On the Periodic Orbits of Four-Particle Time-Dependent FPU Chains

We study the existence of the periodic orbits for four-particle time-dependent FPU chains. To begin with, taking an orthogonal transformation and dropping two variables, we put the system into a lower dimensional system. In this case, the unperturbed system has the linear spectra 2 and√2, which are rational independent.Therefore, the classical averaging theory cannot be applied directly, and the existence and multiplicity of periodic orbits for four-particle FPU chains is proved by using a new result from averaging theory by Buică, Francoise, and Llibre.


Introduction
The Fermi-Pasta-Ulam (FPU) system describes an approximate model for the behaviour of a classical solid at low temperatures [1], whose development dates from the first numerical results of FPU by Fermi et al. in 1955 [2].FPU problem, both ushering in the age of computational science and marking the birth of our own field of nonlinear science [3], has received the widespread attention both in mathematics and in physics, including ergodicity [4,5], integrability [6,7], breathers [8][9][10], chaos, and stability of motion [11][12][13][14].In fact, the FPU problem is much more than a little discovery and will doubtless be challenging scientists and stimulating new research directions for many decades to come [3].
In this paper, we consider the time-dependent Hamiltonian: where   denotes the displacement of the th oscillator with respect to its equilibrium position.The coupled equations of motions are q  +    (,   −  −1 ) −    (,  +1 −   ) = 0,   =  + ,  ∈ N,  ∈ Z, and (, ), denoting the coupling potential between the neighboring particles, is given by  (, ) = Here, h.o.t. is the high order term of  6 , and (), (), (), () are continuous -periodic functions.If  ≡ 0,  ≡ 0,  ≡ 0,  ̸ = 0, the chain is called an -chain, and if  ≡ 0,  ≡ 0,  ≡ 0,  ̸ = 0, it is called a -chain.The FPU model with the time-dependent nonlinear term is used to explore the relation between the thermodynamic entropy change due to irreversible processes and the change of information loss as a chaotic dynamical system [15].For a recent report, we can refer to [16].
In low frequency modes and the transfer of energy to higher modes, periodic orbits of FPU lattices have been widely studied [17,18].In [18], Flach and Ponno discussed the FPU problem including periodic orbits, normal forms, and resonance overlap criteria, compared all these results with each other, and established the similarities and differences.The paper [19] studied weakly nonlinear spatially localized solutions of a FPU model which describe a one-dimensional chain of particles interacting with a number of neighbors depended on the site.When (, ) is asymptotically linear with respect to  both at origin and at infinity, the existence of nontrivial periodic solutions of system (2) has been proved in [20].In this paper, we study existence of periodic orbits for four-particle time-dependent FPU chains by using the averaging theory.
The averaging method, an important method of perturbation theory that has played a rather modest role in nonlinear differential system with a small parameter, gives useful approximate solutions of the equations of motion [21].For instance, the existence of periodic orbits and the stability of system have been studied using averaging method in [22][23][24][25][26].In the previous work [27], we have studied the existence of periodic orbits and their stability by using the classical averaging method for the three-particle timedependent FPU chains, where the system can be regarded as a perturbation of isochronous system.In this paper, we shall investigate the existence and multiplicity of periodic orbits for four-particle time-dependent FPU chains.In this case, the unperturbed system has the linear spectra 2 and √ 2, which are rational independent.Therefore, the classical averaging theory cannot be applied directly.For some specific coefficients being usually used in the physical experiment, we establish the existence and multiplicity of periodic orbits for four-particle time-dependent FPU chains by using the result from Buicȃ et al. [28].
The rest of paper is organized as follows.In Section 2, we present our main results on the periodic solutions of the FPU chains and introduce the basic results from averaging theory in Section 3. The detailed proofs for the existence and multiplicity of periodic orbits of four-particle timedependent FPU chains are given in the last part.
Taking an orthogonal transformation by system (5) transforms into Advances in Mathematical Physics 3 where In system (9), we drop equations of  4 ,  4 in the subsequent discussion since these equations are integrable.Only when  4 ,  4 are equal to zero, system (9) has periodic solutions.Therefore, system (9) can be expressed as The unperturbed system of (11) with  = 0 is given by which has a rest point at the origin with eigenvalues ±2, ± √ 2, these last two with multiplicity two.Consequently, this system in the phase space ( 1 ,  1 ,  2 ,  2 ,  3 ,  3 ) has three planes filled with periodic solutions with the exception of the origin.These periodic solutions have periods  1 =  or  2 = √ 2, according to whether they belong to the plane associated with the eigenvectors with eigenvalues ±2 or ± √ 2.In this paper, we shall study which of these periodic solutions persist for the perturbed system (11) thereby for FPU system (5), when the parameter  is sufficiently small and the perturbed functions   for  = 1, 2, 3 have period either  1 or  2 .
The solution of unperturbed system (12) with initial value We define the functions as follows: A solution ( * 10 ,  * 10 ) of the algebraic equations is called a simple zero of the averaged system associated with (11).Similarly, a solution ( 20 ,  20 ,  30 ,  30 ) of the algebraic equations such that is also called a simple zero of the averaged system associated with (11).
Since scale transformation is used to the change of variables for all  > 0, the transformed system (9) and the original system (5) have the same topological structure.In the following, we state our main results.

