Research On Chaotic Dynamics Of Non-smooth Maps

In recent years there has been a growing interest in the studying of non-smooth dynamical systems both in the mathematics and engineering fields. Many dynamical systems arising in applications are non-smooth, such as vibro-impact systems, stick-slip oscillators with dry frictions, switching circuits and certain control systems, etc. On the other hand, the global Poincare mmaps of many smooth systems is non-smooth,Non-smooth systems can appear new forms of bifurcations which are not found in smooth systems, such as grazing bifurcation and sliding bifurcation. Moreover, these bifurcations provide new routes toward to chaotic motions.In Chapter 1, we survey some recent achievements, developments and open problems of the non-smooth dynamics. Moreover, we introduce the main results of this dissertation.In Chapter 2, we recall some basic notions and results of the dynamical sys-tems and ergodic theory, including Birkhoff's ergodic theorem, measure theoretic entropy, Sinale horseshoes and so on, which will be used in what follows.In Chapter 3, the statistical properties of an interval map having a square-root singular point which characterizes grazing bifurcations of impact oscillators are studied. We show that in some parameter region the map admits an induced Markov structure with exponential decay tail of the return time function. Then we prove that the map has a mixing absolutely continuous invariant probability measure. By applying the Markov tower method we prove that exponential decay of correlations and the central limit theorem hold for Holder continuous observations.In Chapter 4, we investigate the chaotic dynamics of a two-dimensional non-smooth map. The map represents the normal form of a discrete time repre-sentation of impact oscillators near grazing orbits. It is proved that, in certain region of the parameter space, the non-wandering set of the map is contained in a square region and that, restricted to the non-wandering set, the map is topologically conjugate to the shift maps on the two sided symbolic space.In Chapter 5, we investigate the symbolic dynamics of a family of Belykh-type maps (a family of two-dimensional discontinuous piecewise linear maps).The admissibility condition for symbol sequences named the pruning front con-jecture is proved under a hyperbolicity condition. Using this result, a symbolic dynamics model of the map is constructed according to its pruning front and primary pruned region. Moreover, the set of the parameters for which the map is chaotic of horseshoe type is given.In Chapter 6, we consider also the family of Belykh-type maps studied in Chapter 5. We compute the Hausdorff dimension of the strange attractor of the map. First, we construct a trapping region of the map. Since the unstable manifolds of the hyperbolic fixed points are contained in the trapping region,the map has a strange attractor. Then we determine a parameter set for which the map has SRB measures. By estimate the capacity of the attractor, we give a upper bound of its Hausdorff dimension. By applying Young's theorem on Hausdorff dimension and Pesin's entropy formula, we can obtain a lower bound of the Hausdorff dimension of the attractor. We find that the lower bound and the upper bound are equal, so we obtain the precise formula for the Hausdorff dimension of the attractor.