On the accumulation of separatrices by invariant circles

Let f be a smooth symplectic diffeomorphism of ${\mathbb R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if f is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

Quantitative destruction of invariant circles

<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id="M3">\begin{document}$ R_N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \|R_N\|_{C^r}&lt;\epsilon $\end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id="M8">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id="M9">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>

Invariant circles and depinning transition

We associate the existence or non-existence of rotational invariant circles of an area-preserving twist map on the cylinder with a physically motivated quantity, the depinning force, which is a critical value in the depinning transition. Assume that $H:\mathbb{R}^{2}\mapsto \mathbb{R}$ is a $C^{2}$ generating function of an exact area-preserving twist map $\bar{\unicode[STIX]{x1D711}}$ and consider the tilted Frenkel–Kontorova (FK) model: $$\begin{eqnarray}{\dot{x}}_{n}=-D_{1}H(x_{n},x_{n+1})-D_{2}H(x_{n-1},x_{n})+F,\quad n\in \mathbb{Z},\end{eqnarray}$$ where $F\geq 0$ is the driving force. The depinning force is the critical value $F_{d}(\unicode[STIX]{x1D714})$ depending on the mean spacing $\unicode[STIX]{x1D714}$ of particles, above which the tilted FK model is sliding, and below which the particles are pinned. We prove that there exists an invariant circle with irrational rotation number $\unicode[STIX]{x1D714}$ for $\bar{\unicode[STIX]{x1D711}}$ if and only if $F_{d}(\unicode[STIX]{x1D714})=0$. For rational $\unicode[STIX]{x1D714}$, $F_{d}(\unicode[STIX]{x1D714})=0$ is equivalent to the existence of an invariant circle on which $\bar{\unicode[STIX]{x1D711}}$ is topologically conjugate to the rational rotation with rotation number $\unicode[STIX]{x1D714}$. Such conclusions were claimed much earlier by Aubry et al. We also show that the depinning force $F_{d}(\unicode[STIX]{x1D714})$ is continuous at irrational $\unicode[STIX]{x1D714}$.

Dynamical Bifurcation of the Generalized Swift–Hohenberg Equation

In this paper, we prove that the generalized Swift–Hohenberg equation bifurcates from the trivial states to an attractor as the control parameter α passes through critical points. The bifurcation is divided into two groups according to the dimension of the center manifolds. We show that the bifurcated attractor is homeomorphic to S1 or S3 and it contains invariant circles of static solutions. We provide a criterion on the quadratic instability parameter μ which determines the bifurcation to be supercritical or subcritical.