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

Abstract

We study the following two-order differential equation, (Φ_𝑝(𝑥^′))^′+𝑓(𝑥,𝑡)Φ_𝑝(𝑥^′)+𝑔(𝑥,𝑡)=0, where Φ_𝑝(𝑠)=|𝑠|^(𝑝−2)𝑠, 𝑝>0. 𝑓(𝑥,𝑡) and 𝑔(𝑥,𝑡) are real analytic functions in 𝑥 and 𝑡, 2𝑎𝜋_𝑝− periodic in 𝑥, and quasi-periodic in 𝑡 with frequencies (𝜔₁,…,𝜔_𝑚). 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 sup_𝑡∈𝑅|𝑥^′(𝑡)|<+∞.

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 the 𝑃-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∈𝑅^3𝑠,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 𝑅^3𝑠. 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 𝑑𝑥(),𝑑𝑡=𝑓𝑥,𝑡𝑑𝑦()𝑑𝑡=𝑔𝑥,𝑡,𝑥∈𝑅^𝑝,𝑦∈𝑅^𝑞, (1.1) is reversible with respect to the involution 𝐺∶𝑥,𝑦⟶−𝑥,𝑦 (1.2) 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: 𝐴∶𝑥,𝑦⟶𝑥+𝜔+𝑦+𝑓𝑥,𝑦,𝑦+𝑔𝑥,𝑦, (1.3) where 𝑓 and 𝑔 are quasi-periodic in 𝑥 with frequencies 𝜇₁,…,𝜇_𝑚 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 𝑥^′′+𝑓(𝑥,𝑡)𝑥^′+𝑔(𝑥,𝑡)=0 (1.4) and an asymmetric oscillator depending quasi-periodically on time. By some theorems in [12], in this paper, we consider two-order differential equations Φ_𝑝𝑥^′′+𝑓(𝑥,𝑡)Φ_𝑝𝑥^′+𝑔(𝑥,𝑡)=0, (1.5) where Φ_𝑝(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 function 𝑓∶𝑅→𝑅 is called real analytic quasi-periodic with frequencies 𝜇₁,…,𝜇_𝑚 if it can be represented as a Fourier series of the type 𝑓(𝑡)=_𝑘𝑓_𝑘𝑒^{⟨𝑘,𝜇⟩𝑡}, (2.1) where 𝑘=(𝑘₁,…,𝑘_𝑚),𝜇=(𝜇₁,…,𝜇_𝑚),⟨𝑘,𝜇⟩=∑𝑘_𝑗𝜇_𝑗≠0; if 𝑘≠0, the coefficients 𝑓_𝑘 decay exponentially with |𝑘|=|𝑘₁|+⋯+|𝑘_𝑚|.

Definition 2. The vector 𝜇=(𝜇₁,…,𝜇_𝑚) satisfies the Diophantine condition if: ≥𝑐𝑘,𝜇|𝑘|^𝜎,𝑐,𝜎>0 (2.2) 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, 2𝑎𝜋_𝑝 periodic in 𝑥, and quasi-periodic in 𝑡 with frequencies (𝜔₁,…,𝜔_𝑚), where the number 𝜋_𝑝 is defined by 𝜋_𝑝=2^{(𝑝−1)1/𝑝}₀𝑑𝑠1−𝑠^𝑝/𝑝−1^1/𝑝. (2.3) Moreover, we assume 𝑓(−𝑥,−𝑡)=𝑓(𝑥,𝑡),𝑔(−𝑥,−𝑡)=𝑔(𝑥,𝑡). (2.4)

Theorem 2.1. Suppose that (𝜔₁,…,𝜔_𝑚) satisfy the Diophantine condition ≥𝑐𝑘,𝜇⁰|𝑘|^𝜎0,𝑓𝑜𝑟𝑘∈𝑍^𝑚⧵{0}, (2.5) where 𝑐₀,𝜎₀ are positive constants. Then there are infinitely many quasi-periodic solutions with large amplitude, and the solutions of (1.5) satisfy sup_𝑡∈𝑅𝑥^′(𝑡)<+∞. (2.6)

