# Dissipative structures in a parametrically driven dissipative lattice:
chimera, localized disorder, continuous-wave, and staggered state

A. M. Cabanas<EMAIL_ADDRESS>J. A. Velez  L. M. Pérez  P. Díaz
M. G. Clerc  D. Laroze<EMAIL_ADDRESS>B. A. Malomed  Sede Esmeralda,
Universidad de Tarapacá, Av. Luis Emilio Recabarren 2477, Iquique, Chile
Instituto de Alta Investigación, CEDENNA, Universidad de Tarapacá, Casilla 7D,
Arica, Chile  Departamento de Ciencias Físicas, Universidad de La Frontera,
Casilla 54-D, Temuco, Chile  Departamento de Física and Millennium Institute
for Research in Optics, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile, Casilla 4873, Santiago, Chile  Department of Physical
Electronics, School of Electrical Engineering, Faculty of Engineering, and
Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv IL-69978,
Israel

###### Abstract

Discrete dissipative coupled systems exhibit complex behavior such as chaos,
spatiotemporal intermittence, chimera among others. We construct and
investigate chimera states, in the form of confined stationary and dynamical
states in a chain of parametrically driven sites with onsite damping and cubic
nonlinearity. The system is modeled by the respective discrete parametrically
driven damped nonlinear Schrödinger equation. Chimeras feature quasi-periodic
or chaotic dynamic in the filled area, quantified by time dependence of the
total norm (along with its power spectrum), and by the largest Lyapunov
exponent. Systematic numerical simulations, in combination with some
analytical results, reveal regions in the parameter space populated by stable
localized states of different types. A phase transition from the stationary
disorder states to spatially confined dynamical chaotic one is identified.
Essential parameters of the system are the strength and detuning of the
forcing, as well as the lattice’s coupling constant.

††journal: Chaos, Solitons & Fractals

##  1 Introduction

Coupled oscillators are of great interest owing to their wide applicability in
physics, chemistry, and biology (see Refs. [1, 2, 3, 4, 5, 6] and references
therein). Likewise, these systems attract global attention due to their
dynamic behavior, such as synchronization, defects and/or phase turbulence,
defect-mediated turbulence, spatiotemporal intermittency, and coexisting
coherent and incoherent states, among others [1, 6, 7]. In the last few
decades, a great deal of effort has been devoted to understand the coexistence
of coherent and incoherent domains called chimera states. These states were
introduced by Kuramoto and Battogtokh [8]. This finding came as a surprise
because the oscillators were identical and symmetrically coupled.
Nevertheless, along with regular synchronized dynamic, the coupled oscillator
lattices exhibit incoherent or desynchronized behavior. The understanding of
the symmetry breaking, which leads to the emergence of chimeras in dissipative
systems [9], is a significant issue as it appears in different contexts, such
as biological models, delayed systems, metamaterials, coupled map lattices,
quantum systems, networks, and even in a population of social agents, to
mention a few [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30].

As concerns dynamical elements of which lattices may be built, a paradigmatic
example of forced nonlinear oscillators is provided by the parametrically
driven damped nonlinear Schrödinger equation [31]. It models resonant
phenomena in nonlinear dispersive media and gives rise to soliton solutions
with a variety of dynamical behavior [33, 34, 32, 35], Faraday waves [36],
breathers [37, 38], two-soliton states [40, 39], and other soliton complexes
[41], and spatiotemporal chaos [42]. Generalizations of this equation and its
applications can be found in Refs. [43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 54, 41, 55, 56, 57].

In many physical realizations, this equation is replaced by its lattice
counterpart, in the framework of the class of models known as discrete
nonlinear Schrödinger (DNLS) equations [58, 59]. These equations furnish
fundamental models in discrete nonlinear optics [60, 61, 62, 63, 64, 65, 66,
67, 68], as well as for Bose-Einstein condensates trapped and fragmented in
deep optical-lattice potentials [70, 71, 72, 69]. In particular, the stability
of discrete solitons in the parametrically driven DNLS equations, both
conservative and dissipative ones, has been studied in Refs. [74, 75, 76, 77,
78].

The present work aims to study and build up chimera states in arrays of
dissipative parametrically driven coupled oscillators which are described by
the damped DNLS equation with the parametric gain applied at its sites. This
state appear as patched-shaped (self-confined) patterns filled by waves with a
“leaping" spatial structure (see an exact definition below), which are linked
by steep transient layers to the stable zero state. When the parameters are
modified the chimera states are replaced by a spatially disordered localized
one. Such states exist in a stationary form due to the bistability and
discrete nature of the system. In particular, we explore the existence of the
chimera state as a function of the driving force and its detuning, producing a
complete chart for stable stationary and dynamical states in the respective
parameter space. All these dynamic behavior correspond to permanent states of
the system out of equilibrium. Which correspond to dissipative structures in
the terminology introduced by Prigogine [79]. Effects of variation of the
intersite-coupling strength in the underlying DNLS equation are examined too.
We find that, in addition to being regular dynamical states, chimera, and the
system gives rise to multistability.

The manuscript is organized as follows. The model and some analytical findings
for it are presented in Sec. 2. Results from systematic numerical
investigation of the stationary and dynamical states are summarized in Sec. 3.
Conclusions are presented in Sec. 4.

