For a generalized pendulum equation we estimate the number of periodic solutions from below using lower and upper solutions and from above using a complex equation and Jensen’s inequality.
For an arbitrary analytic system which has a linear center at the origin we compute recursively all its Poincare-Lyapunov constants in terms of the coefficients of the system, giving an answer to the classical center problem. We also compute the coefficients of the Poincare series in terms of the same coefficients. The algorithm for these computations has an easy implementation. Our method does not need the computation of any definite or indefinite integral. We apply the algorithm to some polynomial...
In the paper we will be concerned with the topological structure of the set of solutions of the initial value problem of a semilinear multi-valued system on a closed and convex set. Assuming that the linear part of the system generates a -semigroup we show the -structure of this set under certain natural boundary conditions. Using this result we obtain several criteria for the existence of periodic solutions for the semilinear system. As an application the problem of controlled heat transfer...
We prove that the solutions of a sweeping process make up an -set under the following assumptions: the moving set C(t) has a lipschitzian retraction and, in the neighbourhood of each point x of its boundary, it can be seen as the epigraph of a lipschitzian function, in such a way that the diameter of the neighbourhood and the related Lipschitz constant do not depend on x and t. An application to the existence of periodic solutions is given.
We give a clear and systematic exposition of one-parameter families of brake orbits in dynamical systems on product vector bundles (where the fiber has the same dimension as the base manifold). A generalized definition of a brake orbit is given, and the relationship between brake orbits and periodic orbits is discussed. The brake equation, which implicitly encodes information about the brake orbits of a dynamical system, is defined. Using the brake equation, a one-parameter family of brake orbits...