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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., Comment: 20 pages, 12 figures

This article describes a reduction of a nonautonomous Brusselator reaction-diffusion system of partial differential equations on a spherical cap with time dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this nonautonomous normal form. The coefficients of such a normal form are computed and the reduction solutions are compared to numerical solutions., Comment: 25 pages, 9 figures

On a two-dimensional circular domain, we analyze the formation of spatio-temporal patterns for a class of coupled bulk-surface reaction-diffusion models for which a passive diffusion process occurring in the interior bulk domain is linearly coupled to a nonlinear reaction-diffusion process on the domain boundary. For this coupled PDE system we construct a radially symmetric steady state solution and from a linearized stability analysis formulate criteria for which this base state can undergo either a Hopf bifurcation, a symmetry-breaking pitchfork (or Turing) bifurcation, or a codimension-two pitchfork-Hopf bifurcation. For each of these three types of bifurcations, a multiple time-scale asymptotic analysis is used to derive normal form amplitude equations characterizing the local branching behavior of spatio-temporal patterns in the weakly nonlinear regime. Among the novel aspects of this weakly nonlinear analysis are the two-dimensionality of the bulk domain, the systematic treatment of arbitrary reaction kinetics restricted to the boundary, the bifurcation parameters which arise in the boundary conditions, and the underlying spectral problem where both the differential operator and the boundary conditions involve the eigenvalue parameter. The normal form theory is illustrated for both Schnakenberg and Brusselator reaction kinetics, and the weakly nonlinear results are favorably compared with numerical bifurcation results and results from time-dependent PDE simulations of the coupled bulk-surface system. Overall, the results show the existence of either subcritical or supercritical Hopf and symmetry-breaking bifurcations, and mixed-mode oscillations characteristic of codimension-two bifurcations. Finally, the formation of global structures such as large amplitude rotating waves is briefly explored through PDE numerical simulations., Comment: 40 pages and 39 figures