# The Lagrangian Stability for a Class of Second-Order Quasi-Periodic Reversible Systems

Authors: Yanling Shi; Jia Li
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the following two-order differential equation,(Φp(x'))'+f(x,t)Φp(x')+g(x,t)=0, where Φp(s)=|s|(p-2)s, p>0. f(x,t) and g(x,t) are real analytic functions in x and t, 2aπp- periodic in x, and quasi-periodic in t with frequencies (ω1,…,ωm). Under some odd-even property of f(x,t) and g(x,t), we obtain the existence of invariant curves for the above equations by a variant of small twist theorem. Then all solutions for the above equations are bounded in the sense of supt∈R|x′(t)|<+∞.


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## Body

## 1. Introduction

The Kolmogorov-Arnold-Moser (KAM) theory was developed for conservative (Hamiltonian) dynamical systems that are nearly integrable. Integrable systems in their phase space contain lots of invariant tori, and KAM theory establishes persistence of such tori, which carry quasi-periodic motions, whereas parallel results exist for other classes of dynamical systems as well. In particular, the reversible KAM theory (starting with Moser’s paper [1]) is to a great extent parallel to the Hamiltonian ones, see [2–6] and references therein. In the case of reversible diffeomorphisms, however, some special effects are exhibited [7], and the weakly reversible KAM theory has been developed in [8, 9].In this paper, we will consider theP-Laplace equations with quasi-periodic reversible structure. Firstly, we give some concepts of reversible system, a mechanical system of s particles with the interaction forces which are independent of velocities or even functions in velocities; such a system ruled by the Newton equation, d2r/dt2=F(r,v),r∈R3s,v=dr/dt,F(r,-v)≡F(r,v), is time reversible, that is, reversing all the velocities v reverses all the trajectories in the configuration space R3s. More generally, an autonomous differential equation du/dt=V(u) and the corresponding vector field V are said to be reversible if there exists a phase space involution G (a mapping which G2=Id) that reverses the direction of time: TG∘V=-V∘G, where TG is the differential of G; that is, G transforms the field V into the opposite field -V. This u(t), in addition to G(u(-t)), is a solution of the equation.In the above example,G:(r,v)↦(r,-v). A system
(1.1)dxdt=f(x,t),dydt=g(x,t),x∈Rp,y∈Rq,
is reversible with respect to the involution (1.2)G:(x,y)⟶(-x,y)
if and only if f is even in x and g is odd in x.We can refer to [2, 8–11] for more detailed concepts of reversibility. Side by side with reversible vector fields, there are reversible diffeomorphisms. A mapping A is said to be reversible if there exists an involution G that conjugates A with its inverse A-1, that is,AGA=G. The flow map of a reversible vector field for each fixed time is reversible with respect to the same involution, and vice versa.In [12] Liu considered the following quasi-periodic mappings:
(1.3)A:(x,y)⟶(x+ω+y+f(x,y),y+g(x,y)),
where f and g are quasi-periodic in x with frequencies μ1,…,μm and real analytic in x and y, the variable y ranges in a neighborhood of the origin of the real line R, and ω is a positive constant. He supposes that the mapping A is reversible with the involution R:(x,y)↦(-x,y), that is,RAR=A-1. Such a map is often met when the vector field is quasi-periodic in time and reversible with respect to the involution R. In fact, the phase flow induces such a map on a cross-section transversal to the vector field.The invariant curve theorem of reversible systems was first obtained by Moser [1] who then developed it [13] (for continuous systems), it was also developed by  Sevryuk [9] (for both continuous and discrete system). In [1], the author also studied the existence of invariant tori of a reversible system depending quasi-periodically on time. In [12] Liu obtained an invariant curve theorem for reversible quasi-periodic mappings, as application, he studies the existence of quasi-periodic solutions and the boundedness of solutions for a pendulum-type equation
(1.4)x′′+f(x,t)x′+g(x,t)=0
and an asymmetric oscillator depending quasi-periodically on time. By some theorems in [12], in this paper, we consider two-order differential equations
(1.5)(Φp(x′))′+f(x,t)Φp(x′)+g(x,t)=0,
where Φp(s)=|s|(p-2)s,p>0,ω>0;ifp=2, it becomes (1.4).Our main result is the following theorem.


