The Existence Of Multiple Equilibrium Points In Global Attractors For Some Symmetric Dynamical Systems

The purpose of this doctoral paper is to study the geometry structure of the global attractor of a kind of symmetrical Infinite-Dimensional dynamical with Lyapunov function,and to study the existence of multiple equilibrium point in the global attractor.1st-Under hypothesis that an odd symmetrical continuous semigroup has a Lyapunov function,and0is local minimal point of the Lyapunov function on the absorbing neighborhood about origin.We prove that the global attractor has symmetrical structure and get the Theorem A1as follow:Theorem A1Assume that{S(t)}t≥0is a continuous semigroup on X and satisfies that (B1) S(t):X→X is odd for each t≥0;(B2){S(t)}t≥0has a C0Lyapunov even function F on X;(B3) there exists a subset M0of X and enjoys the following properties1. M0is compact symmetric subset of X, and S(t)M0=M0for all t≥0;2.0(?) M0and γ(M0)=n. Then the semigroup{S(t)}t≥0possesses at least n pairs of different equilibrium points in M0.Corollary Under the same hypothesis of above Theorem A,and moreover if{S(t)}t≥0possesses an global attractor (?), then dim(?)≥n.Then we take two kind of equations as examples, validate the existence of an absorbing neighborhood which center is origin,and point out that0is the local minimal value of the Lyapunov function on the neighborhood. As a conclusion,we get the Theorems as follow:Theorem A2:Let (?) is the global attractor of semigroup{S(t)}t≥0of (3.2).Then forβ, big enough,semigroup{S(t)}t≥0possesses at least n pairs different equilibrium points in (?).Theorem A3:Let μ<λ1. Denote (?) is the global attractor of semigroup {S(t)}t≥0of (3.11). β, Then for β big enough,{S(t)}t≥0possess at least2n pairs different equilibrium points in (?).As a corollary,we claim that dimension of the (?) of equations above is infinite.2nd-For an odd continuous semigroup with Lyapunov function,and when origin is not the local minimal point of the Lyapunov function on the absorbing neighborhood which center is origin,we get Theorem B1as follow:Theorem A'1:LetX be a Banach space,{S(t)}t≥0is a continuous semigroup on X and satisfied:(A) S(t):X→X is odd for (?)t≥0.(A2){S(t)}t≥0possesses a global attractor (?) on X.(A3){S(t)}t≥0possesses an even C0Lyapunov function F on X (A4) exist two closed subspace X+and X-of X satisfied:{A4-1) codimX+≤dimX-<∞, and X=X++X-(A4-2) exist α>0, ρ>0,such that F|x+∩(?)B(0,ρ)≥α,(A4-3) exist R and0<ρ<R,such that then,semigroup{S(t)}t≥0possesses at least dimX--codimX+pairs d-ifferent equilibrium points on (?)∩F-1((0,∞)).And then,by use of Theorem B1, we estimate the number of equilibrium points in the global attractor of the a kind of Reaction-Diffusion equations and get:Theorem B2: Let (?) is the global attractor of semigroup{S(t)}t≥0of (4.5). Then for β big enough, semigroup{S(t)}t≥0possess at least dimX-codimX+pair different equilibrium points on (?)∩F-1((0,∞)).