论文标题
折叠分叉的小振幅周期波的不稳定性
Instability of small-amplitude periodic waves from fold-Hopf bifurcation
论文作者
论文摘要
我们研究了一个反应扩散方程的系统中，与一个普通的微分方程相连的系统中，小振幅周期性波从折叠 - hopf平衡中出现的小振幅周期性波的存在和稳定性。该耦合系统包括Fitzhugh-Nagumo系统，漫画钙模型，消费者资源模型和其他应用程序中的其他模型。基于关于平均理论的最新结果，我们在相关的三维系统中求解了周期性解决方案，然后证明了折叠式分叉产生的周期性波的存在。 [J. Tsai，W。Zhang，V。Kirk和J. Sneyd，Siam J. Appl。 dyn。系统。 11（2012），1149--1199]曾经提出，讽刺钙模型中折叠式分叉的周期性波在光谱上是不稳定的，但没有证据。在通过相对有界的扰动分析有关周期性波的线性化后，我们通过对折叠 - hopf equilibria的线性化的不稳定光谱的扰动证明了小振幅周期性的不稳定性。作为一种应用，我们证明了带有施加电流的Fitzhugh-Nagumo系统中的小振幅周期性周期波的存在和稳定性。
We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction-diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh-Nagumo system, caricature calcium models, consumer-resource models and other models in the real-world applications. Based on the recent results on the averaging theory, we solve periodic solutions in related three-dimensional systems and then prove the existence of periodic waves arising from fold-Hopf bifurcations. Numerical computation in [J. Tsai, W. Zhang, V. Kirk, and J. Sneyd, SIAM J. Appl. Dyn. Syst. 11 (2012), 1149--1199] once suggested that the periodic waves from fold-Hopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of small-amplitude periodic waves through a perturbation of the unstable spectra for the linearizations about the fold-Hopf equilibria. As an application, we prove the existence and stability of small-amplitude periodic waves from fold-Hopf bifurcations in the FitzHugh-Nagumo system with an applied current.