##  2 The model and analytical results

###  2.1 The parametrically driven damped DNLS equation

Let us consider a one-dimensional array of coupled nonlinear oscillators under
the action of dissipation and parametric drive. The respective DNLS equation
is (cf. Ref. [76])

$\displaystyle\frac{dA^{j}}{dt}=-i\epsilon(A^{j+1}+A^{j-1}-2A^{j})$
$\displaystyle+\gamma\left(A^{j}\right)^{\ast}-i\nu A^{j}-\alpha
A^{j}-i|A^{j}|^{2}A^{j},$ (1)

where $A^{j}$ is a complex-valued amplitude at site $j$, the asterisk symbol
stands for the complex conjugation, and $t$ is normalized time. If Eq. (1)
models an array of parametrically amplified lossy optical waveguides with the
Kerr nonlinearity [80], $t$, is actually, the propagation distance along each
waveguide. Further, $\epsilon>0$ represents the coupling between adjacent
sites in the array, $\gamma>0$ is the strength of the parametric drive (the
sign of $\gamma<0$ may be inverted by redefining $\tilde{A}^{j}\equiv
iA^{j}$), $\nu$ is the detuning of the drive, and $\alpha>0$ is the damping
constant, while the on-site nonlinearity coefficient is scaled to be $1$. Note
that the sign of the nonlinear term in Eq. (1) corresponds to the self-
focusing onsite nonlinearity; if its original sign is opposite, it may be
inverted by means of the staggering transformation [81], that is,
$A^{j}\equiv(-1)^{j}\exp\left(-2i\epsilon t\right)\left(A^{j}\right)^{\ast}$.
The remaining scaling invariance of Eq. (1) allows us to fix one of the
parameters. We use this option to choose the value of the damping coefficient,
$\alpha=0.35$, which is convenient for the presentation of numerical results.
The analysis is reported below for negative detuning, $\nu<0$, which implies a
possibility of the existence of the dissipative bright discrete solitons,
supported by the self-focusing nonlinearity.

It is relevant to mention that, in addition to the direct realizations of Eq.
(1) as the chain of coupled oscillators with the complex amplitudes, the same
model can be derived as an asymptotic approximation for a parametrically
driven discrete nonlinear Klein-Gordon equation or Frenkel-Kontorava model,
i.e., a chain of nonlinear oscillators with real dynamical variables [78].
Figure 1 illustrates some examples of different physical systems that are
described by parametrically driven damped discrete nonlinear Schrödinger
equation (1). Panel a) represents a one-dimensional array of forcing coupled-
waveguide resonators [12], frame b) illustrates a set of parametrically driven
coupled magnets [52, 38, 39], while panel c) depicts a vertically driven
coupled pendulums [44].

Figure 1: Parametrically damped driven discrete systems. Schematic
representation of a one-dimensional array of forcing coupled-waveguide
resonators a), coupled magnets under a time dependent magnetic field b), and
vertically driven coupled pendulums c).

Stationary solutions to Eq. (1) can be looked for with a constant value of the
phase at all sites:

$A^{j}=B^{j}\exp\left(\mp
i\delta_{0}\right),\delta_{0}\equiv\tan^{-1}\left(\sqrt{\frac{\gamma-\alpha}{\gamma+\alpha}}\right).$
(2)

where real amplitudes $B^{j}$ obey the stationary version of the usual DNLS
equation,

$\displaystyle\epsilon(B^{j+1}+B^{j-1}-2B^{j})+\left(B^{j}\right)^{3}=$
$\displaystyle\left(-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}\right)B^{j}.$ (3)

The top and bottom signs in Eq. (3) correspond to those in Eq. (2).

In simulations reported in this work, we fixed the number of sites in the
lattice as $N=256$, so the discrete coordinate takes values $j=1,...,256$. We
used Neumann boundary conditions at edges of the lattice, which means formally
setting $A^{0}\equiv A^{1}$ and $A^{N+1}\equiv A^{N}$ in Eq. (1) at $j=1$ and
$j=256$, respectively.

###  2.2 Stability of the zero state

Because states considered in this work include zero-field segments, such
states are relevant solutions if $A^{j}=0$ is a stable solution of model Eq.
(1). To address the stability of zero, perturbed solutions to the linearized
version of Eq. (1) are looked for as a combination of terms $\exp\left(\Gamma
t\right)\left\\{\cos\left(kj\right),\sin\left(kj\right)\right\\}$ to derive a
dispersion relation for the instability growth rate, $\Gamma$, as a function
of the perturbation wavenumber $k$

$\Gamma=-\alpha\pm\sqrt{\gamma^{2}-\left[4\epsilon\sin^{2}\left(k/2\right)-\nu\right]^{2}}$
(4)

(cf. Ref. [82]). It follows from Eq. (4) that stability conditions for the
zero state, $\mathrm{Re}\left(\Gamma(k)\right)\leq 0$, which must hold for all
real values of $k$, amount to

$\displaystyle\gamma^{2}$ $\displaystyle\leq$
$\displaystyle\alpha^{2}+\nu^{2}\equiv\gamma_{\max}^{2},\quad~{}\mathrm{at}~{}\nu<0,$
$\displaystyle\gamma$ $\displaystyle\leq$
$\displaystyle\alpha,\quad\quad\quad\quad\quad\quad\quad~{}\mathrm{at}~{}0<\nu<4\epsilon,$
$\displaystyle\gamma^{2}$ $\displaystyle\leq$
$\displaystyle\alpha^{2}+\left(\nu-4\epsilon\right)^{2},\quad~{}\mathrm{at}~{}\nu>4\epsilon.$
(5)

In the continuum limit, which corresponds to $\epsilon\rightarrow\infty$, or,
effectively, to replacement

$4\epsilon\sin^{2}\left(k/2\right)\rightarrow\epsilon k^{2},$ (6)

in Eq. (4), the bottom line in Eq. (2.2) is irrelevant, while the top and
middle ones remain valid, cf. Ref. [36]. In the case of
$\gamma^{2}>\gamma_{\max}^{2}$ at $~{}\nu<0$ [see the top line in Eq. (2.2)],
the instability of the zero state leads to establishment of solutions to Eq.
(1) in the form of Faraday patterns, as [36].

###  2.3 Continuous-wave states: existence conditions

Next, Eq. (3) gives rise to two uniform (“continuous-wave", CW) solutions with
a constant amplitude:

$B=\sqrt{-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}}\equiv B_{0}^{\left(\pm\right)}.$
(7)

Equation (7) also describes the amplitude of lattice solitons, with one or
several excited sites, in the anti-continuum limit, $\epsilon\rightarrow 0$
[83].

