Chaotic Dynamics and Bifurcations in Impact Systems

Bifurcations of dynamical systems described by several second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set. DOI: 10.4018/ijeoe.2012100102 16 International Journal of Energy Optimization and Engineering, 1(4), 15-37, October-December 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. the method, which allows finding homoclinic points, corresponding to grazing. The main idea of the proof is the nonsmoothness of Perron surfaces in the neighborhood of periodic solution. If these manifolds bend in a “good” way (the corresponding sufficient conditions can be written down explicitly) they can intersect. This implies chaos. We study a motion of a point mass, described by system of second order differential equations of the general form and impact conditions of Newtonian type. Unlike the similar author’s paper Kryzhevich (2008) here we study the systems with several degrees of freedom. For these systems a near-grazing periodic point is not automatically hyperbolic, so we need to provide additional conditions to have a classical Smale horseshoe. In order to avoid technical troubles we assume that the delimiter is plain, immobile and slippery. However, there are no obstacles to apply the offered method to the systems with a mobile delimiter (e.g., Holmes, 1982), ones with non-Newtonian model of impacts (Babitsky, 1998; Fredriksson & Nordmark, 1997; Ivanov, 1996; Kozlov & Treshev, 1991) and even to some special cases of strongly nonlinear dynamical systems without any impact conditions. The paper is organized as follows. First, we consider the mathematical model of vibroimpact systems and then define the grazing family of periodic solution. Afterwards, the near-grazing behavior of solutions is described. Next, the main result of the paper is presented and the near-impact behavior of solutions is studied and estimates of Lyapunov exponents are given. In the next section an analogue of the Smale horseshoe was constructed and an analogue of the Smale-Birkhof theorem is proved. An example, illustrating the main result, is considered afterwards. We then discuss some practical applications and experiments and simulations, related to results of the paper and the results, mentioned after are not original; we need them to provide an experimental justification of Theorem 1 that is the main result of the current paper.