Lyapunov functions and isolating blocks

摘要: Let M denote a smooth (Cm) n-dimensional manifold and let $: x R -+ Cr flow (or dynamical system) which is generated by CT vector field 4 (= (d/dt) +(t, x) j t = 0) on (0 max(1, r}. In this case, we can only obtain Ck smoothness for functions M. K compact invariant set ‘p, i.e., subset of if E K, then 9(x, t) all R. Two topological tools have been used to describe the behavior p near are “Lyapunov functions” “isolating blocks.” A Lyapunov function real-valued (V) defined neighborhood whose derivatives r v in direction special properties. We shall be especially interested cases monotone Lyu$unov (77 strictly decreasing every trajectory complement K) hy~erbolt’c