In the case of positive detuning, $\nu>0$, solution (7) exists only with the
top sign, under condition $\gamma^{2}>\alpha^{2}+\nu^{2}$, which is
incompatible with the middle and bottom lines in Eq. (2.2), hence in this
case, the CW solution cannot coexist with the stable zero state. However, in
the case of $\nu<0$, CW solutions with both signs in Eq. (7) exist in the
interval of values of the drive’s strength

$\alpha^{2}<\gamma^{2}\leq\alpha^{2}+\nu^{2},$ (8)

in which the stability condition of the zero solution, given by the top line
in Eq. (2.2), holds. Therefore, in this interval, both the zero state and the
CW solution with a larger amplitude, corresponding to the top sign in Eq. (7),
may be stable. Note that, according to general principles of the bifurcation
theory [86], the latter solution is definitely stable in the anti-continuum
limit, with $\epsilon=0$ in Eq. (1), while the full stability analysis at
$\epsilon>0$ makes necessary to consider the possibility of the modulational
instability, see the following subsection. On the other hand, the intermediate
solution, corresponding to the bottom sign in Eq. (7), is definitely unstable,
playing the role of a separatrix between attraction basins of the two
presumably stable states.

With the increase of $\gamma$, both CW states, given by Eq. (7), emerge
through the saddle-node bifurcation [86] at $\gamma=\alpha$. With the
subsequent increase of $\gamma$, the unstable solution with the smaller
amplitude collides with the zero state, and thus disappears, through the
inverse pitchfork bifurcation [86], at $\gamma^{2}=\alpha^{2}+\nu^{2}$
(formally, the latter bifurcation involves solution $B_{0}^{(-)}$, given by
Eq. (7), and its counterpart $-B_{0}^{(-)}$). The same bifurcation
destabilizes the zero solution, as seen from the top line in Eq. (2.2). At
$\gamma^{2}>\alpha^{2}+\nu^{2}$, only the CW solution with the top sign in Eq.
(7) exists, remaining possibly stable, while the zero state is unstable.

Figure 2: Localized disorder sate. Panels a), b), and c) display the input,
transient, and established localized states, generated by simulations of Eq.
(1) with $\epsilon=0.51$, $\alpha=0.35$, $\gamma=1.01$, and $\nu=-1.15$ at
three instants of time $t_{0}=0$, $t_{i}=11$, and $t_{f}=300$. Panels d), e),
and f) display the corresponding solutions for the extended state that
occupies the entire spatial domain at the same value of the parameters. Both
the localized and extended states are filled by the LW (leaping wave). The red
lines represent CW amplitudes described by Eq (7) for the given values of
$\alpha$, $\gamma$, and $\nu$.

Thus, stable zero and CW states coexist in interval (8), under the condition
of $\nu<0$. Nevertheless, it is impossible to build up a transient layer or
domain wall (front solution) connecting these states, with exponentially
decaying tails approaching each one. Indeed, the tail approaching the zero
state can be looked for as the following solution to the linearized version of
Eq. (3)

$B^{j}=b_{0}\exp\left(-q|j|\right),$ (9)

at $|j|\rightarrow\infty$, with some constant $b_{0}$ and the spatial decay
rate, $q$, determined by expression

$\sinh^{2}\left(q/2\right)=\left(4\epsilon\right)^{-1}\left(-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}\right),$
(10)

where the top and bottom signs correspond to those in Eqs. (7) and (9). On the
other hand, the tail approaching the CW state is looked for as

$B^{j}=B_{0}^{\left(\pm\right)}+\tilde{b}_{0}\exp\left(-p|j|\right),$ (11)

with another constant $\tilde{b}_{0}$ and the decay rate $p$ determined by Eq.
(3) linearized around $B=B_{0}^{\left(\pm\right)}$:

$\sinh^{2}\left(p/2\right)=-\left(2\epsilon\right)^{-1}\left(-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}\right).$
(12)

Obviously, conditions $\sinh^{2}\left(q/2\right)>0$ and
$\sinh^{2}\left(p/2\right)>0$, which are necessary for the existence of the
transient layer, are incompatible, as it follows from Eqs. (10) and (12).
Therefore, localized patterns in the form of finite-width patches of CW
connected by smooth fronts to the stable zero state do not exist.
Nevertheless, we find that, at $\epsilon<0.44$, flat localized states filled
by the CW do exist, being bounded by layers which include sharp edge peaks,
hence they cannot be approximated by ansatz (11) [87, 88].

Localized states produced by Eq. (1) are the main subject of this work. These
states are built as localized patches filled not by flat CW states, but rather
by ones spatially oscillating between different amplitude levels (leaping
waves, LWs). Chimera states are built as spatiotemporal incoherence segments
set on top of the stable zero background. Example of them are shown in Figs. 2
c), 3 a) and 4 below and explained in detail in section 2.5.

In this framework, it is worth to mention that solutions for smooth transient
layers connecting zero and CW states, in spite of being impossible in the
present system, are admitted by models including competing nonlinearities.
Known examples are provided by equations including cubic-quintic [89] or
quadratic-cubic [90] combinations of self-focusing and defocusing terms.

