dot{s}=0$, $\dot{\theta}=0$, we get

$\displaystyle-\upsilon\sqrt{1-s^{2}}\sin\theta$ $\displaystyle=$
$\displaystyle 0,$ (15) $\displaystyle(c+2\lambda)s+\frac{\upsilon
s}{\sqrt{1-s^{2}}}\cos\theta$ $\displaystyle=$ $\displaystyle 0.$ (16)

The steady-state solutions obeyed Eqs. (14) and (15) regard as

$\displaystyle\theta_{1}$ $\displaystyle=$ $\displaystyle 2n\pi,\quad
s_{1}=0\quad\mathrm{for}\quad H=-\upsilon,$ (17) $\displaystyle\theta_{2}$
$\displaystyle=$ $\displaystyle(2n+1)\pi,\quad s_{2}=0\quad\mathrm{for}\quad
H=\upsilon,$ (18) $\theta_{3,4}=(2n+1)\pi,\quad
s_{3,4}=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}}\quad\mathrm{for}\quad
H=\frac{(c+2\lambda)^{2}+\upsilon^{2}}{2(c+2\lambda)^{2}}.$ (19)

According to the linear stability theorem, we look for the perturbed solutions
which are near the steady-state solutions,

$s(t)=s_{i}(t)+\varepsilon_{1}(t),\qquad\theta(t)=\theta_{i}(t)+\varepsilon_{2}(t)$
(20)

where $s_{i}(t)$, $\theta_{i}(t)$ for $i=1,2,3,4$ signify the steady-state
solutions, $|\varepsilon_{1}(t)|\ll|s_{i}(t)|$ and
$|\varepsilon_{2}(t)|\ll|\theta_{i}(t)|$ which is relate to the first-order
perturbed. Inserting the above expression into Eqs. (11) and (12), we can
obtain the linear equations near to the steady-states of the nonlinear
equations as

$\dot{\varepsilon_{1}}=\left(\frac{\partial f_{1}}{\partial
s}\right)_{1}\varepsilon_{1}+\left(\frac{\partial
f_{1}}{\partial\theta}\right)_{1}\varepsilon_{2}\qquad
namely\qquad\dot{\varepsilon_{1}}=a_{11}\varepsilon_{1}+a_{12}\varepsilon_{2}$
(21) $\dot{\varepsilon_{2}}=\left(\frac{\partial f_{2}}{\partial
s}\right)_{2}\varepsilon_{1}+\left(\frac{\partial
f_{2}}{\partial\theta}\right)_{2}\varepsilon_{2}\qquad
namely\qquad\dot{\varepsilon_{2}}=a_{21}\varepsilon_{1}+a_{22}\varepsilon_{2}$
(22)

Now, we make use of the above expression to investigate the stability of the
steady-states of Eqs. (16-18).

(1)For $\theta_{1}=2n\pi,s_{1}=0,H=-\upsilon$, we can calculate the matrix
elements $a_{11}=0$, $a_{12}=-\upsilon$, $a_{21}=(c+2\lambda)+\upsilon$,
$a_{22}=0$. So, the coefficient matrix of the linearized equations (20) and
(21) becomes $A_{1}=\left[\begin{array}[]{cc}{0}&{-\upsilon}\\\
{c+2\lambda+\upsilon}&{0}\\\ \end{array}\right]$ such that the characteristic
equation writes $\det(A_{1}-\lambda
I)=\left[\begin{array}[]{cc}{0-\lambda}&{-\upsilon}\\\
{c+2\lambda+\upsilon}&{0-\lambda}\\\ \end{array}\right]=0$, which reveals that
$\lambda^{2}+\upsilon(c+2\lambda+\upsilon)=0$. We solve the equation to get
the two eigenvalues of the matrix A as
$\lambda_{1}=\sqrt{-\upsilon(c+2\lambda+\upsilon)},\lambda_{2}=-\sqrt{-\upsilon(c+2\lambda+\upsilon)}$.
In response to the forms of the eigenvalues, there exist two cases for the
stabilities:

(a) $\upsilon(c+2\lambda+\upsilon)\geq 0$, that is

$\upsilon>0\quad and\quad(c+2\lambda)\geq-\upsilon$ (23) $\upsilon<0\quad
and\quad(c+2\lambda)\leq-\upsilon$ (24)

so the two eigenvalues are both pure imaginary numbers. Thus, the stability of
the steady-state solutions $(\theta_{1},s_{1})$ corresponds to a critical case
ce26  and the dynamical bifurcations between the unstable and stable steady-
states will appear when the parameters with two- and three-body interactions
are changed.

(b) $\upsilon(c+2\lambda+\upsilon)<0$, namely

$\upsilon>0\quad and\quad(c+2\lambda)<-\upsilon$ (25) $\upsilon<0\quad
and\quad(c+2\lambda)>-\upsilon$ (26)

so the two eigenvalues are real number. It means that $\varepsilon_{1}$ and
$\varepsilon_{2}$ tend to infinity with the increase of time, and the steady-
state solutions $(\theta_{1},s_{1})$ are unstable.

