We propose a new rigorous numerical technique to prove the existence of symmetric homoclinic orbits in reversible dynamical systems. The essential idea is to calculate Melnikov functions by the exponential dichotomy and the rigorous numerics. The algorithm of our method is explained in detail by dividing into four steps. An application to a two dimensional reversible system is also treated and the existence of a symmetric homoclinic orbit is rigorously verified as an example.

We show that for a generic polynomial $H=H(x,y)$ and an arbitrary differential 1-form $\omega =P(x,y)dx+Q(x,y)dy$ with polynomial coefficients of degree $\le d$, the number of ovals of the foliation $H=\mathrm{const}$, which yield the zero value of the complete Abelian integral $I\left(t\right)={\oint }_{H=t}\omega$, grows at most as $exp{O}_H\left(d\right)$ as $d\to \infty$, where $O_H\left(d\right)$ depends only on $H$. The main result of the paper is derived from the following more general theorem on bounds for isolated zeros occurring in polynomial envelopes of linear differential equations. Let $f_1\left(t\right),\cdots ,f_n\left(t\right)$, $t\in K⋐ℝ$, be a fundamental system of real solutions...

We establish a polynomial normal form for a vector field having a limit cycle of multiplicity 2. The smooth classification problem for such fields is closely related to the problem of classification of germs $\Delta :({ℝ}^1,0)\to ({ℝ}^1,0)$, $\Delta \left(x\right)=x+c{x}^2+\cdots$, solved by F. Takens in 1973. Such germs appear as the germs of Poincaré return maps for semistable cycles, and a smooth conjugacy between any two such germs may be extended to a smooth orbital equivalence between the original fields.If one deals with smooth conjugacy of flows rather than...

Let $ℰ$ be a disjoint decomposition of ${ℝ}^n$ and let $X$ be a vector field on ${ℝ}^n$, defined to be linear on each cell of the decomposition $ℰ$. Under some natural assumptions, we show how to associate a semiflow to $X$ and prove that such semiflow belongs to the o-minimal structure ${ℝ}_{\mathrm{an},exp}$. In particular, when $X$ is a continuous vector field and $\Gamma$ is an invariant subset of $X$, our result implies that if $\Gamma$ is non-spiralling then the Poincaré first return map associated $\Gamma$ is also in ${ℝ}_{\mathrm{an},exp}$.

In this paper, the feedback control for a class of bilinear control systems with a small parameter is proposed to guarantee the existence of limit cycle. We use the perturbation method of seeking in approximate solution as a finite Taylor expansion of the exact solution. This perturbation method is to exploit the “smallness” of the perturbation parameter $\epsilon$ to construct an approximate periodic solution. Furthermore, some simulation results are given to illustrate the existence of a limit cycle for...

The period function of a planar parameter-depending Hamiltonian system is examined. It is proved that, depending on the value of the parameter, it is either monotone or has exactly one critical point.