###  2.4 Modulational stability of the CW state

It is also relevant to address the modulational (in)stability of the CW state
given by expressions (2) and (7). As it is well known, the constant-amplitude
solution of the usual DNLS equation with the self-focusing sign of the onsite
nonlinearity is always unstable, in the absence of the damping and parametric
drive [84]. In the present case, a perturbed form of the CW solution is sought
for as

$A^{j}(t)=B_{0}^{\left(\pm\right)}\left[1+a_{j}(t)\right]\exp\left[\mp
i\delta_{0}+i\chi_{j}(t)\right],$ (13)

where $\mp$ has the same meaning as in Eq. (2). The substitution of this
expression in Eq. (1) and the corresponding linearization with respect to
small perturbations, $a_{j}(t)$ and $\chi_{j}(t)$, leads to a system with the
following evolution equations for the perturbations,

$\displaystyle\frac{da_{j}}{dt}$
$\displaystyle=\epsilon\left(\chi_{j+1}+\chi_{j-1}-2\chi_{j}\right)$
$\displaystyle-2\gamma\sin\left(2\delta\right)\chi_{j},$
$\displaystyle\frac{d\chi_{j}}{dt}$
$\displaystyle=-\epsilon\left(a_{j+1}+a_{j-1}-2a_{j}\right)$
$\displaystyle-2\left(B_{0}^{\left(\pm\right)}\right)^{2}a_{j}-2\gamma\cos\left(2\delta\right)\chi_{j}.$
(14)

Eigenmodes of the small perturbations, with instability growth rate $\sigma$
(that may be complex-valued) and real wavenumber $k$, are looked for in the
usual form,

$\left\\{a_{j}(t),\chi_{j}(t)\right\\}=\left\\{a^{(0)},\chi^{(0)}\right\\}\exp(ikj+\sigma
t).$ (15)

The substitution of ansatz (15) in Eq. (14) yields a dispersion equation for
$\sigma(k)$

$\displaystyle\sigma\left(\sigma+2\alpha\right)+\left[4\epsilon\sin^{2}\left(k/2\right)\mp
2\gamma\sin\left(2\delta\right)+2\nu\right]$
$\displaystyle\times\left[4\epsilon\sin^{2}\left(k/2\right)\mp
2\gamma\sin\left(2\delta\right)\right]=0.$ (16)

It is easy to see that the continuum limit of Eq. (16), which is defined as
per Eq. (6), always gives rise to instability (represented by a root with
$\sigma>0$). On the other hand, the analysis reveals a region in the parameter
space in which Eq. (16) secures stability of the CW in the above-mentioned
relevant case, with $\nu<0$ and the top sign in Eqs. (2), (7), and (16)

$\alpha^{2}+4\epsilon^{2}\equiv\left(\gamma_{\min}^{2}\right)_{\mathrm{CW}}<\gamma^{2}<\alpha^{2}+\nu^{2}\equiv\gamma_{\max}^{2}.$
(17)

The self-consistency condition for the double inequality in Eq. (17) is
$\epsilon<|\nu|/2$. Here the left inequality is the CW modulational-stability
condition proper, while the right one is copied from the top line of Eq.
(2.2). Therefore, the CW state (2) with amplitude $B_{0}^{(+)}$ and zero
solution are _simultaneously stable_ in the parameter window defined by Eq.
(17). The possibility of the respective bistability is relevant to the present
work, which aims to build up stable nonzero modes on top of the stable zero
background (although, as shown above, these two stable states cannot be
connected by a transient-layer solution).

On the other hand, straightforward consideration of Eq. (16) demonstrates
that, in the same case, the CW state with smaller amplitude, which corresponds
to the bottom sign in Eqs. (2), (7), and (16), is always subject to the
modulational instability. Another obvious corollary of Eq. (17) is that (as it
is well known [84]) the stability interval (17) does not exist in the absence
of the drive and damping, $\gamma=\alpha=0$.

Figure 3: Disorder localized state. Panels a) and b) show the squared absolute
value, $|A^{j}|^{2}$, for the robust confined localized state and the extended
one, respectively, filled by the LW (leaping wave). Panels c) and d) show the
spatiotemporal evolution of $\mathrm{Re}\left(A^{j}(t)\right)$ from the flat-
amplitude inputs towards the final states depicted in a) and b), respectively.
The solutions are produced by Eq. (1) at $\gamma=0.55$, $\nu=-1.0$ with
$\alpha=0.35$ and $\epsilon=0.51$.

###  2.5 Staggered states

In addition to the stationary solutions with the fixed phase considered above
in the form of ansatz (2), it is possible to introduce staggered solutions of
Eq. (1), with alternating signs of amplitudes at adjacent sites of the
lattice, $A^{j}\equiv(-1)^{j}\tilde{A}^{j}$. This transformation casts Eq. (1)
in the form of

$\displaystyle\frac{d\tilde{A}^{j}}{dt}=i\epsilon(\tilde{A}^{j+1}+\tilde{A}^{j-1}+2\tilde{A}^{j})-i|\tilde{A}^{j}|^{2}\tilde{A}j$
$\displaystyle-i\nu\tilde{A}^{j}+\gamma\left(\tilde{A}^{j}\right)^{\ast}-\alpha\tilde{A}^{j}.$
(18)

Further, the substitution of

$\tilde{A}^{j}=\tilde{B}^{j}\exp\left(\mp i\delta_{0}\right),$ (19)

with the same $\delta_{0}$ as in Eq. (2) and real discrete field
$\tilde{B}^{j}$, leads to the respective stationary equation,

$\displaystyle-\epsilon(\tilde{B}^{j+1}+\tilde{B}^{j-1}+2\tilde{B}^{j})+\left(\tilde{B}^{j}\right)^{3}=$
$\displaystyle\left(-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}\right)\tilde{B}^{j}.$
(20)

Solutions to Eqs. (18) and (20) can be obtained from the above unstaggered
ones by substitution

$\epsilon\rightarrow-\epsilon,\nu\rightarrow\nu-4\epsilon.$ (21)

In particular, a solution to Eq. (20) with constant $\tilde{B}^{j}$ gives rise
to the following staggered version of the CW state, which can be obtained from
the CW solution (7) via substitution (21) (the solution is written here in
terms of the original lattice field $A^{j}$, rather than $\tilde{A}^{j}$):

$A^{j}=(-1)^{j}\sqrt{4\epsilon-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}}\exp\left(\mp
i\delta_{0}\right).$ (22)

Two different solutions given by Eq. (22) with opposite signs $\pm$ exist if,
as above, condition $\gamma>\alpha$ holds, and the intersite coupling strength
is subject to constraint

$\epsilon>(1/4)\left(\nu+\sqrt{\gamma^{2}-\alpha^{2}}\right).$ (23)

In particular, for $\nu<0$ condition (23) holds if $\gamma$ satisfies
inequality (8), the same one which is necessary for the coexistence of two
unstaggered CWs given by Eqs. (2) and (7).

