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SOE: Fachverband Physik sozio-ökonomischer Systeme

SOE 9: Symposium: Synchronization Patterns in Complex Dynamical Networks (organized by Jakub Sawicki, Sabine Klapp, Markus Bär and Jens Christian Claussen) (joint session DY/SOE)

SOE 9.4: Hauptvortrag

Freitag, 1. Oktober 2021, 15:30–16:00, ESS

A bridge between the fractal geometry of the Mandelbrot set and partially synchronized dynamics of chimera states. — •Ralph G Andrejzak — Universitat Pompeu Fabra, Barcelona, Catalonia, Spain

A simple quadratic map with a complex-valued parameter c allows one to generate enormously rich dynamics and patterns. Fractal Julia sets and the Mandelbrot set divide the complex plane into stable and divergent regions of the map's initial conditions and parameters c. What happens if one couples several quadratic maps? We address this question using a minimal two-population network of two pairs of two quadratic maps. In dependence on c, the network enters into qualitatively different dynamical states. The network iterates can diverge to infinity or remain bounded. Bounded solutions can get fully synchronized, fully desynchronized, or enter into different partially synchronized states, including a symmetry-broken chimera state. We will at first inspect examples for these different dynamical states in the domain of the complex-valued iterates of the network. We then illustrate that the boundaries between different dynamical states form intriguing fractal patterns in the domain of the complex-valued c.