Some Results from Averaging Theory
In this section, we present the basic result from the averaging theory that we apply to prove our main results.
We consider the problem of the bifurcation of -periodic solutions from differential systems of the following form: with  ̸ = 0 being sufficiently small, where  0 ,  1 : R × Ω → R  and  2 : R × Ω × (− 0 ,  0 ) → R  are  2 smoothness and -periodic in , and Ω is an open subset of R  .When  = 0, the unperturbed system is given by and we assume that it has a submanifold of periodic solutions.
Using the classical averaging theory, the existence of periodic solutions of system ( 27) can be established with nondegenerate equilibrium points of the associated averaged system.We can refer to [29] for details.Denote the linearization of the unperturbed system (28) along (, ) by where (, ) is the periodic solution of the system (28) such that (0, ) = .For simplicity, we assume that   () is a fundamental matrix of the linear differential system (29), and  : In the following, we state a new averaging theorem by Buicȃ et al. in [28].

Proofs of Theorems 1 and 2
In this section, we will show how to apply Theorem 3 given in Section 3 to prove our main theorems.The key step is to find a bounded and open set , where there exist simple zeroes of the associated averaged system.We firstly prove the existence and multiplicity of periodic solutions for system (11), thereby for system (9), and then using the orthogonal transformation (8) return periodic solutions to the desired FPU system (5).

The Proof of Theorem 1.
In order to apply Theorem 3, we rewrite system (11) in the following form: where ) . ( For the particular case of the differential system ( 27), we describe the different conditions which appear in the statement of Theorem 3. Therefore, we have that  = 6,  = 2, and the set Ω = R 6 .Let  1 > 0 be arbitrarily small and let  2 > 0 be arbitrarily large.Define an open and bounded subset  on the projection plane  1  1 by  = { ∈ R 2 :  = ( 10 ,  10 ) ,  1 < √  2  10 + The function  :  → R 4 is taken by () ≡ (0, 0, 0, 0).Obviously,  is of class  2 .We define the set Z by As the origin is a periodic solution with arbitrary period of the unperturbed system, the aim of defining  1 is to exclude the equilibrium point (0, 0, 0, 0, 0, 0).Obviously, for each   ∈ Z and  = 0, the solution (,   ) of (31) with the initial value (0,   ) =   is -periodic with respect to .In fact, every solution of (31) with  = 0 starting from Z can be written by coordinates as (,   ) = ( 1 (,   ),  1 (,   ),  2 (,   ),  2 (,   ),  3 (,   ),  3 (,   )), where Along the periodic solution (,   ), the variational equation of the unperturbed system is given by where () is a 6 × 6 matrix function.Now we present the linearization matrix of the unperturbed system along the periodic solution by The fundamental matrix    () of the variational equation (36), where    (0) is the identity matrix, adopts the simple following form: with ) . ( Notice that    () does not depend on the initial value   of periodic solution (,   ).Therefore, we drop the subscript   and reset () =    ().Computing the matrix  −1 (0) −  −1 (), we get This matrix has in the upper right corner 2 × 4 null matrix and in the lower right corner a 4 × 4 matrix Δ  satisfying det Therefore, the assumption (ii) of Theorem 3 holds.Following the notations of Theorem 3, we have In the following, we will calculate the following function: Let () =  +  cos(4), () =  0 with , ,  0 ∈ R and (), () be  2 periodic function with the period .Then it follows that  1 (,  1 (),  1 (), 0, 0, 0, 0) is a -periodic function only depending on () and (), which is not associated with the functions () and ().
Therefore, we obtain ) . (44) Finally, we shall find the simple zeroes of the function G() in the open and bounded subset  ⊂ R 2 .A straightforward computation shows that the solutions of F 1 ( 10 ,  10 ) = 0, F 2 ( 10 ,  10 ) = 0 in  are given by whenever the radical expressions of (45) are well defined.We leave out of account the solution  1 , since it may be corresponding to the trivial periodic solution, zero solution.

