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Edited by: Ulrich Parlitz, Max-Planck-Institute for Dynamics and Self-Organisation, Max Planck Society (MPG), Germany

Reviewed by: Isao T. Tokuda, Ritsumeikan University, Japan; Anastasiia Panchuk, Institute of Mathematics (NAN Ukraine), Ukraine

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Amplitude chimera (AC) is an interesting chimera pattern that has been discovered recently and is distinct from other chimera patterns, like phase chimeras and amplitude mediated phase chimeras. Unlike other chimeras, in the AC pattern all the oscillators have the same phase velocity, however, the oscillators in the incoherent domain show periodic oscillations with randomly shifted origin. In this paper we investigate the effect of local filtering in the coupling path on the occurrence of AC patterns. Our study is motivated by the fact that in the practical coupling channels filtering effects come into play due to the presence of dispersion and dissipation. We show that a low-pass or all-pass filtering is actually detrimental to the occurrence of AC. We quantitatively establish that with decreasing cut-off frequency of the filter, an AC transforms into a synchronized pattern. We also show that the symmetry-breaking steady state, i.e., the oscillation death state can be revoked and rhythmogenesis can be induced by local filtering. Our study will shed light on the understanding of many biological systems where spontaneous symmetry-breaking and local filtering occur simultaneously.

Networks of coupled identical oscillators show various cooperative behaviors. From the symmetry considerations they can be categorized into two broad types: (i) symmetric (or symmetry preserving) states, like synchronization, phase locking, and amplitude death (AD) state [

Although chimeras were discovered in phase oscillators, later on the notion was extended to the general class of oscillators having both phase as well as amplitude dynamics. Those oscillators may show amplitude mediated phase chimeras (AMC) [

In contrast to other chimera patterns, AC has strong connections to another symmetry-breaking

As amplitude chimeras are a recently discovered variant of chimera patterns, therefore, it is less explored: the effect of node dynamics and coupling on the occurrence of AC demands further investigations. Specifically, in realistic networks, where signals often suffer from time delay [

Motivated by the above discussion, in this paper we study the effect of local filtering on the occurrence of amplitude chimera (AC) in a network of nonlocally coupled Stuart-Landau oscillators. By local filtering we mean that the filtering effect is considered in the self-feedback path only. We consider local low-pass and all-pass filters in the network and for the first time we show that both types of filtering have a detrimental effect on the occurrence of amplitude chimeras: filtering always suppresses amplitude chimeras. With the variation of a filtering parameter (namely, the corner or cut-off frequency) we observe transitions from the oscillation death and amplitude chimera state to the globally synchronized state.

We consider

with

To explore the dynamics of the coupled network we numerically solve Equation (1) using the fourth-order Runge-Kutta method (step size = 0.01). Throughout this paper we consider ω = 2 and use the following initial conditions [_{i} = 1 and _{i} = −1 for _{i} = −1 and _{i} = 1 for

Without filtering:

Without filtering: Phase-space plot of a few nodes of the network from the coherent and incoherent domains

We consider

Equation ((2c)) is the mathematical equation of a low-pass filter whose input is _{i} and output is _{i}. This _{i} is fed to the coupling part of Equation (2a). Here α represents the corner or cut-off frequency of the LPF: the lower is the value of α, the higher is the effect of filtering. For larger α, filtering effects become lesser: if we put α → ∞ in Equation (2a), it simply gives _{i} = _{i}, i.e., no filtering effect is present and Equation (2) reduces to the original Equation (1). Since in the literature of filters we are conversant with the frequency domain representation, therefore, at first it is difficult to realize the role of α in Equation (2c). However, a close inspection reveals that α controls both phase and amplitude of the output signal _{i} by the following way: the phase shift between input and output is given by

We investigate the effect of local low-pass filtering on the occurrence of amplitude chimera. Since α is the only control parameter, we will explore the effect of α on the dynamics of the network. We keep all the parameters and initial conditions the same as in the unfiltered case; the initial conditions for the filter variable _{i} are chosen the same as those of _{i} for the unfiltered case.

_{c}) the AC and OD state disappear and only the synchronized state prevails in the whole

With local low-pass filtering: Phase diagram in the

The scenario can be understood more clearly in the ε − α space for a fixed _{c} of α, below which the synchronized state is the only possible state.

With local low-pass filtering:

In the above results we use suitable measures, such as the measure of spatial correlation (_{0}) and the center of mass (_{cmi}) to ensure the occurrence of the synchronized state and AC state and also to distinguish them (distinction of the OD state is relatively simple as we have to check whether a steady state is reached or not). According to Kemeth et al. [

Here

where _{i}}. In our present case the state variable ψ_{i}(_{i} (one can use _{i} as well). In Equation (3) we consider a threshold value δ_{th} = 0.01_{max}, where _{max} is the maximum curvature in the network [_{0}(_{0}(_{0}(_{0}(

where _{i} represents the state of the _{cmi} gives a measure of the shift of a limit cycle from the origin. Therefore, it can distinguish the homogeneous limit cycles from inhomogeneous ones.