(2)For $\theta_{2}=(2n+1)\pi$, $s_{2}=0$, $H=\upsilon$, the matrix elements
write as $a_{11}=0,a_{12}=-\upsilon,a_{21}=(c+2\lambda)-\upsilon,a_{22}=0$.
The corresponding eigenvalues of the matrix $A_{2}$ become
$\lambda_{1}=\sqrt{-\upsilon(\upsilon-(c+2\lambda))},\lambda_{2}=-\sqrt{-\upsilon(\upsilon-(c+2\lambda))}$.
Similarly, there are two cases of the stabilities:

(a) $\upsilon(\upsilon-(c+2\lambda))>0$, that is

$\displaystyle(c+2\lambda)>0\quad$
$\displaystyle\mathrm{and}\quad\upsilon>(c+2\lambda)$ (27)
$\displaystyle(c+2\lambda)<0\quad$ $\displaystyle\mathrm{and}\quad\upsilon>0.$
(28)

so the two eigenvalues are both pure imaginary numbers. And the stability of
the steady-state solutions $(\theta_{2},s_{2})$ of the nonlinear equations are
reviewed as critical and the dynamical bifurcations will occur.

(b) $\upsilon(\upsilon-(c+2\lambda))\leq 0$, that is

$\displaystyle\upsilon>0\quad$ $\displaystyle\mathrm{and}$
$\displaystyle\quad(c+2\lambda)\geq\upsilon$ (29)
$\displaystyle(c+2\lambda)<\upsilon\quad$ $\displaystyle\mathrm{and}$
$\displaystyle\quad\upsilon<0$ (30)

so the two eigenvalues are positive or negative real number, respectively.
$\varepsilon_{1}$, $\varepsilon_{2}$ tend to infinity as increasing the time
to infinity, and the steady-state solutions $(\theta_{2},s_{2})$ are losing
their stability.

(3)For
$\theta_{3,4}=(2n+1)\pi,s_{3,4}=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}},H=\frac{(c+2\lambda)^{2}+\upsilon^{2}}{2(c+2\lambda)^{2}}$,
the matrix elements read $a_{11}=0$, $a_{12}=\upsilon^{2}/(c+2\lambda)$,
$a_{21}=(c+2\lambda)-(c+2\lambda)^{3}/\upsilon^{2}$, $a_{22}=0$, and the
eigenvalues $\lambda_{1}=\sqrt{\upsilon^{2}-(c+2\lambda)^{2})}$,
$\lambda_{2}=-\sqrt{\upsilon^{2}-(c+2\lambda)^{2})}$. In Eq. (18) the
population $s_{3,4}$ are both real quantities which implies

$(c+2\lambda)^{2}>\upsilon^{2}$ (31)

Figure 1: Plots of the tuning-fork bifurcation from Eqs. (17) and (18), where
$s_{2},s_{3},s_{4}$ are the steady-state solutions and the bifurcation point
is $\frac{\upsilon}{c+2\lambda}=1$

Therefore, the two eigenvalues are pure imaginary numbers. The stability of
the steady-state solutions$(\theta_{3,4},s_{3,4})$ of the nonlinear equations
are regarded as critical and the dynamical bifurcations will emerge at the
bifurcation point $(c+2\lambda)=\upsilon$, $s=0$. Obviously, the existence of
three-body interaction can change the bifurcation point of the system. It
plays a important role for stability analysis of the system, as shown in Fig.
1. For $\frac{\upsilon}{c+2\lambda}>1$, the system is in the critically stable
steady-state ($\theta_{2},s_{2}$), and for $\frac{\upsilon}{c+2\lambda}<1$,
($\theta_{2},s_{2}$) is unstable and the two steady-state solutions
($\theta_{3,4},s_{3,4}$) are critically stable. This is a typical tuning-fork
bifurcation, and the bifurcation point is $\frac{\upsilon}{c+2\lambda}=1$

According to the above analysis, we conclude that three steady-state solutions
possess different stability for different parameter regions. And it is very
interesting to arrive at the critically stable steady-state solution in
experiment which relate to the stable stationary MQST ce26 . In the following
section, we will illustrate the MQST of the non-stationary states in detail by
two different methods.

##  IV The macroscopic quantum self-trapping of BEC with two- and three-atom
interactions

In this section, we investigate the macroscopic quantum self-trapping by
plotting the phase trajectories and the time evolution of the relative
population of the system.

###  IV.1 The phase trajectories diagram

The macroscopic quantum self-trapping refers to the phase space trajectories
whose the relative population is not equal zero. This can be well understood
from the analysis Eqs. (8)-(10), corresponding to the critically stable
steady-state solutions discussed in sec.II. Three kinds of cases occur with
different three-body interaction parameters, as shown in Fig.2.

(1) In the case of $\upsilon=0.2,c=0.1,0<\lambda<0.05$ in the phase space ,
there are two stable points $P_{1},P_{2}$ at $s=0,\theta=\pi$ and
$s=0,\theta=0$ respectively [Fig. 2(a)], from the circumstance described by
Eqs. (22) and (26). Obviously, for the stable points $P_{1}$, $P_{2}$, the
atoms distributions are equal in the two adjacent wells, the relative
population of the trajectories around them is equal to $0$. It means that
atoms oscillate between two adjacent wells and the macroscopic quantum self-
trapping phenomenon does not emerge in this case.

(2) When parameter is set to $\upsilon=0.2$, $c=0.1$, $0.05\leq\lambda<0.15$,
two more fixed points emerge in the line $\theta=\pi$ marked by $P_{3}$,
$P_{4}$. Among them, $P_{1}$, $P_{3}$ are steady which is corresponding to
condition of Eq. (30). They are located in
$s=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}}$, hence, $P_{4}$ is unstable
point which lies in $s=0$ and corresponds to condition of Eq.(26). As seen
from Fig. 2(b), for the stable points $P_{1},P_{3}$, the atoms distributions
are not equilibrium between two adjacent wells, and the relative population of
the trajectories around them is not equal to $0$. It indicates that atoms are
self-trapped in one well. We take it as oscillating-phase-type because the
relative population $s$ and the relative phase $\theta$ oscillate around the
fixed points.

