Weakly Nonlinear Theory for Oscillatory Dynamics in a One-Dimensional PDE-ODE Model of Membrane Dynamics Coupled by a Bulk Diffusion Field

TL;DR

This paper develops a weakly nonlinear theory for oscillatory dynamics in a coupled PDE-ODE system modeling membrane dynamics with bulk diffusion, analyzing bifurcations, synchronization, and chaos in biological and physical systems.

Contribution

It derives amplitude equations near bifurcation points for coupled PDE-ODE systems and applies them to membrane kinetics and Lorenz oscillators, revealing stability and chaos transitions.

Findings

Supercritical Hopf bifurcations occur with varying coupling and diffusivity.

Coupling can stabilize or destabilize oscillations, affecting bifurcation points.

Transition to synchronous chaos can be predicted via Lyapunov exponents.

Abstract

We study the dynamics of systems consisting of two spatially segregated ODE compartments coupled through a one-dimensional bulk diffusion field. For this coupled PDE-ODE system, we first employ a multi-scale asymptotic expansion to derive amplitude equations near codimension-one Hopf bifurcation points for both in-phase and anti-phase synchronization modes. The resulting normal form equations pertain to any vector nonlinearity restricted to the ODE compartments. In our first example, we apply our weakly nonlinear theory to a coupled PDE-ODE system with Sel'kov membrane kinetics, and show that the symmetric steady state undergoes supercritical Hopf bifurcations as the coupling strength and the diffusivity vary. We then consider the PDE diffusive coupling of two Lorenz oscillators. It is shown that this coupling mechanism can have a stabilizing effect, characterized by a significant increase in the Rayleigh number required for a Hopf bifurcation. Within the chaotic regime, we can distinguish between synchronous chaos, where both the left and right oscillators are in-phase, and chaotic states characterized by the absence of synchrony. Finally, we compute the largest Lyapunov exponent associated with a linearization around the synchronous manifold that only considers odd perturbations. This allows us to predict the transition to synchronous chaos as the coupling strength and the diffusivity increase.