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

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