Figure 2: Trajectories on the phase space of the system with three-body
interaction varying from $\lambda=0$ to $\lambda=0.25$(the first row).
Corresponding to in the second row we plot the energy profiles for the
relative phase $\theta=0$ (red dashed) and $\theta=\pi$ (blue solid)

(3) For $\upsilon=0.2,c=0.1$, $\lambda\geq 0.15$ , It emerges new trajectories
, i.e.the trajectories across point $P_{c}$ [Fig. 2(c)]. Only the fixed point
$P_{2}$ is stable which is relate to Eq. (22). So for these trajectories, $s$
varies with time from region of $[-1,0]$ to $[0,1]$, Apparently $\langle
s\rangle\neq 0$, atoms are self-trapped in one well. We regard it as running-
phase-type macroscopic quantum self-trapping, as described in Refs. ce27 ;
ce28  and observed in experiment ce29 .

The above changes on the topological structure of the phase space are
concerned with the change of the energy profile. When the relative phase is
zero or $\pi$, energy relying on the parameter with three-body interaction and
the average population $s$ can be derived from Eq. (10). Seeing Fig. 2 , the
transition from case(1)to case(2) corresponds to the bifurcation of the energy
profile of $\theta=\pi$: energy curve bifurcates from a single minimum to the
curve of two minima. It means the system goes from the Rabi regime into the
self-trapping regime through this bifurcation. The lowest order of energy
profile with $\theta=0$ is $-\frac{c+2\lambda}{2}$, and the energy of the
unstable point $P_{4}$ is $-\upsilon$ which is located on the maximal order of
energy profile with $\theta=\pi$. The results displayed by the phase space
trajectories conform to the case of steady-state solutions discussed in
Sec.III. The transition from case (2) to case(3) is signified by the overlap
of the two energy regions of the profile. In this condition the trajectory
stared from $s=-1$, $\theta=0$ should be confined to the lower half of phase
plane, corresponding to the running-phase-type macroscopic quantum self-
trapping.

Connecting the analysis of the steady-state solutions to the above analysis on
the energy profile, it concludes that stable behaviors of the system change
constantly with the increase of $\lambda$ and we obtain a general criterion
for the macroscopic quantum self-trapping trajectories, namely,
$H(s,\theta)<-\upsilon$. It plays a critical role to find the transition
parameters of macroscopic quantum self-tapping.

###  IV.2 Numerical simulations of the MQST

Now, we focus on the dynamic behavior which dominated by Eq. (8) and (9)
without the time-dependent system parameters. We study the effect parameters
of the system on the MQST with numerical method starting form Eq. (8) and (9).

Figure 3:  The time evolution of the relative population from Eqs.(8) and (9)
with initial conditions $s(0)=0$, $\theta_{0}=\pi/2$ and parameter: (a)
$c=0.1$, $\lambda=0.45$, $v=0.2$, and $\gamma=0$; (b) $c=0.1$, $\lambda=0.95$,
$v=0.2$, and $\gamma=0$; (c) $c=0.1$, $\lambda=0.45$, $v=0.8$, and $\gamma=0$;
(d) $c=0.1$, $\lambda=0.45$, $v=0.2$, and $\gamma=0.5$; (e) $c=0.1$,
$\lambda=0.95$, $v=0.2$, and $\gamma=0.5$; (f) $c=0.1$, $\lambda=0.45$,
$v=0.8$, and $\gamma=0.5$;

Figure 4: the time evolution of the relative population from Eqs. (8) and (9).
(a) initial conditions $s(0)=0.8$, $\theta_{0}=\pi$ (b) initial conditions
$s(0)=0.8$, $\theta_{0}=\pi/2$, and the other parameters $c=0.1$,
$\lambda=0.45$, $v=0.2$, and $\gamma=0$.

Choosing initial condition $s(0)=0$, $\theta(0)=\pi/2$, the time evolutions of
the relative population Fig. (3a)-(3d) show some very absorbing features. In
Fig. 3(a), the oscillations are regular and the average the relative
population $\bar{s}$ is zero for symmetric well case ($\gamma=0$) with a
special parameter, but the corresponding MQST does not appear. If we increase
$\lambda$ from 0.45 to 0.95 in Fig. 3(b), the MQST does not still appear, but
the oscillating period becomes short. Similarly, rising $\upsilon$ , we obtain
the same result as shown in Fig. 3(c).

Here, we study impacting asymmetric well case ($\gamma\neq 0$) on the MQST.
when we enhance the level bias to $\gamma=0.5$ the average the relative
population is changed to $-0.41$ in Fig.3(d). Correspondingly, the oscillating
period of $s$ is longer and the MQST emerges. Note that parameter $c$,
$\lambda$ and $\upsilon$ impact greatly on the MQST which are plotted in Fig.
3(e) and (f). In fig. 3(e), when $\lambda$ is from $0.45$ to $0.95$, the MQST
is suppressed with shorter oscillating period. Similarly, with increasing
$\upsilon$, the average relative population are changed to $-0.21$ and the
oscillating period becomes shorter again, as seen in Fig. 3(f). Thus, the
influence of parameter $c$ ,$\lambda$ ,$\upsilon$ and $\gamma$ on the MQST of
the system is very dramatic. In the case of $\gamma=0$, fixing the other
parameters and changing the initial condition from $s(0)=0,\theta(0)=\pi/2$ of
Fig.3 to $s(0)=0.8,\theta(0)=\frac{\pi}{2}$ and $s(0)=0.8,\theta(0)=\pi$, we
observe that the MQST always emerges with varying $s(0),\theta(0)$. The
oscillating period is decreased comparing to Fig.3(a)and Fig. 3(d), but the
$\bar{s}$ is increased to $-0.86,-0.72$ as shown in Fig. 4.

