反应扩散系统中基于不变流形和全局分支的多重行波解案例研究

Hamiltonian pitchfork bifurcation in transition across index-1 saddles

We study the effect of changes in the parameters of a two-dimensional potential energy surface on the phase space structures relevant for chemical reaction dynamics. The changes in the potential energy are representative of chemical reactions such as isomerization between two structural conformations or dissociation of a molecule with an intermediate. We present a two degrees of freedom quartic Hamiltonian that shows pitchfork bifurcation when the parameters are varied and we derive the bifurcation criteria relating the parameters. Next, we describe the phase space structures - unstable period…

Codimension-two Bifurcations Induce Hysteresis Behavior and Multistabilities in Delay-coupled Kuramoto Oscillators

Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the…

Reaction rates and the noisy saddle-node bifurcation: Renormalization group for barrier crossing

Barrier crossing calculations in chemical reaction-rate theory typically assume that the barrier is large compared to the temperature. When the barrier vanishes, however, there is a qualitative change in behavior. Instead of crossing a barrier, particles slide down a sloping potential. We formulate a renormalization group description of this noisy saddle-node transition. We derive the universal scaling behavior and corrections to scaling for the mean escape time in overdamped systems with arbitrary barrier height. We also develop an accurate approximation to the full distribution of barrier es…

Invariant manifolds of the Bonhoeffer-van der Pol oscillator

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.

Amplitude Equations and Chemical Reaction-Diffusion Systems

The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction-diffusion system based on an Oregonator model of the Belousov-Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction-diffusion equation can be approximated by a complex Ginzburg-Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction-diffusion equation with the corresponding solutio…

Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward…

Bifurcations of Normally Hyperbolic Invariant Manifolds and Consequences for Reaction Dynamics

In this paper we study the breakdown of normal hyperbolicity and its consequences for reaction dynamics; in particular, the dividing surface, the flux through the dividing surface (DS), and the gap time distribution. Our approach is to study these questions using simple, two degree-of-freedom Hamiltonian models where calculations for the different geometrical and dynamical quantities can be carried out exactly. For our examples, we show that resonances within the normally hyperbolic invariant manifold may, or may not, lead to a `loss of normal hyperbolicity'. Moreover, we show that the onset o…

Center manifold reduction for large populations of globally coupled phase oscillators

A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is $\sin θ$, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin θ$, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.

The Phase Space Mechanism for Selectivity in a Symmetric Potential Energy Surface with a Post-Transition-State Bifurcation

Chemical selectivity is a phenomenon displayed by potential energy surfaces (PES) that is relevant for many organic chemical reactions whose PES feature a valley-ridge inflection point (VRI) in the region between two sequential index-1 saddles. In this letter we describe the underlying dynamical phase space mechanism that qualitatively determines the product distributions resulting from bifurcating reaction pathways. We show that selectivity is a consequence of the heteroclinic and homoclinic connections established between the invariant manifolds of the families of unstable periodic orbits (U…

阻尼驱动双原子颗粒晶体中的全局分岔

Global Bifurcations in a Damped-Driven Diatomic Granular Crystal

本文研究了阻尼驱动二聚体颗粒晶体的动力学，通过识别鞍周期解的不稳定流形，揭示了同宿缠结和倍周期分岔导致混沌吸引器出现的机制，为高维实验系统中混沌的产生提供了更具体的见解。

不稳定性分岔场景的粒子在非对称双矮星系中的情况

The NHIM bifurcation scenario of a particle in an asymmetric binary system of dwarf galaxies

研究了非对称双矮星系中粒子的分岔场景，揭示了NHIMs的双曲性丧失与KAM环破裂导致的瞬态混沌动力学及分岔场景间的协调性。