Let be a disjoint decomposition of and let be a vector field on , defined to be linear on each cell of the decomposition . Under some natural assumptions, we show how to associate a semiflow to and prove that such semiflow belongs to the o-minimal structure . In particular, when is a continuous vector field and is an invariant subset of , our result implies that if is non-spiralling then the Poincaré first return map associated is also in .
We prove that every diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.
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 to construct an approximate periodic solution. Furthermore, some simulation results are given to illustrate the existence of a limit cycle for...
Given a real analytic vector field tangent to a hypersurface with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra associated with the singularity of the vector field on . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...
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.