Riddling: Chimera’s dilemma

Abstract

We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Hénon maps. It is known that coexisting basins of attraction lead to a hysteretic behaviour in the diagrams of the density of states as a function of a varying parameter. Chimera states, for which coherent and incoherent domains occur simultaneously, emerge as a consequence of the coexistence of basin of attractions for each state. Consequently, the distribution of chimera states can remain invariant by a parameter change, and it can also suffer subtle changes when one of the basins ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimera's dilemma is a consequence of the fractal and riddled nature of the basin boundaries.

Chimeras confined by fractal boundaries in the complex plane

“…In qualitative terms, the fractals generated by our network of coupled quadratic maps seem less filigree, more disordered, but even fuller of variety than the Mandelbrot set. Complementing previous work [21][22][23] , our study establishes an intriguing link between the dynamics of partially synchronized networks and the geometry of fractals in the complex plane.…”

“…21-23. These studies showed that the set of initial conditions leading to chimeras 21,22 or sub-chimeras 23 form riddled and fractal basins of attraction for networks of Rössler dynamics 22 , Henon maps 21 , logistic maps 23 , and sine-squared maps 23 (see also Ref. 56).…”

“…Chimera states are characterized by the coexistence of synchronization and desynchronization in networks of coupled dynamics 19,20 . The set of initial conditions leading to chimera states can form riddled and fractal basins of attraction [21][22][23] . Initial conditions outside of these basins of attraction can result in fully synchronized or fully desynchronized dynamics, among others.…”

“…While chimera states were first described 19 and continue to be studied for networks of time-continuous oscillators, an increasing interest is paid to networks of time-discrete maps. 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 .…”

“…The boundaries between these different states in the domain of the complex-valued c form fractal patterns. In contrast to the high-dimensional fractal basins of attraction [21][22][23] , these fractals reside in the complex plane and are therefore straightforward to visualize. In qualitative terms, the fractals generated by our network of coupled quadratic maps seem less filigree, more disordered, but even fuller of variety than the Mandelbrot set.…”