## 2. Main Result

We first give some definitions [12].Definition 1.
A functionf:R→R is called real analytic quasi-periodic with frequencies μ1,…,μm if it can be represented as a Fourier series of the type
(2.1)f(t)=∑kfke〈k,μ〉t,
where k=(k1,…,km),μ=(μ1,…,μm),〈k,μ〉=∑kjμj≠0; if k≠0, the coefficients fk decay exponentially with |k|=|k1|+⋯+|km|.Definition 2.
The vectorμ=(μ1,…,μm) satisfies the Diophantine condition if:
(2.2)|〈k,μ〉|≥c|k|σ,c,σ>0
for all integer vector k≠0.For our study of (1.5), where f(x,t) and g(x,t) are real analytic in x and t, 2aπp periodic in x, and quasi-periodic in t with frequencies (ω1,…,ωm), where the number πp is defined by
(2.3)πp=2∫0(p-1)1/pds[1-sp/(p-1)]1/p.
Moreover, we assume
(2.4)f(-x,-t)=f(x,t),g(-x,-t)=g(x,t).Theorem 2.1.
Suppose that(ω1,…,ωm) satisfy the Diophantine condition
(2.5)|〈k,μ〉|≥c0|k|σ0,fork∈Zm∖{0},
where c0,σ0 are positive constants. Then there are infinitely many quasi-periodic solutions with large amplitude, and the solutions of (1.5) satisfy
(2.6)sup⁡t∈R|x′(t)|<+∞.


## 3. Coordination Transformation

In this section we first make a coordination transformation then study the boundedness of all solutions of the new system.Equation (1.5) is equivalent to the planar system:
(3.1)x′=Φq(y),y′=-f(x,t)y-g(x,t).
It is easy to verify that planar system (3.1) is reversible with respect to the involution R:(x,y)↦(-x,y).Since we are concerned with the boundedness of solutions and the existence of quasi-periodic solutions with large amplitude, we may assume|y|≥1. Instead of considering planar system (3.1), we are concerned with the following system:
(3.2)dtdx=1Φq(y),dydx=-f(x,t)yΦq(y)-g(x,t)Φq(y).
We will prove that if (t(x),y(x)) is a solution of system (3.2), then |y| is bounded.


## 4. Poincaré Map

In this section, we first introduce new action variable then give an expression for the Poincaré map of the new system.Introduce a new action variablev and a small parameter ε as follows: (4.1)y=1Φp(ε)v,v∈[1γ,γ],γ>1.
So system (3.2) is changed into the following form:
(4.2)dtdx=εΦq(v),dvdx=ε|v|qf(x,t)+|ε|pvq+1g(x,t).
It is easy to verify that system (4.2) is reversible with respect to the involution (t,v)↦(-t,v). We make the ansatz that the solution (t(x,v0,t0,ε),ρ(x,v0,t0,ε)) has the following form:
(4.3)t=t0+εT(x,v0,t0;ε),v=V0+εV(x,v0,t0;ε).
The functions T and V satisfy
(4.4)dTdx=Φq(V0+εV)=Φq(V0)+O(ε),dVdx=f(x,t0+εT)|V0+εV|q+Φp(ε)|V0+εV|q+1g(x,t0+εT)=f(x,t0)V0q+O(ε).
Denote by P the Poincaré map of (4.2); then, from the above equations, it is easy to see that
(4.5)P(t0,V0)=(t0+2aπpεΦq(v0)+O(ε2),V0+εV0q∫02aπpf(x,t0)dx+O(ε2)).


