Quantifying the transition from spiral waves to spiral wave chimeras in a lattice of self-sustained oscillators. (English) Zbl 1480.34053
Summary: The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the network under consideration are the van der Pol oscillator or the FitzHugh-Nagumo neuron. Both of the models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity, which enables us to evaluate the sensitivity of each oscillator at a finite time. Spiral waves are observed in both lattices when the interaction between elements has a local character. The dynamics of all the elements is regular. There are no pronounced high-sensitive regions. We have discovered that, when the coupling becomes nonlocal, the features of the system change significantly. The oscillation regime of the spiral wave center element switches to a chaotic one. Besides, a region with high sensitivity occurs around the wave center oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. A sharp increase in the values of the maximal Lyapunov exponent in the positive region leads to the formation of the incoherence cluster. Furthermore, we find that the system can even switch to a hyperchaotic regime when several Lyapunov exponents become positive.
MSC:
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34C26 | Relaxation oscillations for ordinary differential equations |
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
| 34D08 | Characteristic and Lyapunov exponents of ordinary differential equations |
Keywords:
spatiotemporal pattern; chimera state; van der Pol oscillator; FitzHugh-Nagumo neuron; spiral wave; spiral wave chimera; nonlocal interaction; Lyapunov exponentReferences:
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