Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Abstract

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case where the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

“…The present framework can also be extended to such situations and can be used to derive the optimal coupling schemes between two coupled spatiotemporal oscillations. A study in this direction is reported in our forthcoming article [51], where improvement in the stability of synchronized states between reaction-diffusion systems by introducing linear spatial filters into mutual coupling is considered.…”

“…We seek for the optimal K opt that gives the maximum stability of the synchronized state, and compare the results for K opt with those for the identity coupling, i.e., K I given by Eq. (51). The overall intensity P is fixed at P = 0.1 in the following.…”

Synchronization results for a class of fractional-order spatiotemporal partial differential systems based on fractional Lyapunov approach

“…Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals. Synchronization dynamics of reaction-diffusion systems has been studied in [21,22] using phase reduction theory. It has been shown that reaction-diffusion systems can exhibit synchronization in a similar way to low-dimensional oscillators.…”

Phase reduction of limit-torus solutions to partial differential algebraic equations

“…We recently demonstrated a phase reduction method for limit-cycle solutions to the following partial differential equations: (i) nonlinear Fokker-Planck equations representing collective oscillations of globally coupled noisy dynamical elements [20,21], (ii) fluid equations representing oscillatory thermal convection in ordinary Hele-Shaw cells [22,23], (iii) reaction-diffusion equations representing spatiotemporal rhythms in chemical and biological systems [24,25], and (iv) fourth-order nonlinear partial differential equations representing beating flagella [26]; more precisely, the first and second equations are partial differential integral equations. This method can be considered as a generalization of the conventional phase reduction method for ordinary differential equations.…”