SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1520-1545, January 2020.
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 multiscale asymptotic expansion to derive amplitude equations near codimension-one Hopf bifurcation points for both in-phase and antiphase 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.

中文翻译：

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一维PDE-ODE膜动力学与体扩散场耦合的振动动力学的弱非线性理论

SIAM应用数学杂志，第80卷，第3期，第1520-1545页，2020年1月。
我们研究由一维整体扩散场耦合的两个空间隔离的ODE隔室组成的系统的动力学。对于这种耦合的PDE-ODE系统，我们首先采用多尺度渐近展开来导出同相和反相同步模式的共维一Hopf分叉点附近的振幅方程。所得的法线形式方程与限制在ODE格中的任何矢量非线性有关。在我们的第一个示例中，我们将弱非线性理论应用于具有Sel'kov膜动力学的PDE-ODE耦合系统，并表明随着耦合强度和扩散系数的变化，对称稳态会经历超临界Hopf分叉。然后，我们考虑两个Lorenz振荡器的PDE扩散耦合。已经表明，该耦合机构可以具有稳定作用，其特征在于霍普夫分叉所需的瑞利数显着增加。在混沌状态下，我们可以区分同步混沌和左右同步状态，在同步混沌中，左右振荡器都是同相的；在混沌状态中，没有同步。最后，我们计算了与仅考虑奇数摄动的同步流形周围的线性化相关的最大Lyapunov指数。这使我们能够预测随着耦合强度和扩散率增加，向同步混沌的过渡。以及缺乏同步的混乱状态。最后，我们计算了与仅考虑奇数摄动的同步流形周围的线性化相关的最大Lyapunov指数。这使我们能够预测随着耦合强度和扩散率增加，向同步混沌的过渡。以及缺乏同步的混乱状态。最后，我们计算了与仅考虑奇数摄动的同步流形周围的线性化相关的最大Lyapunov指数。这使我们能够预测随着耦合强度和扩散率增加，向同步混沌的过渡。