Chaos & Oscilations II

In the Kuramoto model of infinitely many globally coupled phase oscillators with different frequencies, beyond the synchronization transition the oscillators separate neatly into two groups: an ordered one locked to the mean field, and a disordered one rotating at different frequencies. We show that this picture, valid in the thermodynamic limit of infinite populations, is not exact for finite-sized ensembles, where the mean field fluctuates due to the finite-size effects. We demonstrate that these fluctuations lead to cross-correlations in the disordered group on a microscopic scale.

The Belousov Zhabotinsky reaction, a self-organized oscillatory color-changing reaction, can show complex behavior when left unstirred in a closed cuvette environment. The most intriguing behavior is the transition from periodicity to chaos and back to periodicity as the system evolves in time. It was discovered that this complex behavior is due to the decoupling of reaction, diffusion and convection [1]. We have recently discovered that, as the so-called chaotic transient takes place, periodic bulk motions like convective rolls are visible in the reaction solution.

We discuss the occurrence of different kinds of chimera states such as transient amplitude chimera, stable amplitude chimera and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibit oscillations with large amplitudes while the desynchronized group of oscillators oscillate with small amplitudes and this behavior of coexistence of synchronized and desynchronized oscillations fluctuate with time.

For generic non-linear dynamical systems, we develop a theory to show that the exact edge of chaos is between the periodic cycle phase due to Neimark-Sacker bifurcation and the chaotic phase. The asymptotic Jacobian norm determines the location of the edge, and such theoretical result can be extrapolated to simple systems like the logistic map.