| The invariant sets of an underlying dynamical system are always part of its global attractors. So the global attractor has a lot of interesting dynamical structure, and specially, it contains every (global) unstable manifold. It is universally acknowledged that the unstable manifolds of the invariant sets have a crucial influence of the complexity of the dynamical behaviors of the underlying dynamical system. Such as the transverse intersection of an unstable manifold with a stable manifold leads to the existence of complicated dynamical behavior close to this intersection and many other cases. Considering this, it is quite necessary to develop a numerical algorithm for approximating global attractors. In the literature cite [1] written by Michael Dellnitz, and Adreas Hohmann in 1997, a new subdivision algorithm for calculating global attractors has been established. This algorithm can effectively approximate the unstable manifolds and the global attractor of the underlying system. Different with the local method applied by many algorithm for calculating unstable manifolds in the past, which starts form a hyperbolic periodic point and then computes part of the unstable manifold by some sort of continuation procedure, the new subdivision algorithm mentioned in the literature cite [1] proposes a global approach by using an adaptive subdivision process.More precisely, suppose that the dynamical system is defined on Rn. We start by specifying a box in Rn in which we want to analyze the dynamical behavior and then repeatedly subdivide the area and delete the boxes which don't contain the global attractor to continue the algorithm. By this method, the algorithm provides in each step a covering of the relative global attractor under consideration. This covering, in the sense of decreasing Hausdorff distance, becomes more and more accurate to the global attractor in the course of iteration. Moreover, assume that the relative global attractor is a hyperbolic set which has a local product structure we can even obtain an error estimate depending on the quotient of the contraction rate and the speed of refinement. We avoid several problems which we may encounter when using old algorithms by approximating (global) unstable manifolds as sets (using a global approach) rather than as objects which consist of single trajectories.The major process for calculating unstable manifolds and global attractors by subdivision algorithm is as follows:The algorithm generates a sequence B0,B1,..., of finite collections of compact subsets of Rn such that the diameterconverges to zero for k→∞.Given an initial collection B0, we inductively obtain Bk from Bk-1,k=1,2,... for in two steps.1. Subdivision: Construct a new collection (?)k such atand2. Selection: Define the new collection Bk byIn the first three chapters, based on the result of literature cited [1], we provided detailed descriptions about the background, construction consideration,calculation process, convergence proof, and the detailed application skill of the subdivision algorithm. We demonstrated that the subdivision algorithm is indeed an effective numerical method to approximate the unstable manifolds and the global attractors of the dynamical system. On the other side, considering the importance of the piecewise smooth dynamical system in the real engineering, such as the fact that many describing equations of the dynamics, electrical engineering, autonomous control theory were involved with such piecewise system. It is also quite important for both theoretical research and real value to develop a numeral method for approximating unstable manifolds and global attractors of the piecewise smooth dynamical system.Considering this, in the fourth chapter of the article, we trying to apply the subdivision algorithm to the piecewise smooth dynamical system, this is also the major part of this article. First by applying the algorithm to Henon mapping under continuous condition, and comparing the results of both numerical experiments of this article and literature cite [1], we proved that the effectiveness of the algorithm and the correctness of the numerical realization of the algorithm in this article. In the following sections, we built several piecewise smooth Henon-like dynamical system, and again proved that the algorithm can effectively approximate the unstable manifolds and the global attractors of the piecewise smooth system.For further broadening the application space of the subdivision algorithm,we continue to raise the possibility that we can also approximate the stable manifold of dynamical systems by using the algorithm. In brief, by serious research of the results of numerical realization, it is cleared that the subdivision algorithm is truly effective, and can apply to most of both continuous and piecewise smooth dynamical systems. It also can approximate both stable and unstable manifolds of the dynamical system, which can generate great practical value to the research work of bifurcation problems and homoclinic orbit problems. Finally, we showed some of the possible flaws or deficiencies of the algorithm under particular situations and have given some possible suggestions for further promotions.
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