Two populations of coupled quadratic maps exhibit a plentitude of symmetric and symmetry broken dynamics

Abstract

We numerically study a network of two identical populations of identical real-valued quadratic maps. Upon variation of the coupling strengths within and across populations, the network exhibits a rich variety of distinct dynamics. The maps in individual populations can be synchronized or desynchronized. Their temporal evolution can be periodic or aperiodic. Furthermore, one can find blends of synchronized with desynchronized states and periodic with aperiodic motions. We show symmetric patterns for which both populations have the same type of dynamics as well as chimera states of a broken symmetry. The network can furthermore show multistability by settling to distinct dynamics for different realizations of random initial conditions or by switching intermittently between distinct dynamics for the same realization. We conclude that our system of two populations of a particularly simple map is the most simple system that can show this highly diverse and complex behavior, which includes but is not limited to chimera states. As an outlook to future studies, we explore the stability of two populations of quadratic maps with a complex-valued control parameter. We show that bounded and diverging dynamics are separated by fractal boundaries in the complex plane of this control parameter.

Chimeras confined by fractal boundaries in the complex plane

“…There are several important advances achieved by the present study with regard to our previous work, 39 in which we used two large populations of each 100 quadratic maps f c . In the main part of Ref.…”

“…This parsimonious approach to use a minimal two-population network of only two maps per population is different from Ref. 39, where each population consisted of one hundred maps. For previous work on chimera states in networks consisting of a minimal number of nodes see for example Refs.…”

“…Otherwise, the Julia set is a Cantor space. The Mandelbrot set is defined by the set of c for which the Julia set is connected, or equivalently, by the set of c for which iterates z n of f c initiated at z 0 = 0 remain bounded for all n. [42][43][44] We have shown previously that two identical populations of quadratic maps f c yield a plentitude of dynamics which can be symmetric or symmetry broken with regard to the two populations 39 . In particular, these dynamics included chimera states for which one population was identically synchronized while the other remained incoherent.…”

“…Chimera states were found for networks of coupled logistic maps [4][5][6][7][8][9][23][24][25][26][27][28][29][30][31] , Henon maps [10][11][12][13][14][15]21 , cubic maps 7,16,17 , map-based neuron models 18,32 , sine-circle maps [33][34][35] , sine-squared maps 23,26,36 , cosine maps 37 , piecewise linear maps 38 , and piecewise logistic maps 38 . Recently, two populations of quadratic maps with real-valued parameter c were shown to yield a plentitude of dynamics including chimera states 39 . Quadratic maps, however, reveal their full complexity only for complexvalued parameters c. In particular, they allow one to generate fractal Julia sets and Mandelbrot sets in the complex plane [40][41][42][43][44] .…”

Chimeras and Clusters Emerging from Robust‐Chaos Dynamics

“…In addition, the study of chimera states currently attracts much interest (for reviews see [21,22]). Chimera states have been found in networks of nonlocally coupled phase oscillators [23,24], in systems with local [25][26][27][28] and global [29][30][31][32][33][34] interactions, and in networks of time-discrete maps [35][36][37][38]. ese states have been investigated in a diversity of contexts [39][40][41][42][43][44][45][46][47].…”