_{0}(_{cmi} of each oscillator corresponding to the synchronized state of _{0}(_{cmi} = 0 indicating that the whole network is synchronized. On the other hand, _{0}(_{cmi} corresponding to the AC state of _{0}(_{cmi} in the incoherent region exhibits a random sequence of shifts to positive and negative values, however, in the coherent region _{cmi} = 0 indicating that the resulting chimera is indeed an AC pattern.

With local low-pass filtering: _{0} corresponding to the synchronized _{cmi}) for the above two points, showing synchronized

According to Tumash et al. [

with ^{n} and also consider that a periodic solution ψ(

where

Here δ

where _{k}). Each Floquet multiplier can be expressed as μ_{k} = _{k} + _{k})_{k} + _{k}) is the Floquet exponent. The stability of the periodic orbit can be analyzed by determining the sign of the real part of these exponents. When the real parts of all the Floquet exponents are less than zero (i.e., Λ_{k} < 0) except the Goldstone mode (which is equal to zero) then the periodic solution is stable indicating a synchronized solution [_{k} > 0), then the solution becomes unstable indicating a saddle cycle in phase space which corresponds to an AC state. In our computation we average the exponents over 200_{k}s have small (< 0.5) positive values, which means that the system is in the AC state. Note the agreement between

With local low-pass filtering: Phase diagram of the periodic solutions (Sync and AC) in ε − α space based on the Floquet exponent. For the synchronized region (black), at each point, the largest real part of the Floquet exponents (Λ_{max}) is negative (for the Goldstone mode it is approximately equal to zero). For the AC region (orange) at each point it is greater than zero (i.e., Λ_{max} > 0). Other parameters are

Next we consider the effect of all-pass filtering (APF) in the network of Stuart-Landau oscillators described in Equation (1). The mathematical model of the coupled system is given by

Equations (10c, 10d) jointly represent the differential algebraic equation of an all-pass filter, whose input is _{i} and output is _{i} [_{i}: Here α does not affect the amplitude of _{i}, it only affects the phase part by introducing a phase shift between the input and output signals, given by θ = 2^{−1}). Note that for the same α the phase shift introduced by a LPF (i.e., ϕ) is half of that of an APF (i.e., θ).

In _{c}, the critical value below which AC and OD are completely suppressed, is much higher for an APF compared to that of a LPF (not shown here): therefore even a relatively weak all-pass local filtering is equivalent to a stronger local low-pass filtering, as far as suppressing AC and OD is concerned. This is the consequence of the fact that at a particular value of α, the phase shift introduced by an APF is twice of that of a LPF [

With all-pass filtering: Phase diagram in the

In this paper, we have revealed that the presence of local filtering (either low-pass or all-pass) suppresses the amplitude chimera state and therefore gives rise to global synchrony (coherent traveling waves). Further, it has been shown that local filtering causes rhythmogenesis by suppressing the steady state behavior (i.e., OD state), which has immense importance in many biological and engineering systems [

Our study reveals that the cut-off frequency α of the local filter acts as an efficient control parameter of the network that can be tuned to achieve a desired symmetry-breaking state or synchronized state without changing coupling strength or range. Several control methods to stabilize phase or amplitude mediated phase chimeras have recently been proposed [

From the perspective of dynamical systems the role of α can be understood in the following way: α actually controls the dissipative property of the whole network by controlling the dissipation and dispersion in the coupling path; a smaller α imposes a larger filtering effect and therefore smaller dissipation, which favors synchrony and rhythmogenesis. In this context we observe that filtering does not affect the pattern of phase chimera appreciably. This may be due to the fact that additional phase shift and/or attenuation caused by filtering has lesser effect on the mean frequency than on the amplitude dynamics (note that in the phase chimera the mean frequency is the determining factor that distinguishes the coherent and incoherent domains, whereas in the amplitude chimera, the amplitude of the nodes matters).

In this paper we have considered a network of Stuart-Landau oscillators. However, we verified that the filtering affects the amplitude chimera in a similar way in other systems also, for example, in a network of Rayleigh oscillators [

TB, ES, and AZ formulated the problem. BB and TB carried out the analysis. BB performed the computations. All authors discussed the results and contributed to writing the manuscript, read and approved the final manuscript.

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

BB acknowledges University of Burdwan for providing financial support through the state funded research fellowship. AZ and ES acknowledge the financial support by DFG in the framework of SFB 910.