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: 𝑥′=Φ_𝑞𝑦,𝑦′=−𝑓(𝑥,𝑡)𝑦−𝑔(𝑥,𝑡). (3.1) 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: 𝑑𝑡=1𝑑𝑥Φ_𝑞𝑦,𝑑𝑦𝑓()𝑦𝑑𝑥=−𝑥,𝑡Φ_𝑞𝑦−𝑔(𝑥,𝑡)Φ_𝑞𝑦. (3.2) 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 variable 𝑣 and a small parameter 𝜀 as follows: 1𝑦=Φ_𝑝(𝜀)𝑣1,𝑣∈𝛾,𝛾,𝛾>1. (4.1) So system (3.2) is changed into the following form: 𝑑𝑡𝑑𝑥=𝜀Φ_𝑞(𝑣),𝑑𝑣|𝑣|𝑑𝑥=𝜀^𝑞𝑓(𝑥,𝑡)+|𝜀|^𝑝𝑣^𝑞+1𝑔(𝑥,𝑡). (4.2) 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: 𝑡=𝑡₀+𝜀𝑇𝑥,𝑣₀,𝑡₀;𝜀,𝑣=𝑉₀+𝜀𝑉𝑥,𝑣₀,𝑡₀;𝜀. (4.3) The functions 𝑇 and 𝑉 satisfy 𝑑𝑇𝑑𝑥=Φ_𝑞𝑉₀+𝜀𝑉=Φ_𝑞𝑉₀+𝑂(𝜀),𝑑𝑉𝑑𝑥=𝑓𝑥,𝑡₀+𝜀𝑇𝑉₀+𝜀𝑉^𝑞+Φ_𝑝(𝜀)𝑉₀+𝜀𝑉^𝑞+1𝑔𝑥,𝑡₀+𝜀𝑇=𝑓𝑥,𝑡₀𝑉^𝑞₀+𝑂(𝜀). (4.4) Denote by 𝑃 the Poincaré map of (4.2); then, from the above equations, it is easy to see that 𝑃𝑡₀,𝑉₀=𝑡₀+2𝑎𝜋_𝑝𝜀Φ_𝑞𝑣₀+𝑂𝜀²,𝑉₀+𝜀𝑉^𝑞₀^{2𝑎𝜋𝑝}₀𝑓𝑥,𝑡₀𝑑𝑥+𝑂𝜀². (4.5)

5. The Proof of Theorem 2.1

In this part, we will prove the map 𝑃 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 𝑀_𝛿∶𝑥,𝑦⟶𝑥+𝛼+𝛿𝐿𝑥,𝑦+𝛿𝑓𝑥,𝑦,𝛿,𝑦+𝛿𝑀𝑥,𝑦+𝛿𝑔𝑥,𝑦,𝛿, (5.1) where 𝐿,𝑀,𝑓,𝑔 are quasi-periodic in 𝑥 with the frequencies 𝜇₁,…,𝜇_𝑚,𝑓(𝑥,𝑦,0)=𝑔(𝑥,𝑦,0)=0.

We also assume that these functions are real analytic in a complex neighborhood of the domain 𝑅×[𝑎₁,𝑏₁]. The functions 𝐿 and 𝑀 can be represented in the form𝐿𝑥,𝑦∶=𝐿𝑥,𝑦+𝐿𝑥,𝑦=∑_{𝑘∈𝑍𝑚⧵𝐾}𝐿_𝑘𝑦𝑒^{𝑖⟨𝑘,𝜇⟩𝑥}+∑_𝑘∈𝐾𝐿_𝑘𝑦𝑒^{𝑖⟨𝑘,𝜇⟩𝑥},𝑀𝑥,𝑦∶=𝑀𝑥,𝑦+𝑀𝑥,𝑦=∑_{𝑘∈𝑍𝑚⧵𝐾}𝑀_𝑘𝑦𝑒^{𝑖⟨𝑘,𝜇⟩𝑥}+∑_𝑘∈𝐾𝑀_𝑘𝑦𝑒^{𝑖⟨𝑘,𝜇⟩𝑥}. (5.2) Note that 𝑒^{𝑖⟨𝑘,𝜇⟩𝛼}−1≠0,for𝑘∈𝑍^𝑚⧵𝐾,𝐿(𝑥+𝛼)≡𝐿(𝑥),𝑀(𝑥+𝛼)≡𝑀(𝑥). (5.3)

Lemma 5.1 ([12, Theorem 4]). Suppose the function 𝐿 satisfies 𝐿𝜕𝑥,𝑦>0,𝐿𝜕𝑦>0, (5.4) and there is a real analytic function 𝐼(𝑥,𝑦)=𝐼(𝑥+𝛼,𝑦) satisfying 𝜕𝐼𝜕𝑦>0,𝐿𝑥,𝑦𝜕𝐼+𝜕𝑥𝑥,𝑦𝑀𝑥,𝑦𝜕𝐼𝜕𝑦𝑥,𝑦≡0. (5.5)

Moreover, suppose that there are two numbers 𝑎 and 𝑏̃ such that 𝑎₁<𝑎<𝑏̃<𝑏₁ and 𝐼_𝑀𝑎₁<𝐼_𝑚(𝑎)≤𝐼_𝑀(𝑎)<𝐼_𝑚𝑏̃≤𝐼_𝑀𝑏̃<𝐼_𝑚𝑏₁, (5.6) where 𝐼_𝑀𝑦=max_𝑥∈𝑅𝐼𝑥,𝑦,𝐼_𝑚𝑦=min_𝑥∈𝑅𝐼𝑥,𝑦. (5.7)

