Spatially Organized Dynamical States in Chemical Oscillator Networks: Synchronization, Dynamical Differentiation, and Chimera Patterns

Abstract

Dynamical processes in many engineered and living systems take place on complex networks of discrete dynamical units. We present laboratory experiments with a networked chemical system of nickel electrodissolution in which synchronization patterns are recorded in systems with smooth periodic, relaxation periodic, and chaotic oscillators organized in networks composed of up to twenty dynamical units and 140 connections. The reaction system formed domains of synchronization patterns that are strongly affected by the architecture of the network. Spatially organized partial synchronization could be observed either due to densely connected network nodes or through the ‘chimera’ symmetry breaking mechanism. Relaxation periodic and chaotic oscillators formed structures by dynamical differentiation. We have identified effects of network structure on pattern selection (through permutation symmetry and coupling directness) and on formation of hierarchical and ‘fuzzy’ clusters. With chaotic oscillators we provide experimental evidence that critical coupling strengths at which transition to identical synchronization occurs can be interpreted by experiments with a pair of oscillators and analysis of the eigenvalues of the Laplacian connectivity matrix. The experiments thus provide an insight into the extent of the impact of the architecture of a network on self-organized synchronization patterns.

Chimera patterns in two-dimensional networks of coupled neurons

“…First found in identical and symmetrically coupled phase oscillators [2], chimera states have been the focus of extensive research for over a decade now. Both theoretical and experimental works have shown that this counter-intuitive collective phenomenon may arise in numerous systems including mechanical, chemical, electro-chemical, electrooptical, electronic, and superconducting coupled oscillators [3][4][5][6][7][8][9][10][11][12][13][14].…”

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

“…Recently, in a purely mechanical experiment involving two groups of identical metronomes, it was shown that chimeras emerge naturally as a coexistence of two competing synchronization patterns [25]. Chimeras were also realized in experiments involving electronic nonlinear oscillators with delay [26] and electrochemical oscillator systems [27,28].…”

Chimera states in coupled Kuramoto oscillators with inertia

“…This so-called chimera state is counter-intuitive as it appears even when the oscillators are identical, but, since its discover, this state has become a relevant subject of investigation for experimental and theoretical scientists active in different fields ranging from laser dynamics to chemical oscillators, from mechanical pendula to (computational) neuroscience. In particular chimera states have been shown to emerge in various numerical/theoretical studies [26][27][28][29][30][31][32][33][34][35] and in various experimental settings, including mechanical [36][37][38], (electro-)chemical [39][40][41] lasing systems [42,43] and BOLD fMRI signals detection during resting state activity [44], among others. Therefore chimera states are an ubiquitous phenomenon in nature much like synchronization itself and may often have been overlooked or dismissed in previous studies.…”