Further, staggered decaying-tail solutions of the linearized version of Eq.
(20), in the form of $\tilde{B}^{j}=\tilde{b}_{0}\exp\left(-q|j|\right)$,
which make possible to connect zero and nonzero states [cf. Eq. (9)], are
obtained from Eq. (10) with the help of substitution (21):

$\cosh^{2}\left(q/2\right)=\left(4\epsilon\right)^{-1}\left(\nu\mp\sqrt{\gamma^{2}-\alpha^{2}}\right),$
(24)

while the stability conditions for the zero solution keep the form of Eq.
(2.2). In the case of $\nu<0$ (negative detuning), the tail solution (24)
exists solely under condition $\gamma^{2}>\alpha^{2}+\nu^{2}$, which makes the
zero state unstable, as per the top line in Eq. (2.2). Hence, the staggered CW
solution cannot coexist with stable zero, as parts of a chimera or localized
state. In the case of positive detuning, $\nu>0$, a possibility of building a
chimera combining the stable zero and a stable staggered CW is not promising
either, because conditions necessary for the existence of tail (24), i.e.,
$\cosh^{2}\left(q/2\right)>1$, and of two CW states, as given by Eq. (22),
contradict each other.

Lastly, as a generalization of the staggered solutions, it is possible to
introduce “twisted" ones (so named following Ref. [85]), in which stationary
field $B^{j}$ [defined as in Eq. (2)] takes opposite values at sites separated
by ones with $B^{j}=0$:

$B^{j}=\left\\{\begin{array}[]{c}\sqrt{2\epsilon-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}},~{}j=4n,\\\
0,~{}j=1+2n,\\\
-\sqrt{2\epsilon-\nu\pm\sqrt{\gamma^{2}-\alpha^{2}}},~{}j=2\left(1+2n\right),\end{array}\right.$
(25)

cf. Eqs. (7) and (22). In this work, we do not aim to consider patterns of
this type in detail.

Figure 4: Chimera states. Spatiotemporal evolution of
$\mathrm{Re}\left(A^{j}(t)\right)$ for three species of self-confined chimera
states, generated by simulations of Eq. (1). (a) A disorder localized state,
found for $\gamma=0.67$, $\nu=-1.0$, with $\lambda_{\max}=-0.011$. (b) A
quasi-periodic chimera, for $\gamma=0.75$ and $\nu=-0.80$, with
$\lambda_{\max}=-5.54\times 10^{-5}$. (c) A chaotic chimera, for $\gamma=0.90$
and $\nu=-1.2$, with $\lambda_{\max}=0.07$.

##  3 Localized states

###  3.1 Localized disorder states

The underlying equation (1) with the above-mentioned boundary conditions was
solved by means of a variable-step fifth-order Runge-Kutta scheme that ensures
the relative precision of $10^{-7}$ [91]. The integration was performed up to
$t=4.8\times 10^{3}$, which is essentially larger than the sufficient time for
the establishment of various static and dynamical states. To identify the
region of existence of chimera states, we have first produced stable patterns
filling the entire solution domain. Then, the numerical solution produced
chimeras by taking parameters from the patterns’ stability area, and running
simulations of Eq. (1) with inputs in the form of a localized segment of the
extended pattern, set in the central part of the integration domain. The same
localized states could be produced from a simpler input, taken as a
rectangular box with a constant amplitude in a central segment of the
integration domain, letting it to evolve into a static or dynamical chimera of
approximately the same width, see an example below in Fig. 4(c).

The forcing, damping, and detuning terms play a crucial role in the formation
of various states in the present model. As said above, the damping coefficient
is fixed to be constant through rescaling, $\alpha=0.35$, therefore results
were collected by varying $\gamma$ and $\nu$, while (in most cases) the
intersite coupling constant was kept constant, at value $\epsilon=0.51$, which
allows to present generic findings.

Figure 2 shows examples of stationary states, produced as numerical solutions
of Eq. (1) with $\gamma=1.01$ and $\nu=-1.15$. Panel a) displays the above-
mentioned input, taken as a rectangular box with a flat amplitude. Panel b)
represents a transient state produced in the course of the evolution, that
leads to the establishment of the stationary localized disorder depicted in
panel c). It is a finite-length patch of a quasi-regular LW separated by steep
edges (fronts) from the zero state. This localized disorder state is stable
and motionless at the present values of the parameters, according to the top
line in Eq. (2.2).

Further, panel f) displays a stationary LW pattern occupying the entire
integration domain, obtained via an intermediate state (panel e)) from the
input shown in panel d). It was introduced as a small-amplitude periodic wave
with the mean value close to the CW amplitude $B_{0}^{+}$, given by Eq. (7)
for current values of the parameters: $B_{0}^{\left(+\right)}\approx 1.448$
and $B_{0}^{\left(-\right)}\approx 0.450$. The periodic component was added to
the input to initiate the development of the extended LW state. Note that both
values $B_{0}^{\left(\pm\right)}$ are clearly visible as the limits, shown by
red lines in panels c) and f), between which the LW oscillates in both the
confined (localized disorder state) and extended states. The numerical data
demonstrate that the stationary solutions displayed in panels c) and f) have a
spatially uniform phase.