According to the above analysis, we can draw conclusion that when the initial
conditions $s(0)=0$, $\theta(0)=\pi/2$ are read, the parameter $c$, $\lambda$,
$\upsilon$ can impact on the MQST for asymmetric well case($\gamma\neq 0$). In
addition, in the symmetric case, the MQST does not appear and those parameters
only affect the oscillating period of the system. Besides, the initial
conditions can impact the MQST for anyone parameter set.

##  V Numerical simulation of chaos by applying periodic modulation on the
lever bias

As a whole, the elementary features of chaos is that the dynamic behaviors are
unpredictable for a deterministic system. It is very sensitive for the initial
conditions and parameters of the system. So, according to these
characteristics, we can adjust the parameters to make the system get into or
get out of the chaos, in other words, we can control the regime appearing
chaos. In this section We discuss the chaotic behaviors of the system by
numerical method.

Figure 5: Dynamical phase orbits of the dimensionless variables ($s,ds/dt$)
from Eqs. (31) and (32) with parameters $\upsilon=0.001$, $c=0.1$,
$\lambda=0.45$, $\omega=0.1$, $s(0)=0$, $\theta_{0}=\pi$, and (a)
$A_{1}=0.002$, (b) $A_{1}=0.009$, (c) $A_{1}=0.04$, (d) $A_{1}=0.12$, (e)
$A_{1}=0.3$, (f)=$A_{1}=1$. Here, $A_{1}$ denotes the amplitude of the time-
dependent relative energy.

If we apply periodic modulation on the lever bias
$\gamma=A_{0}+A_{1}sin(\omega t)$, the chaos will appear in a special region,
where $A_{0},A_{1}$ stand for initial phase and amplitude respectively.
Inserting this into Eqs. (8)and (9), one derives the below dynamic equation.

$\displaystyle\dot{s}$ $\displaystyle=$
$\displaystyle-\upsilon\sqrt{1-s^{2}}\sin\theta$ (32)
$\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle
A_{0}+A_{1}\sin(\omega t)+(c+2\lambda)s+\frac{\upsilon
s}{\sqrt{1-s^{2}}}\cos\theta$ (33)

Figure 6: (a) and (b): The time evolution of the relative population of the
relative population from Eqs. (31)and (32) with the parameters
$\upsilon=0.001$, $A_{0}=0.4$, $c=0.1$, $\lambda=0.45$, $\omega=0.1$,
$s(0)=0$, $\theta(0)=\pi$, and (a) $A_{1}=0.002$, (b) $A_{1}=0.3$ (c) and (d):
The corresponding power spectrum, where the parameters in Fig. 6(c)are the
same with Fig. 6(a) and the parameters in Fig. 6(d)are the same with Fig.
6(b).

Starting from Eqs. (32), It is found that the dynamics behavior of the system
is periodic in some special parameters region and it will vary from order to
chaos with the increase of $A_{1}$ , as shown in Fig.5. With initial
conditions $s(0)=0,\theta(0)=\pi$, the phase orbit is a period-one cycle and
the corresponding oscillation is a Rabi oscillation for the set of parameters
with amplitude $A_{1}=0.002$, as in Fig. 5(a). In this case, we set the
oscillating period of the relative population $T$. When $A_{1}=0.009$, the
phase orbit becomes period-two in Fig. 5(b). It means the oscillating period
of $s$ arriving at $2T$. Then the phase orbit increases from that of period-
four to period-eight with increasing $A_{1}$ as shown in Fig. 5(c)and (d).
Fig. 5(e) and 5(f) are plotted for $A_{1}=0.3$ and $A_{1}=1$, where the phase
orbit does not show a clear periodicity which signifies the emergence of
chaos.

From the above analysis, we find that the oscillating period of the relative
population varies from a period-one limit-cycle to period-two to period-four
and then to period-eight and finally all limit-cycles tend to infinity with
$\gamma$ increasing. It exhibits a process from order to chaos, through the
period-doubling bifurcations ce26 . That is to say, for a set of fixed
parameter $\upsilon$, $c$, $\lambda$, $A_{0}$, $A_{1}$, $s(0)$, $\theta(0)$
and $\omega$, the first-order derivative of relative population transform from
the single period to multiple period and get into chaos at last with the
increase of vibration amplitude $A_{1}$.

For the aim of showing the chaotic MQST, we present the plots of the time
evolution of the relative population and corresponding plots of power spectra
from Eqs. (31) and (32) in Fig. 6. And the parameter of Fig.5(a) is accord
with Fig. 6(a) and 6(c) where the system oscillates periodically. Making use
of those parameters of Fig. 5(e), we plot Fig. 6(b)and 6(d). It shows that the
power spectrum appears confusion and the average value of the relative
population is less than zero, which implies the existence of the chaotic
behaviors .

