Instability of homogeneous state and two-domain spatiotemporal structures in reaction-diffusion systems with global coupling

V. A. Kostin ; G. V. Osipov

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 29 (2021) no. 1, pp. 186-207.

Résumé

The purpose of this work was to study the typical instability of a homogeneous state resulting in two-domain spatiotemporal patterns in reaction-diffusion systems with global coupling. Methods. The linear stage of instability was analyzed based on the method of separation of variables for a one-dimensional two-component system of general form on a finite interval with Neumann boundary conditions. The development of instability at the nonlinear stage was simulated numerically using the method of lines for specific systems. Results. It was shown that the introduction of a global coupling can lead to a loss of stability of initially stable homogeneous states. The instability criteria are determined for the two-component systems in general case. A case is singled out when, even in long media, the spatial mode with a wavelength equal to twice the size of the system has the largest growth rate, which can lead to the formation of distinctive two-domain patterns as a result of the instability developing at the nonlinear stage. In this case, the interdomain boundary can both be stationary or oscillate, and the corresponding dynamical regimes can be interpreted as trigger waves with zero or alternating velocity. This interpretation made it possible to analytically estimate the steady-state sizes of domains in the distributed FitzHugh-Nagumo system, as well as to construct simple examples of systems in which the interdomain boundary oscillates harmonically with arbitrary amplitude or chaotically in way similar to the motion of the Rossler system. Conclusion. The investigated instability of a homogeneous state exists in a wide range of systems and differs from the well-known diffusion-driven instabilities (in particular, the Turing instability), where the spatial scale of growing disturbances in the long-medium limit is determined exclusively by the local properties of the system, but not by its dimensions.

Détail

Keywords: reaction-diffusion systems, instability of homogeneous state, Turing instability, wave instability, global instability, FitzHugh-Nagumo system, Rossler system, trigger waves, Zeldovich-Frank-Kamenetskii equation.