Advances in
) . (51 Then system (11) has the following form: Following the notations in Theorem 3, we have that  = 6,  = 4, and the set Ω = R 6 .Let  1 > 0 be arbitrarily small and let  2 > 0 be arbitrarily large.Define an open and bounded subset  on the projection plane  2  3  2  3 by The function  :  → R 2 is taken by () ≡ (0, 0).Consequently, we define the set Z by Obviously, for each   ∈ Z and  = 0, the solution (,   ) of (52) with the initial value (0,   ) =   is √ 2-periodic with respect to .In fact, every solution (,   ) of ( 52) with  = 0 starting from Z is given by Accordingly, the variational equation of the unperturbed system, along the periodic solution (,   ), is given by ) where () is a 6×6 matrix function.The fundamental matrix    () of (57), which satisfies that    (0) is the identity matrix, adopts the simple following form: with Note that    () does not depend on the initial value   of periodic solution (,   ).Therefore, we drop the subscript   and reset () =    ().With a direct calculation, we get 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 We observe that the matrix  −1 (0) −  −1 ( √ 2) has in the upper right corner the 4 × 2 null matrix and in the lower right corner a 2 × 2 matrix Δ  satisfying det Therefore, assumptions (i) and (ii) of Theorem 3 hold.

Numerical Simulation
In order to support our results, we perform some numerical simulations.Although we cannot find the exact periodic solutions, according to proposed theorems (see Theorems 1 and 2) and their proofs, the initial values of periodic solutions of ( 11) are near its simple zeroes.The differences between the simple zeroes and the initial values of corresponding periodic solutions are of one order .

Conclusions and Further Discussions
The paper studies the existence and multiplicity of periodic orbits of four-particle time-dependent FPU chains by averaging theory which is one of important methods in the area of Advances in Mathematical Physics    5): the red curve is an approximate √ 2-periodic solution starting from  3 , while the blue dashed curve and the green dashed curve are the reference solutions starting from  31 and  32 , respectively.dynamical system.In this case, the classical averaging theory cannot be applied directly by reason of the unperturbed system with multiple irrational frequencies.Therefore, to overcome this difficulty, we apply a new result from the averaging theory as the main mathematical tool.Theorems 1 and 2 are our main results on the periodic orbits for fourdimensional time-dependent FPU chains.We conclude that four-particle time-dependent FPU system (5) has multiplicity nontrivial periodic solution.
Moreover, the method used in this paper is easy to apply for  ( > 4) particles FPU chains.Of course, with the increase of the dimension of system, the amount of calculation may increase much more.The method used for proving the existence and multiplicity of periodic orbits of fourparticle time-dependent FPU chains can be applied to other Hamiltonian systems with multiple irrational frequencies.We expect the numerical and theoretical results in the near future.

Figure 1 :
Figure 1:The curves of solutions of (5): the red curve is an approximate -periodic solution starting from  1 , while the blue dashed curve and the green dashed curve are the reference solutions starting from  11 and  12 , respectively.

Figure 2 :
Figure 2:The curves of solutions of (5): the red curve is an approximate -periodic solution starting from  2 , while the blue dashed curve and the green dashed curve are the reference solutions starting from  21 and  22 , respectively.

Figure 3 :
Figure 3:  The curves of solutions of (5): the red curve is an approximate √ 2-periodic solution starting from  3 , while the blue dashed curve and the green dashed curve are the reference solutions starting from  31 and  32 , respectively.