##  VI Summary and conclusion

In this paper, we study the stability and chaos of BEC with repulsive two- and
three-body interactions immersed in a one-dimensional optical lattice. The
stability of the steady-state solution are analyzed with the linear stability
theorem. The analytical results show: (1) For $\upsilon>0$ and
$c+2\lambda\geq-\upsilon$ or $\upsilon<0$ and $c+2\lambda\leq-\upsilon$, the
stability of the steady-state solution($\theta_{1}=2n\pi,s_{1}=0$) is in the
critical case. (2) For $c+2\lambda>0$ and $\upsilon>c+2\lambda$ or
$c+2\lambda<0$ and $\upsilon>0$, the steady-state
solution($\theta_{2}=(2n+1)\pi,s_{2}=0$) is the critical stability. (3) For
$(c+2\lambda)_{2}>\upsilon_{2}$, the steady-state solution
($\theta_{3,4}=(2n+1)\pi,S_{3,4}=\pm\sqrt{1-(\frac{\upsilon}{c+2\lambda})^{2}}$)
is also critically stable. When these relationship are not satisfied, the
corresponding steady-state solution are unstable. A typical tuning-fork
bifurcation of steady-state relative population appears in special parameter
region. And the existence of three-body interaction can change the bifurcation
point of the system, which is shown as Fig. 1. It plays a important role for
stability analysis of the system.

The critically stable steady-state solution indicates the existence of the
stationary MSQT. The stable behaviors of the system change constantly with the
increase of $\lambda$ and get a general criteria for the self-trapping
trajectories, $H<-\upsilon$. In addition, we also investigate the effects of
the initial conditions, a set of parameters $c,\upsilon,\lambda,\gamma$ on
MQST. It shows that $c,\upsilon,\lambda$ could affect on the MQST when
$s(0)=0,\theta_{0}=\pi$ for $\gamma\neq 0$. Particularly, the initial value
$s(0)=0,\theta_{0}=\pi$ or $s(0)=0,\theta_{0}=\pi/2$ can directly impact on
the MQST. Finally, we discuss the chaos behaviors by applying the modulation
on the energy bias ($\gamma=A_{0}+A_{1}sin\omega t$). In this case, the system
will go into chaos through the period-doubling bifurcations with the
increasing of $\lambda$, and the time evolution of the relative population and
power spectra indicate the existence of the chaos MQST. It suggests that one
can adjust the lasing detuning and intensity to change the values of the
parameters in experiments. This adjustable parameters supply the possibility
for controlling the instabilities of the system, MQST state and the chaotic
behaviors.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
and by the Open Project of Key Laboratory for Magnetism and Magnetic Materials
of the Ministry of Education, Lanzhou University.

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# On lower estimations of square-linear ratio for plane Peano curves.

E. V. Shchepin , E. Yu. Mychka  Steklov Math. Institute, scepin@mi-
ras.ruLomonosov Moscow State University<EMAIL_ADDRESS>
(February, 2021)

###### Abstract

It is proved that for any mapping of a unit segment to a unit square, there is
a pair of points of the segment for which the square of the Euclidean distance
between their images exceeds the distance between them on the segment by at
least $3\frac{5}{8}$ times. And the additional condition that the images of
the beginning and end of the segment belong to opposite sides of the square
increases the estimate to $4+\varepsilon$.

In the article [1] the problem of finding the smallest number of $\kappa$ for
which there exists a mapping $p(t)$ of a unit segment to a unit square such
that for any pair of distinct points the inequality is satisfied for the
segment

$\frac{|p(t_{1})-p(t_{2})|^{2}}{t_{1}-t_{2}}\leq\kappa$ (1)

The numerator of the fraction in this inequality is the square of the
Euclidean distance between the points $p(t_{i})$, and the fraction itself is
called the square-linear ratio of a pair of points on the curve. In the same
paper, it was proved that $\kappa\geq 3.5$. Essentially the same result was
somewhat earlier obtained in the paper [3] in discrete form, that is,
concerned with the linear ordering of two-dimensional lattices. In the last
paper, also in discrete form, it is essentially proved that $\kappa\leq 4$. In
continuous form, the proof of the latter inequality can be found, for example,
in the paper [4].

The square-linear ratio (abbreviated SLR) is defined for a plane curve $p(t)$
as the supremum of the square-linear ratios of all possible pairs of its
points and is denoted by $\kappa(p)$. The main result of this paper is to
prove the inequality $\kappa(p)\geq 3\frac{5}{8}$ for any curve $p(t)$ that
maps a segment to a square, whose area is numerically equal to the length of
the segment. In addition, we also consider the question of the lower SLR
estimate for maps with fixed origin and the end. It is proved that if the ends
of the curve described above also belong to opposite sides of the square, then
her SLR will be strictly greater than 4. This allows us to improve the result
from [2] (Theorem 4), also giving a lower bound of 4 for fractal maps, but in
the form of a non-strict inequality.

######  Lemma 1.

Let $f\colon[a,b]\to\mathbb{R}^{2}$ be a continuous mapping of the segment to
the plane. And let us give the sequence $t_{0}<t_{1}<\dots<t_{n-1}<t_{n}$ of
the points of the segment. Then the inequality is satisfied

$\max_{i\leq
n}\frac{|f(t_{i})-f(t_{i-1})|^{2}}{t_{i}-t_{i-1}}\geq\frac{\sum\limits_{i=1}^{n}|f(t_{i})-f(t_{i-1})|^{2}}{t_{n}-t_{0}},$
(2)

which turns into equality only if, for any $i$, the equality is satisfied

$\max_{i\leq
n}\frac{|f(t_{i})-f(t_{i-1})|^{2}}{t_{i}-t_{i-1}}=\frac{|f(t_{i})-f(t_{i-1})|^{2}}{t_{i}-t_{i-1}}$
(3)

###### Proof.

Left to the reader. ∎

######  Lemma 2.

Let $f\colon[a,b]\to\mathbb{R}^{2}$ is a continuous mapping of the line
segment in the plane for which the maximum square-linear ratio is achieved on
a pair of end points $p(a)p(b)$. Then the image $f(I)$ is contained in the
circle, the diameter of which is a pair of images of the endpoints of the
interval, and for every $t$, with the image of $p(t)$ that belong to the
boundary of a circle, a square-linear ratios of pairs $p(a),p(t)$ and
$p(t),p(b)$ be the maximum.