## 5. The Proof of Theorem2.1

In this part, we will prove the mapP has an invariant curve, and then boundedness of solutions of (1.5) follows from the standard arguments [14–18]. In the following, we will apply the invariant curves of quasi-periodic reversible mapping theorem to prove our conclusion.Now we state Liu's result [12].Consider the map(5.1)Mδ:(x,y)⟶(x+α+δL(x,y)+δf(x,y,δ),y+δM(x,y)+δg(x,y,δ)),
where L,M,f,g are quasi-periodic in x with the frequencies μ1,…,μm,f(x,y,0)=g(x,y,0)=0.We also assume that these functions are real analytic in a complex neighborhood of the domainR×[a1,b1]. The functions L and M can be represented in the form(5.2)L(x,y)∶=L̃(x,y)+L¯(x,y)=∑k∈Zm∖KLk(y)ei〈k,μ〉x+∑k∈KLk(y)ei〈k,μ〉x,M(x,y)∶=M̃(x,y)+M¯(x,y)=∑k∈Zm∖KMk(y)ei〈k,μ〉x+∑k∈KMk(y)ei〈k,μ〉x.
Note that
(5.3)ei〈k,μ〉α-1≠0,fork∈Zm∖K,L¯(x+α)≡L¯(x),M¯(x+α)≡M¯(x).Lemma 5.1 ([12, Theorem 4]).
Suppose the functionL satisfies
(5.4)L¯(x,y)>0,∂L¯∂y>0,
and there is a real analytic function I(x,y)=I(x+α,y) satisfying
(5.5)∂I∂y>0,L¯(x,y)∂I∂x(x,y)+M¯(x,y)∂I∂y(x,y)≡0.Moreover, suppose that there are two numbersã and b̃ such that a1<ã<b̃<b1 and
(5.6)IM(a1)<Im(ã)≤IM(ã)<Im(b̃)≤IM(b̃)<Im(b1),
where
(5.7)IM(y)=max⁡x∈R⁡I(x,y),Im(y)=min⁡x∈R⁡I(x,y).Then there existε>0 and Δ>0 such that if δ<Δ and
(5.8)‖f(⋅,⋅,δ)‖+‖g(⋅,⋅,δ)‖<ε,
the mapping Mδ has an invariant curve which is of the form y=ϕ(x), and ϕ is quasi-periodic in x with frequencies μ1,…,μm. The constants ε and Δ depend on a,ã,b̃,b,L,M, and I. In particular, ε is independent of δ. If α=0, the conclusion also holds; in this case, K=Zm,L̃=M̃=0, L¯=L,andM¯=M.For the Poincaré mapP(t0,V0) of (4.2), let
(5.9)I(t0,V0)=V0e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt.
Since f(-t,-x)=-f(t,x), we know that I(t0,V0) is even and quasi-periodic in t0 with frequencies μ1,…,μm. And it is easy to see that
(5.10)∂I∂V0=e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt>0,L¯(V0,t0)∂I∂t0(V0,t0)+M¯(V0,t0)∂I∂V0(V0,t0)=2aπpΦq(v0)⋅V0(-12aπp)∫02aπpf(x,t0)dx⋅e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt+V0q∫02aπpf(x,t0)dx⋅e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt=V0q∫02aπpf(x,t0)dx⋅e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt-V0q∫02aπpf(x,t0)dx⋅e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt=0.
Moreover, if we define  α and β by
(5.11)I(t0,V0)=V0e-(1/2aπp)∫0t0∫02aπpf(x,t)dxdt,α=min⁡t0∈Rexp⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt),β=max⁡t0∈Rexp⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt),
then β≫α>0. Now we choose the constants γ,γ1,γ2 as
(5.12)γ=(2βα)3>1,γ1=(2βα)-1,γ2=(2βα).
Then
(5.13)IM(1γ)=1γmax⁡t0∈Rexp⁡⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt)=α38β2,Im(γ)=γmin⁡t0∈Rexp⁡⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt)=8β3α2,Im(γ1)=γ1min⁡t0∈Rexp⁡⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt)=α22β,IM(γ1)=γ1max⁡t0∈Rexp⁡⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt)=α2,Im(γ2)=γ2min⁡t0∈Rexp⁡⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt)=2β,IM(γ2)=γ2max⁡t0∈Rexp⁡(-12aπp∫0t0∫02aπpf(x,t)dxdt)=-2β2α.We have already demonstrated that the mapP satisfies all the conditions in Lemma 5.1; hence, P has an invariant curve; thus, all solutions of (1.5) are bounded.



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*Source: 489148-2011-11-17.xml*