Transition to Chaos: From Small to Large Systems in the Nikolaevskiy Model

Abstract

We investigate the dynamics and the transitions to spatiotemporal chaos observed in a partial differential equation known as Nikolaevskiy equation in the regime of small domains while applying periodic boundary conditions. In contrast to generic chaotic solutions in large domains called soft-mode turbulence, the Nikolaevskiy model exhibits a rich variety of different chaotic and nonchaotic dynamics if the considered domain size is constrained to only a few characteristic wavelengths. Extending the work by Tanaka, we provide (i) an in-depth numerical analysis including maps of solution types for several parameter subspaces and (ii) results from the numerical continuation of selected types of regular dynamics. Doing this, we detect and classify the highly elaborate scenario of different transitions from regular dynamics to chaos that occur if the system size is varied. Due to the model’s simplicity, we expect those results to be adaptable in similar partial differential equations.