###### Proof.

Indeed, for any $t\in(a,b)$, by virtue of the lemma 1, there is an inequality

$|f(t)-f(a)|^{2}+|f(b)-f(t)|^{2}\leq(b-a)\max\left\\{\frac{|f(t)-f(a)|^{2}}{t-a},\frac{|f(b)-f(t)|^{2}}{b-t}\right\\}\leq|f(b)-f(a)|^{2},$
(4)

that shows that $f(t)$ lies in the specified circle. If instead of the
inequality we have equality, then by the same Lemma, we conclude maximal
condition on the ratios $\frac{|f(t)-f(a)|^{2}}{t-a}$
$\frac{|f(b)-f(t)|^{2}}{b-t}$. ∎

######  Lemma 3.

Let $f\colon I\to C$ be a continuous map of a segment on a convex two-
dimensional plane set that is centrally symmetric with respect to zero. Then
there is a point $x$ belonging to the boundary of the set $C$, such that the
convex hull of its preimage $f^{-1}(x)$ contains some of the preimages of its
antipode $f^{-1}(-x)$.

figure 1CBDAEF

###### Proof.

Assuming the opposite, we come to the conclusion that the boundary points are
divided into two types. The first — in which all the points of the preimage
precede the points of the preimage of the antipode, and the second — in which
follow them. But it is quite clear that the set of points of the first type is
open. The same is true for the second type. But then these sets, due to the
connectivity of the segment $I$, must intersect. This gives the desired
contradiction. ∎

For the curve $f\colon I\to I^{2}$ and the point of the image $V$, we denote
by $V_{f}$ the first moment of the curve passing through $V$, and by $V^{f}$ —
the last such moment. Thus we have the following relations:

  * •

$V_{f}\leq V^{f}$

  * •

$f(V_{f})=V=f(V^{f})$

  * •

$f^{-1}(V)\subseteq[V_{f},V^{f}]$

######  Lemma 4.

If unclosed curve $f\colon[0,1]\to[ABCD]$ maps the unit interval to the unit
square with vertices $ABCD$, so that $A_{f}<B_{f}<C_{f}<D_{f}$, and has a
square-linear ratio $\kappa(f)\leq 4$, then one has

  1. 1.

$|AB|=|BC|=|CD|=1$

  2. 2.

$A^{f}<B_{f}<B^{f}<C_{f}<C^{f}<D_{f}$

  3. 3.

there is a point $E$ on the side $AD$, such that $E_{f}<B_{f}$ and
$E^{f}>C^{f}$.

###### Proof.

We prove the first statement. If we assume that $AB$ is the diagonal of the
square, then so is $CD$. In this case, the sum of the squares of the links of
the chain $ABCD$ is $2+1+2$, which leads to a violation of the condition
$\kappa(f)\leq 4$. Similarly, it is shown that the diagonal cannot be $CD$.
Suppose now that the diagonal is $BC$. In this case, the sum of the squares of
the lengths of the $ABCD$ chain is 4. And the Lemma 1 allows us to state that
the square-linear ratios of the pairs $A_{f},B_{f})$, $B_{f},C_{f}$,
$C_{f},D_{f}$ are all maximal and equal to four, and in addition $A_{f}=0$,
$D_{f}=1$. According to the Lemma 2, the image of the segment $[C_{f},1]$ is
contained in a semicircle with $CD$ as a diameter. Since this semicircle does
not contain any neighborhood of the vertex $D$ in the square $ABCD$, then any
neighborhood of this vertex contains points belonging to the image of the
segment $[0,C_{f}]$. And the closeness of this image allows us to say that in
this case, it contains the point $D$ itself, which contradicts the choice of
$D_{f}$ as the smallest element of the preimage $f^{-1}(D)$. Thus, the
validity of the first statement of the lemma is established.

To proof the second statement suppose the opposite. Suppose, for example, that
$D_{f}>A^{f}>B_{f}$. In this case, we get increasing chains
$A_{f}<B_{f}<A^{f}<C_{f}<D_{f}$ or $A_{f}<B_{f}<C_{f}<A^{f}<D_{f}$, with the
sum of the squares of the distances between their images equal to five. If we
assume that $D_{f}<A^{f}$, then we get $A^{f}=1$ and $A_{f}=0$, that is, that
the curve $f$ is closed, contrary to the conditions of the lemma. If we assume
that $B^{f}>C_{f}$, we get the chains $A_{f}<B_{f}<C_{f}<B^{f}<D_{f}$ or
$A_{f}<B_{f}<C_{f}<D_{f}<B^{f}$ with the corresponding sum of the squares of
the distances equal to five. And the inequality, $C^{f}<D_{f}$ it is justified
in exactly the same way as the inequality $A^{f}<B_{f}$.

Let us proceed to the proof of the third statement of the lemma. By virtue of
the lemma 3, there are points $E$ and $F$ that belong to the boundary of the
square and are centrally symmetric, that $E_{f}<F_{f}<E^{f}$. Suppose that $E$
lies on the side of $AB$. If we assume that $E_{f}>B_{f}$, then for
$E^{f}<C_{f}$ we have a chain $ABEFECD$ on the curve with the sum of squares
greater than four, and for $E^{f}>C_{f}$, we have chains $ABCED$ or $ABCDE$,
both with the sum of squares greater than four. Hence, if the square-linear
ratio of the curve does not exceed four, then $E_{f}$ must lie between $A_{f}$
and $B_{f}$. But in this case, $F_{f}$ no longer lies in this interval,
because otherwise the chain $AFBCD$ would give the sum of squares more than
four. If we assume that $E^{f}<C_{f}$, then the sum of squares is greater than
four gives the chain $ABFECD$. All possible chains of $AEB(FC)(ED)$, obtained
by rearranging the letters inside the brackets in places, give the sum of
squares greater than four. Thus, in all cases, the assumption that $E$ lies on
$AB$ leads us to find a chain with a sum of squares of more than four. To the
same conclusion, even shorter, we are led by the assumption that $E$ belongs
to $BC$ or $CD$. So the only remaining possibility for $E$ — is on the
$AD$side.

