Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

TL;DR

This paper investigates how competing instabilities in coupled nonlinear equations lead to complex spatio-temporal patterns, including coexistence, traveling waves, and chaos, through linear stability analysis and numerical simulations.

Contribution

It introduces two models with asymmetrically coupled Swift-Hohenberg and Cahn-Hilliard equations, revealing new pattern formation phenomena and bifurcation structures.

Findings

Identified co-dimension two bifurcation points for Turing and wave instabilities.

Discovered coexistence of stationary and traveling patterns in the models.

Showed that weak coupling arrests coarsening, leading to periodic patterns.

Abstract

Pattern formation often occurs in spatially extended physical, biological and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular revealing for the Swift-Hohenberg equations a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of an weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.