Then there exist 𝜀>0 and Δ>0 such that if 𝛿<Δ and 𝑓(⋅,⋅,𝛿)+𝑔(⋅,⋅,𝛿)<𝜀, (5.8) the mapping 𝑀_𝛿 has an invariant curve which is of the form 𝑦=𝜙(𝑥), and 𝜙 is quasi-periodic in 𝑥 with frequencies 𝜇₁,…,𝜇_𝑚. The constants 𝜀 and Δ depend on 𝑎,𝑎,𝑏̃,𝑏,𝐿,𝑀, and 𝐼. In particular, 𝜀 is independent of 𝛿. If 𝛼=0, the conclusion also holds; in this case, 𝐾=𝑍^𝑚,𝐿=𝑀=0, 𝐿=𝐿,𝑎𝑛𝑑𝑀=𝑀.

For the Poincaré map 𝑃(𝑡₀,𝑉₀) of (4.2), let 𝐼𝑡₀,𝑉₀=𝑉₀𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡}. (5.9) Since 𝑓(−𝑡,−𝑥)=−𝑓(𝑡,𝑥), we know that 𝐼(𝑡₀,𝑉₀) is even and quasi-periodic in 𝑡₀ with frequencies 𝜇₁,…,𝜇_𝑚. And it is easy to see that 𝜕𝐼𝜕𝑉₀=𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡}>0,𝐿𝑉₀,𝑡₀𝜕𝐼𝜕𝑡₀𝑉₀,𝑡₀+𝑀𝑉₀,𝑡₀𝜕𝐼𝜕𝑉₀𝑉₀,𝑡₀=2𝑎𝜋_𝑝Φ_𝑞𝑣₀⋅𝑉₀−12𝑎𝜋_𝑝^{2𝑎𝜋𝑝}₀𝑓𝑥,𝑡₀𝑑𝑥⋅𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡}+𝑉^𝑞₀^{2𝑎𝜋𝑝}₀𝑓𝑥,𝑡₀𝑑𝑥⋅𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡}=𝑉^𝑞₀^{2𝑎𝜋𝑝}₀𝑓𝑥,𝑡₀𝑑𝑥⋅𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡}−𝑉^𝑞₀^{2𝑎𝜋𝑝}₀𝑓𝑥,𝑡₀𝑑𝑥⋅𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡}=0. (5.10) Moreover, if we define 𝛼 and 𝛽 by 𝐼𝑡₀,𝑉₀=𝑉₀𝑒^{−(1/2𝑎𝜋𝑝)∫𝑡00∫𝑝02𝑎𝜋𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡},𝛼=min_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝^𝑡0₀^{2𝑎𝜋𝑝}₀𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡,𝛽=max_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝^𝑡0₀^{2𝑎𝜋𝑝}₀𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡, (5.11) then 𝛽≫𝛼>0. Now we choose the constants 𝛾,𝛾₁,𝛾₂ as 2𝛽𝛾=𝛼³>1,𝛾₁=2𝛽𝛼⁻¹,𝛾₂=2𝛽𝛼. (5.12) Then 𝐼_𝑀1𝛾=1𝛾max_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝∫^𝑡0₀∫^{2𝑎𝜋𝑝}₀𝑓()=𝛼𝑥,𝑡𝑑𝑥𝑑𝑡³8𝛽²,𝐼_𝑚𝛾=𝛾min_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝∫^𝑡0₀∫^{2𝑎𝜋𝑝}₀𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡=8𝛽³𝛼²,𝐼_𝑚𝛾₁=𝛾₁min_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝∫^𝑡0₀∫^{2𝑎𝜋𝑝}₀𝑓()=𝛼𝑥,𝑡𝑑𝑥𝑑𝑡²,𝐼2𝛽_𝑀𝛾₁=𝛾₁max_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝∫^𝑡0₀∫^{2𝑎𝜋𝑝}₀𝑓()=𝛼𝑥,𝑡𝑑𝑥𝑑𝑡2,𝐼_𝑚𝛾₂=𝛾₂min_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝∫^𝑡0₀∫^{2𝑎𝜋𝑝}₀𝑓()𝐼𝑥,𝑡𝑑𝑥𝑑𝑡=2𝛽,_𝑀𝛾₂=𝛾₂max_𝑡0∈𝑅−1exp2𝑎𝜋_𝑝∫^𝑡0₀∫^{2𝑎𝜋𝑝}₀𝑓(𝑥,𝑡)𝑑𝑥𝑑𝑡=−2𝛽²𝛼. (5.13)

We have already demonstrated that the map 𝑃 satisfies all the conditions in Lemma 5.1; hence, 𝑃 has an invariant curve; thus, all solutions of (1.5) are bounded.

Acknowledgments

The authors would like to thank the referees, anonymous editors, and reviewers very much for their valuable comments and suggestions.