Similar dynamic is observed at other values of the parameters. In particular,
Fig. 3 displays the full picture of the establishment of the patch-shaped
(localized disorder state) state with the LW structure, and its counterpart
filling the entire domain, at $\gamma=0.55$, $\nu=-1.0$, $\alpha=0.35$, and
$\epsilon=0.51$. Figures 3 c) and d) corroborate the modulational instability
of the above-mentioned CW state, with $B_{0}^{\left(+\right)}\approx 1.448$,
in the confined and fully extended forms alike, in agreement with the fact
that the current parameter values do not satisfy the left-side inequality in
the CW stability condition given by Eq. (17). The observation that the
instability onset and development are essentially the same in panels c) and d)
is explained by the fact that the instability maximum determined by Eq. (16)
corresponds to $k=\pi$, hence the corresponding perturbation wavelength,
$2\pi/k=2$, is much smaller than the width of the initial CW domains in both
panels.

Another essential conclusion suggested by Figs. 2 and 3 is that a sharp edge
separating the zero and nonzero parts of the finite-size patch (localized
stated) is always stable, which makes possible to build up a stationary
localized disorder state of an arbitrary size, confined by a pair of edges.
Thus, the system demonstrates multistability, including localized states of
different sizes, as well as configurations with two or several localized
domains separated by zero-amplitude segments (not shown here).

Figure 5: The power spectrum, $S(f)$, of the time-dependent total norm,
$Q(t)$, defined as per Eq. (27). Panels a) and b) correspond to the quasi-
periodic and chaotic chimera, displayed in Figs. 4 b) and c), respectively.
The corresponding time dependences $Q(t)$ [see Eq. (26)] are displayed in the
insets.

###  3.2  Dynamical indicators

To characterize different nature of dynamical behavior of localized states, we
use, first, the scaled total norm, which (along with the Hamiltonian) is a
dynamical invariant of the DNLS equation in the absence of the damping and
drive [58], while in the present case it is a function of time, in non-
stationary states

$Q(t)=\frac{1}{N}\sum_{j=1}^{N}|A^{j}(t)|^{2}.$ (26)

In the case of time-dependent $Q(t)$, we computed its power spectrum,

$S(f)={|\mathfrak{F}(f)|}^{2},$ (27)

as a function of frequency $f$ [related to the angular frequency as
$f=\omega/\left(2\pi\right)$], where the Fourier transform of $Q(t)$ is
defined as

$\mathfrak{F}(f)=\frac{1}{\sqrt{2\pi}}\int_{0}^{t_{\max}}Q(t)\exp\left(-ift\right)dt.$
(28)

A general principle is that ${S(f)}$ features a quasi-discrete spectrum, with
a set of narrow peaks, for regular solutions, which exhibit (quasi-) periodic
evolution in time. On the other hand, the spectrum is expected to be an
essentially continuous one if the underlying time-dependent solution can be of
chaotic nature [92], see also a recent realization of the principle for the
nonlinear Schrödinger equation with a trapping potential [93].

Another relevant indicator characterizing the dynamical behavior is the
largest Lyapunov exponent (LLE), $\lambda_{\max}$ [92, 94]. When the system is
chaotic, two trajectories, that were infinitesimally close initially, separate
in the phase space exponentially, the distance between them growing
$\sim\exp\left(\lambda_{\max}t\right)$, with $\lambda_{\max}>0$. On the other
hand, close trajectories converge to the same non-chaotic attractor if
$\lambda_{\max}<0$. The marginal case of $\lambda_{\max}=0$ implies that the
behavior is time-periodic, quasi-periodic, or complex behavior with power
sensitivity to initial conditions. LLE is widely used to quantify regular and
chaotic evolution in diverse dynamical systems, see, e.g., Refs. [94, 95, 98,
96, 100, 99, 97, 101, 102].

Figure 6: Dynamical indicators. a) LLE and b) time-average of the total norm
$\left\langle Q\right\rangle$ as functions of $\gamma$ for fixed negative
values of $\nu$, which are indicated in the panels. Other parameters in Eq.
(1) are $\mu=0.35$, $\epsilon=0.51$, and $\alpha=0.35$. In shaded areas the
stationary LW (leaping wave), filling the entire lattice, is unstable.

###  3.3 Chimera states

To collect systematic numerical data, we ran simulations of model Eq. (1) with
the initial state in the form of a state with a flat amplitude, filling a
central segment of the integration domain, as shown above in Figs. 2 a) and 3
c). Generic results, obtained at different values of the driving strength,
$\gamma$, and negative detuning, $\nu$, are displayed in Fig. 4, in the form
of established dynamical regimes, observed in the course of a long simulation
interval, $0<t<1000$. All these self-sustained states were created in a finite
lattice segment, $100<j<160$.

In panel 4 a), the evolution at $\gamma=0.67$ and $\nu=-1.0$ leads to the
establishment of a stable time-independent localized disorder state. In this
case, the respective value given by the top line in Eq. (2.2) is
$\gamma_{\max}\approx\allowbreak 1.06$, hence the zero background is indeed
stable. The regular behavior of this localized state as a whole is
corroborated by the computation of the respective LLE, which is found to be
negative, $\lambda_{\max}=-0.011$.

Next, increasing the strength of the forcing, a chimera featuring quasi-
periodic temporal dynamic is displayed in panel 4 b) for $\gamma=0.75$ and
$\nu=-0.80$, which is closer to the respective stability boundary for the zero
state, viz., $\gamma_{\mathrm{\max}}\approx 0.87$, as per the top line in Eq.
(2.2). Notice that this solution accounts for a coexistence of coherent and
incoherent domains. In this case, LLE is $\lambda_{\max}=-5.54\cdot 10^{-5}\pm
5.72\cdot 10^{-6}$, which is actually a numerical zero, as may be expected for
quasi-periodic states. Finally, a chaotic chimera, featuring apparently random
spatiotemporal dynamic at $\gamma=0.90$ and $\nu=-1.2$ in a self-confined
segment of the lattice, is displayed in panel c). In the latter case, Eq.
(2.2) yields $\gamma_{\max}\approx 1.25$, hence the zero background remains
stable. The numerically computed LLE is positive, $\lambda_{\max}=0.07$, which
clearly corroborates the chaotic character of the state as a whole [92].