We will show that $E_{f}<B_{f}$. The opposite assumption gives a chain
$A_{f},B_{f},E_{f},F_{f},E^{f}$ with the sum of the squares of the distances
between images greater than four. And the inequality $E^{f}>C^{f}$ is
justified by the appearance of the chain $E_{f},B_{f},E^{f},C^{f},D_{f}$
otherwise. Thus, $E$ is the desired point of the curve. ∎

######  Theorem 1.

If mapping a unit segment to a unit square translates the ends of the segment
to opposite sides, then there is a pair of points whose square-linear ratio is
greater than four.

figure 2BCADO

###### Proof.

Suppose, on the contrary, that the curve $p(t)$ satisfies the conditions of
the lemma and has a SLR of at most four. In this case, based on the lemma 2,
we denote the vertices of the square of the image in the order of their
passage through the curve with the letters $ABCD$, and we have
$AB=BC=CD=AD=1$. In addition, it is clear that the beginning of the curve is
on the side of $AB$, and the end — on $CD$. According to the Lemma 2, $AD$ has
a point $O$ traversed twice. First time before $B$, second time — after $C$.
If we denote by $S$ the starting point of the curve, and by $E$ — the end
point, then depending on the order of passage points $O$ we get four options
for passing the seven listed points SOABCDOE, SAOBCDOE, SOABCODE, SAOBCODE

Since $SO^{2}+OA^{2}+AB^{2}=SA^{2}+AO^{2}+OB^{2}$ and
$CO^{2}+OD^{2}+DE^{2}=CD^{2}+DO^{2}+OE^{2}$, so far in all embodiments, the
sum of squares lengths of chain to be expressed in the following way

$SA^{2}+2x^{2}+1+1+1+2(1-x)^{2}+ED^{2},$ (5)

which takes its minimum value of 4 only when

$S=A,E=D,x=\frac{1}{2}$ (6)

In this case, the square-linear ratio for all links of the chain is the
maximum possible and is equal to 4 by virtue of the lemma 1. So the (6)
conditions are met anyway.

We now prove that the vertices A and D are traversed twice, namely, the point
A must be traversed again after O, and the point $D$ — before the point $O$,
for all the described traversal options.

Consider the $AOB$ movement option. Since the square-linear ratio of the pair
$AO$ is maximal, then according to the Lemma 2, the image of the segment
$[A_{p},O_{p}]$ is contained in a circle with a diameter of AO. Since this
circle does not contain any neighborhood of the vertex A in the square ABCD,
the points arbitrarily close to A belong to the image $p[O_{p},B_{p}]$. And
the closeness of this image implies the existence of such a
$\alpha\in[O_{p},B_{p}]$ that $p(\alpha)=A$. Similarly, we justify the
existence of $\delta\in[C_{p},O^{p}]$, for which $p(\delta)=D$.

By virtue of the lemma 2 on pairs of curve points corresponding to the
arguments $O_{p},A_{p}$, $A_{p},B_{p}$, $O^{p},D_{p}$ and $C_{p},D_{p}$ the
square-linear ratio is maximal, i.e. equal to four. Since

$[O_{p},B_{p}]=[O_{p},A_{p}]]\cup[A_{p},B_{p}],$

in so far as, by virtue of the lemma 2, we obtain that the image of the
segment $[O_{p},B_{p}]$ lies in the union of two circles, the first of which
has a diameter of the side the square $AB$, and the second — segment $AO$. The
last circle also contains the image $[A_{p},O_{p}]$. Similar reasoning shows
that the segment of the curve $[C_{p},D_{p}]$ is contained in the union of two
circles: the first has a diameter of the side $CD$, and the second — the
segment $OD$. And as cut curve $[B_{p},C_{p}]$ is contained in the circle with
a diameter of $CD$, we get that the whole curve must lie in the Union of five
circles with centres on the sides of the square ABCD: the three circles having
the diameters of the side $AB$, $BC$ and $CD$, and two with diameters $AO$ and
$OD$. But the totality of these the circle does not cover the entire square.
For example, no neighborhood of the point $O$ is covered. Thus, we have
obtained a contradiction that proves the theorem. ∎

Similar arguments to the above allow us to briefly prove that the SLR of any
mapping of a unit segment to a unit square is strictly greater than $3.5$. The
following theorem allows us to refine this result.

######  Theorem 2.

If the continuous map $f\colon I\to I^{2}$ maps a unit segment to a unit
square, then there is a pair of points $x,y\in I$ such that

$|f(x)-f(y)|^{2}>3\frac{5}{8}|x-y|.$

figure 3CBDAEFG

###### Proof.

By virtue of the lemma2, in the future we will assume that $E$ belongs to the
side of $AD$. Let the point $E$ be shifted from the center of the side of the
square to the left by $x$. The described situation is shown in the figure.

