By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.

中文翻译：

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具有双饱和项的激活剂-底物模型的时空动力学和模式形成

本文利用中心流形理论、Poincaré-Bendixson定理、线性算子的时空谱和色散关系，研究了齐次Neumann边界条件下具有双饱和项的活化剂-基质模型的时空动力学。令人惊讶的是，该系统可以引发新的动力学，例如亚临界 Hopf 分岔和两个极限环的共存。此外，平衡中的图灵不稳定性主要产生条纹图案，而齐次周期解主要产生斑点图案或斑点条纹图案，其图案形成与理论结果非常一致。有趣的是，图灵不稳定性可以在亚临界 Hopf 分岔中同时产生平衡解和周期解，这是扩散驱动不稳定性的新发现。事实上，这些理论方法也适用于在一维空间中寻找其他模型的模式。