Figure 7: Different species of robust self-confined states, generated by
simulations of Eq. (1) with the same localized input (shown in the inset), are
chartered in the parameter plane of the detuning, $\nu$, and parametric-
drive’s strength, $\gamma$, for a fixed inter-site coupling constant,
$\epsilon=0.51$, and the value of the dissipation coefficient fixed by the
rescaling, $\alpha=0.35$. Blue diamonds, green circles, and red squares
represent, respectively, stable localized disorder state, chimera with quasi-
periodic evolution, and chaotic spatiotemporal dynamic in the spatially
confined area. The dashed horizontal line, $\gamma=\alpha\equiv 0.35$, is the
boundary above which two CW states (7) exist at $\nu<0$. Only the CW state may
exist, as a stable one, above the green continuous curve, i.e., at
$\gamma>\gamma_{\max}=\alpha^{2}+\nu^{2}$ [see the top line in Eq. (2.2)],
where the zero solution and, hence, any finite-width chimera as a whole are
unstable. Indeed, for $\alpha=0.35$ and $\epsilon-0.51$, the stable CW exists
at $\gamma>\sqrt{\alpha^{2}+4\epsilon^{2}}\approx 1.08$ [as per the left-hand
inequality in Eq. (17)], i.e., above the area displayed in the present figure.
No stable nonzero states are produced by the simulations in the white area to
the left of the filled region.

To gain deeper understanding of the dynamic, the quasi-periodic and chaotic
character of the chimeras displayed in Figs. 4 b) and c) is quantified by the
corresponding power spectra, defined as per Eq. (27), which are presented in
Figs. 5 a) and b), respectively. As expected, the spectrum is quasi-discrete
for the quasi-periodic oscillations and continuous for the chaotic dynamic.
The previous analysis allows us to conclude that the emergence of quasi-
periodic and chaotic chimeras follows a route of extended quasi-periocity
[103].

The effect of the strength of the parametric forcing, $\gamma$, on properties
of the states under the consideration is represented by Fig. 6, which shows
LLE and the time-average value of the total norm,

$\left\langle Q\right\rangle=t_{\max}^{-1}\int_{0}^{t_{\max}}Q(t)dt,$ (29)

as functions $\gamma$, at different fixed values of detuning $\nu$, for the
self-confined states created in the above-mentioned lattice segment,
$100<j<160$. The transition from regular to chaotic dynamic with the increase
of the forcing strength, $\gamma$, i.e., from $\lambda_{\max}\approx 0$ to
definitely positive values of $\lambda_{\max}$, coincides with the transition
from nearly constant to growing values of $\left\langle Q\right\rangle$. Both
phenomena are manifestations of the phase transition from regular dynamic to
chaos, which takes place at a critical point, $\gamma=\gamma_{\mathrm{crit}}$.
For example, $\gamma_{\mathrm{crit}}(\nu=-0.90)\approx 0.92$ and
$\gamma_{\mathrm{crit}}\left(\nu=-1.05\right)\approx 0.90$, while the
respective values given by Eq. (2.2) are
$\gamma_{\max}\left(\nu=-0.9\right)\approx 0.966$ and
$\gamma_{\max}\left(\nu=-1.05\right)\approx 1.107$ [recall we have fixed the
dissipation constant in Eq. (1) as $\alpha=0.35$, by means of scaling]. Thus,
the transition to the chaotic dynamic happens when the zero state is still
stable, at $\gamma<\gamma_{\max}$. Generally, ratio
$\gamma_{\mathrm{crit}}/\gamma_{\max}$ takes values between $0.81$ and $0.90$.

Moreover, in the shaded area in Fig. 6 direct simulations demonstrate that the
extended LW state is unstable. Thus, in an interval of the drive’s strength
between $\gamma_{\mathrm{crit}}$ and the left edge of the shaded area the
dynamic is categorized as chaotic, while the explicit LW instability does not
set in, as yet. The “delay" of the onset of the instability after the
transition to the positive LLE may be a consequence of the finite width of the
self-confined patch (localized state) filled by LW. Finally, stable CWs are
easily found at $\gamma>\left(\gamma_{\min}\right)_{\mathrm{CW}}$, where, for
the current values of the parameters, $\alpha=0.35$ and $\epsilon=0.51$, the
left-hand inequality in Eq. (17) gives
$\left(\gamma_{\min}\right)_{\mathrm{CW}}\approx 1.08$.

Figure 8: Existence regions of different stable states as function of the
intersite coupling constant, $\epsilon$, at fixed values of the drive’s
strength, $\gamma=0.95$, and detuning, $\nu=-1.11$. The localized states
(localized disorder and chimeras) are stable in the interval of
$0.44<\epsilon<0.81$. At $\epsilon<\epsilon_{\mathrm{crit}}\approx 0.44$ [see
Eq. (30)], stable states, depending on the initial conditions, are uniform CWs
or broad flat patches of CWs, bounded by narrow transient layers with sharp
(but low) peaks, see Fig. 9. At $\epsilon>0.81$, Eq. (1) gives rise to stable
delocalized patterns with zero crossings (not considered here in detail).
Figure 9: $\mathrm{Re}\left(A^{j}\right)$ for stable states in the form of
confined flat (CW) patches, bounded by narrow layers with sharp peaks. The
states are obtained at $\gamma=0.95$, $\nu=-1.11$, and $\alpha=0.35$ (the same
as in Fig. 8), with $\epsilon=(0.40,0.31,0.22,0.18)$, as color coded in the
right panel (which provides a zoom of the sharp peaks). Note that all these
values of the lattice coupling constant belong to the region (30).

