Quasi-effective Stability For Nearly Twist Mappings

KAM theory [1,4,5] is one of the most achievement in the 20th century. KAM methods not only can be applied into the existence of invariant tori for Hamiltonian sys-tems[4-12], generalized Hamiltonian systems[13-16], and infinite dimensional Hamil-tonian systems[17-20], but also can be applied into reducing quasi-periodic linear sys-tems[21-30] and the problem of volume-preserving mappings[31-36].Many problems in mathematical physics can be reduced to some dynamical sys-tems of volume-preserving mappings [8,37-39]. As all well known, Moser laid the foundation of the KAM type theorem for volume-preserving mappings [1]. This out-standing work leads to some further developments [2,33,34,40,41].Moser [1] considered a kind of mappings in the end of his celebrated paper as follows where (r,θ)∈[a, b] x R, and F and G are smooth and 2π-periodic functions in θ, and γ is a small parameter. Moser proved that (1) survive a family of invariant curves under the condition dα/dr> 0 for sufficiently small γ.In this thesis, we have explored connection between the KAM tori, nearly invariant tori and the effective stability, and finally we proved that the nearly integrable volume-preserving mapping possesses quasi-effective stability under the classical KAM-type nondegeneracy, that is, there is an open subset of the phase space whose measure is nearly full, such that the considered mapping is effective stable on this subset.We considered the following nearly integrable volume-preserving mapping (x, y)= (?)(x,y), small parameter.For the above nearly integrable volume-preserving mapping, we achieved the fol-lowing theorem,Assume (H1)f, g and ω are real analytic andon (Tn×(a, b))+δ for some positive constant M. Theorem 1. Assume that (H1) holds and ω(y) satisfies Then, there exist positive constants β, γ, η,(?) and ∈0 such that, for any ∈∈ (0, ∈0], there is an open subset E∈ of (a, b) suiting the following satisfies the estimateA mapping (?) is said to be quasi-effective stable if it satisfies the conclusions (i) and (ii) in Theorem 2.1. Similarly, β and γ are called stable exponents of the mapping, T(∈)= exp(η∈-β) stable time; R(∈)=η∈γ stable radius.In Nekhoroshev’s theorem, the effective stability of volume-preserving mapping relies on the steepness of the integrable system, but in Theorem 2.1, we get quasi-effective stability without using the steepness condition.It directly follows from the above definitions that effective stability implies quasi-effective stability.From Theorem 1, This shows that, under the classical KAM-type nondegeneracy, the nearly integrable volume-preserving mapping possesses effective stability on the open subset of the phase space whose measure is nearly full. By the constructive principle of open set in R, there are at most countable open subinterval Ii o f (a, b),1≤i≤T, T∈N∪{oo}, satisfying such that (?) is effective stable in each Tn×Ii.For a mapping with multiple action variables, we have the following definition about its orbits, Defination 1. The mapping is said to be quasi-effective stable, if there exists an open subset G∈on G, satisfying and a small parameter ∈0, such that, for any ∈ ∈ (0,∈0] and any (x0,y0) ∈Ge, while r ∈[0, exp(η∈-β)], the orbits starting with (xo,yo) satisfying the following estimateWe consider the volume-preserving mapping with 1-dimension action variables in Theorem 1. By replacing volume-preserving condition with the intersection property, we could get a similar result for the nearly twist mapping with multiple action variables.Generally, we consider the following small twist mapping with multiple action variables under the intersection property (?)t:Tn× G→Tn×Rm are 1-periodic functions with respect to x,∈>0 is a small parameter, in chapter 4, we have the following theorem: Theorem 2. If mappingi(5) suits the following assumptions(H1) (Analyticity) ω, f and g are real analytic mappings defined on T" x G, that is there exists δ>0, such that ω, f and g are analytic on (Tn×G)+δ.(H2) (Regularity)For some positive constant M. (H3) (Intersection property)(?)possesses intersection property on G×Tn.(H4) (Non-degeneracy) ω(p) satisfies Russmann’s nondegenerate conditionThe problem of the stability in nearly integrable mappings is important in the area of dynamical systems. There is a large number of works on KAM theory and effective stability. Here we would like to mention some results which are closely related with the problems in this paper. In 2002, Hairer, Lubich and Wanner developed the theorems of near-invariant tori for a nearly integrable Hamiltonian system and a nearly integrable symplectic mapping, respectively [38]. Their results show that the orbit starting near the diophantine tours of the unperturbed system is nearly invariant, up to exponentially small deviations, over exponentially long times. Later on, Cong, Hong and Han [15] obtained the theorem of near-invariant tori for nearly integrable Poisson systems.In Chapter 3, we concerns the following nearly twist mapping(?)t:Tn×G→Rm×Tn defined by Here Tn=Rn/(2πZ)n is the usual torus, and G is an open and bounded set of Rm; f(x, y) and g(x, y) are 2π-periodic with respect to x. Moreover, t is a small parameter to satisfy 0<t≤1.If Jacobian matrix δω/δγ is of full rank, mapping (7) with twist parameter t and per-turbation parameter ∈ is said to be small twist. This kind of mappings are often met in compute mathematics [38]. In this paper we consider a general small twist map-ping with intersection property because, generally, volume-preserving mappings do not possess the intersection property when dimension of action variables m is bigger than 1.Assume (H1) (Analyticity)ω,fand g are real analytic mappings defined on Tn × G, that is, thereexists ρ> 0 such that ω, f and g are analytic on Bρ×Eρ, whereand ω, f and g take real value when x and y are real variables. (H2) (Intersection property)For any t ∈ (0,1], (?)t is a mapping with intersection prop-erty on ∑ρ×Gρ. (H3) (Non-degeneracy)On G (the closure of G), the following relation holds:Theorem 3. Let τ> (max{m,n}+1)max{m,n}-1 be a given positive constant. Under the above assumptions there exists a constant ε*>0 such that for any ε∈(0,ε*,] and for any t∈(0,1] there exists a nonempty Cantor set G(ε,t) c G satisfying that if ||f||+||f||<ε then the following conclusions hold.(i) (?)t admits a family of invariant tori Ty, y∈G(ε, t) with frequency tω∞ (y, t) satisfying(ii) The Lebesgue measure of G(ε, t) suits meas( uniformly on (O,ε*]×(0,1].Different from the past, in this thesis, we consider the small twist mappings with different dimensional action-angle variables. We should have similar results in the sym-plectic mappings.