If we assume that $E_{f}>A_{f}$, then there are four variants of the chains
$A(BE)C(DE)$. For each of them, the sum $\sigma$ of squares of the lengths of
the chain links is estimated from below as:

$ABECDE:$ $\sigma\geq 4$;

$ABECED:$ $\sigma\geq 4$;

$AEBCDE:$ $\sigma=3x^{2}-x+3\frac{3}{4}\geq 3\frac{2}{3}$;

$AEBCED:$ $\sigma\geq 4$.

So we are left to consider the case where $E_{f}<A_{f}$. In the same way, the
analysis of cases with a possible location of the moment $E^{f}$ leaves only
the case $E^{f}>D_{f}$ to consider.

The sets $f[A_{f},D_{f}]$ and $f[0,A_{f}]\cup f[D_{f},1]$ are closed and cover
the entire square and, in particular, cover the segment
$\\{0.5\\}\times[0,1]$. The first of them does not contain the middle of the
bottom side of the square, and the mid — upper. Otherwise, by adding this
midpoint in the chain $EABCDE$ between the corresponding letters, we will get
an increase in the sum of the squares of the chain by at least $\frac{1}{2}$,
that is, up to four. Therefore, there is a point $G$ on this segment that
belongs to the intersection of these sets. Let the point $G$ be located at the
distance $y$ from the lower side of the square.

The point $G$ belongs to $f[A_{f},D_{f}]\cap f[0,A_{f}]$ or
$f[A_{f},D_{f}]\cap f[D_{f},1]$. Let’s analyze the case when $G$ belongs to
$f[A_{f},D_{f}]\cap f[0,A_{f}]$. The other case is treated similarly.

There are two possible cases: $G_{f}<E_{f}$ and $G_{f}>E_{f}$.

1\. Consider the case of $G_{f}<E_{f}$, which in turn is divided into three
sub-cases, depending on which vertices of the square $G$ was traversed a
second time.

1a. Let’s analyze the sub-case when $G^{f}$ is between $A_{f}$ and $B_{f}$.
Let’s estimate the sum of squares of the $GEAGBCDE$ chain from below. The sum
$\sigma$ is numerically expressed in the following quadratic form:

$(x^{2}+y^{2})+(1/2-x)^{2}+(1/4+y^{2})+(1/4+(1-y)^{2})+2+(1/2+x)^{2},$

whose minimum is reached at $x=0$ and $y=\frac{1}{3}$ and is equal to
$3\frac{2}{3}$.

1b. Let’s analyze the sub-case when $G^{f}$ is between $B_{f}$ and $C_{f}$.
Let’s estimate the sum of squares of the chain $GEABGCDE$ from below. The sum
$\sigma$ is numerically expressed in the following quadratic form:

$(x^{2}+y^{2})+(1/2-x)^{2}+1+(1/4+(1-y)^{2})+(1/4+(1-y)^{2})+1+(1/2+x)^{2},$

whose minimum is reached at $x=0$ and $y=\frac{2}{3}$ and is equal to
$3\frac{2}{3}$.

1b. Let’s analyze the sub-case when $G^{f}$ is between $C_{f}$ and $D_{f}$.
Let’s estimate the sum of squares of the chain $GEABCGDE$ from below. The sum
$\sigma$ is numerically expressed in the following quadratic form:

$(x^{2}+y^{2})+(1/2-x)^{2}+2+(1/4+(1-y)^{2})+(1/4+y^{2})+(1/2+x)^{2},$

the minimum of which is reached at $x=0$ and $y=\frac{1}{3}$ and is equal to
$3\frac{2}{3}$.

2\. Consider the case $G_{f}>E_{f}$, which in turn is divided into three sub-
cases, depending on which vertices of the square $G$ was traversed a second
time.

2a. Let’s analyze the sub-case when $G^{f}$ is between $A_{f}$ and $B_{f}$.
Let’s estimate the sum of squares of the chain $EGAGBCDE$ from below. The sum
$\sigma$ is numerically expressed in the following quadratic form:

$(x^{2}+y^{2})+2(1/4+y^{2})+(1/4+(1-y)^{2})+2+(1/2+x)^{2},$

whose minimum is reached at $x=-\frac{1}{4}$ and $y=\frac{1}{4}$ and is equal
to $3\frac{5}{8}$.

2b. Let’s analyze the sub-case when $G^{f}$ is between $B_{f}$ and $C_{f}$.
Let’s estimate the sum of squares of the chain $EGABGCDE$ from below. The sum
of $\sigma$ is numerically expressed in the following quadratic form:

$(x^{2}+y^{2})+(1/4+y^{2})+1+2(1/4+(1-y)^{2})+1+(1/2+x)^{2},$

whose minimum is reached at $x=-\frac{1}{4}$ and $y=\frac{1}{2}$ and is equal
to $3\frac{7}{8}$.

2b. Let’s analyze the sub-case when $G^{f}$ is between $C_{f}$ and $D_{f}$.
Let’s estimate the sum of squares of the chain $EGABCGDE$ from below. The sum
$\sigma$ is numerically expressed in the following quadratic form:

$(x^{2}+y^{2})+(1/4+y^{2})+2+(1/4+(1-y)^{2})+(1/4+y^{2})+(1/2+x)^{2},$

the minimum of which is reached at $x=-\frac{1}{4}$ and $y=\frac{1}{4}$ and is
equal to $3\frac{5}{8}$.

∎

## References

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  * [3] Niedermeier R., Reinhardt K., Sanders P., Towards optimal locality in mesh-indexings, Discrete Applied Mathematics 117 (2002) 211237.
  * [4] E. V. Shchepin, On the SierpinskiKnopp curve, Russian Math. Surveys, 75:2 (2020), 377379