Results of the systematic numerical analysis are summarized in Fig. 7, which
is a chart designating different established states in the parameter plane of
$\left(\nu,\gamma\right)$. The states are categorized, with the help of the
value of LLE, $\lambda_{\max}$, and the character of the power spectrum of
$Q(t)$ (quasi-discrete or continuous), as stationary, temporarily quasi-
periodic, or chaotic self-confined chimeras (blue diamonds, green circles, and
red squares, respectively, in Fig. 7). The results were generated by means of
sufficiently long simulations of Eq. (1), initiated, at all values of $\nu$
and $\gamma$, by the rectangular excitation field displayed in the inset to
Fig. 7. The localized states, which include extended segments of the stable
zero solution, exist between the dashed horizontal line, $\gamma=\alpha\equiv
0.35$, and the solid green curve, $\gamma=\gamma_{\max}$ [see the top line in
Eq. (2.2)]. These boundaries correspond, respectively, to the above-mentioned
saddle-node and inverse pitchfork bifurcations. In particular, at values of
the drive’s strength $\gamma<0.48$ and $0.55<\gamma<0.65$ for $\nu>-1.0$ the
chart contains only stationary chimeras.

We stress that the model obviously features multistability, as different
species of the stable state coexist at the same values of the parameters. In
particular, as mentioned above (see Figs. 2, 3, 4), Eq. (1) admits bistability
between localized and delocalized states, which are readily found as stable
solutions at the same values of the parameters, the choice between them being
determined by the initial conditions.

Figure 10: Stable randomly shaped patterns produced by the simulations at the
same fixed values of the parameters, $\gamma=0.95$, $\nu=-1.11$, and
$\alpha=0.35$, as in Figs. 8 and 9. The patterns are produced at gradually
increasing values of the lattice coupling constant, $\epsilon$, which leads to
deeper zero crossing. The left and right columns display, severally, confined
and extended patterns, produced by respective inputs.

The above results were obtained for the fixed value of the intersite coupling
constant, $\epsilon=0.51$. It is also relevant to consider effects of
variation of $\epsilon$. In Fig. 8 we report the results for fixed
$\gamma=0.95$ and $\nu=-1.11$, the respective value given by the top line of
Eq. (2.2) being $\gamma_{\max}\approx\allowbreak 1.16$. The figure
demonstrates that, at $\epsilon<0.44$, the simulations produce no chimeras,
which are ousted by the stable uniform CW, in exact agreement with the left-
hand inequality in Eq. (17), which shows that CWs are stable at

$\epsilon<\epsilon_{\mathrm{crit}}\equiv(1/2)\sqrt{\gamma^{2}-\alpha^{2}}\approx
0.44,$ (30)

for $\gamma=0.95$ and $\alpha=0.35$. Detailed simulations demonstrated that,
in the same region at $\epsilon<\epsilon_{\mathrm{crit}}$, the established
numerical solution depends on initial conditions. If an extended pattern is
taken as the input, simulations establish a uniform CW with amplitude
$B_{0}^{+}$, see Eq. (7). On the other hand, the input in the form of a
localized region filled by the LW gives rise to robust solutions in the form
of wide but confined flat CW patches, which are bounded by narrow transient
layers including sharp (but relatively low) peaks, such as the ones shown in
Fig. 9. Because of the presence of the peaks, these states cannot be predicted
by ansatz (11) considered above. The right panel on Fig. 9 demonstrates that
the height of the peaks diminishes with the decrease of $\epsilon$.

As shown in Fig. 8, for the same fixed values $\gamma=0.95$ and $\nu=-1.11$,
stable finite-width chimeras are produced by numerical simulations in an
interval of values of the coupling constant $0.44<\epsilon<0.81$. Lastly, at
$\epsilon>0.81$, the former chimera patterns acquire zero points (originally
falling to $A_{j}=0$, and then crossing zero), as shown in Fig. 10. However,
these states are different from the zero-crossing twisted ones, given by Eq.
(25), as they are apparently randomly built patterns. As shown in Fig. 10,
they may feature both confined and extended shapes, depending on the input.

##  4 Conclusion

In this work, we have presented chimera states and localized disorder state
found in the parametrically driven discrete dissipative system, modeled by the
damped DNLS equation with the onsite parametric drive. The system can be
implemented as an array of optical fibers in which losses are compensated by
the parametric amplification. The localized states are built as finite
segments filled by a pattern which may be a stationary LW (a spatially leaping
wave oscillating between lattice sites), or, depending on parameters, an LW
featuring quasi-periodic or chaotic time dependence. The segment is connected
by steep fronts to stable zero states (in this sense, it is a chimera
combining stable nonzero and zero modes). The dynamic of the chimeras is
characterized by the time dependence of the total norm, and by LLE (the
largest Lyapunov exponent). Our analysis reveals a specific region of
parameters in which robust localized disorder and chimeras state exist, being
the strength and detuning of the parametric drive the essential control
parameters. The dependence of the existence region on the strength of the
intersite coupling, $\epsilon$, is considered as well. Besides, for smaller
values of $\epsilon$, stable states are represented by flat CW patches, which
are confined by narrow transient layers containing sharp peaks. A part of the
numerical findings is explained by analytical results that address the
existence and stability of the zero and uniform CW states.

As an extension of the analysis, it may be relevant to consider chimera states
in the two-dimensional version of the present system. Work in this direction
is in progress.

## Acknowledgments

LMP, PD, and DL acknowledge partial financial support from FONDECYT 1180905.
DL acknowledges the partial financial support from Centers of excellence with
BASAL/CONICYT financing, Grant AFB180001, CEDENNA. The work of BAM is
supported, in a part, by the Israel Science Foundation, through grant No.
1286/17. This author appreciates hospitality of Instituto de Alta
Investigación at Universidad de Tarapacá (Arica, Chile). MGC thanks for the
financial support of FONDECYT projects 1180903 and Millennium Institute for
Research in Optics ANID-Millennium Science Initiative Program–ICN17_012.

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