Periodic Dynamics of a Class of Non-autonomous Contact Hamiltonian Systems

In this paper, we investigate the existence, number and stability of periodic orbits for the following contact Hamiltonian system H(p,q,s,t)=p22m+G(t,q,m)-mdq+cs(c>0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(p,q,s,t)=\frac{p^{2}}{2m}+G(t,q,m)-mdq+cs(c>0)$$\end{document}. At the same time, unbounded conditions of each solution are also given. The contact Hamiltonian system actually represents a kind of physical phenomenon with non-conservation of energy, but the contact Hamiltonian system studied in this paper represents a one-dimensional damped oscillator system with constant variable sign damping coefficient under certain conditions. Therefore, it is of great physical significance to study the periodic dynamic properties of such system.


Introduction
In recent years, the autonomous Hamiltonian system without considering the dissipation effect caused by the environment has attracted the interest of many physicists and mathematicians. In the case of autonomy, previous studies on such Hamiltonian system H(p, q), which does not depend on time variables t, have made great achievements (see e.g. [1][2][3][4]). Every solution curve of such a system corresponds to a constant quantity (i.e. the value at each moment), which is defined as energy in physics. Hence, each solution curve of Hamiltonian system H(p, q) is a curve of energy conservation. In real life, the development of various things is more or less influenced by environmental factors, including the dissipation effect of the Hamiltonian system. In this case, the energy conservation of the Hamiltonian system will be broken. Recently, for this kind of system the researchers gave a new physical concept called contact Hamiltonian system (see [5]). Contact Hamiltonian system includes autonomous contact Hamiltonian system H(p, q, s) independent on time variable t and non-autonomous contact Hamiltonian system H(p, q, s, t) dependent on time variable t. For the discussions on autonomous contact Hamiltonian system H(p, q, s) (see [6][7][8]). The origin of non-autonomous contact Hamiltonian system H(p, q, s, t) (see [9][10][11]), and its corresponding differential equations are as follows: In [11], the author studies some periodic dynamic behaviors of the differential system corresponding to a one-dimensional damped oscillator with variable symbolic damping coefficient through the properties of the solution of Ermakov Pinney-equation where (t) represents variable sign damping coefficient, m represents mass and (t) represents time-varying frequency. Besides, Theorems 5.1 and 5.2 are given respectively by different choices of I(t), and the stability of related periodic solutions is discussed in Theorems 5.3 and 5.4 under the condition of average = 1 2 ∫ T 0 (s)ds. Non-autonomous contact Hamiltonian system is the epitome of one-dimensional damped oscillator (System (1.1) is a special case of one-dimensional damped oscillator under the condition G(t, q, m) = 1 2 m 2 (t)q 2 ) and it ′ s not completely equivalent to one-dimensional damped oscillator, the corresponding differential system is as follows: Recently, many scholars have studied the related nonlinear physical problems and obtained some practical conclusions (See [12][13][14]). In [12], the authors gave a series of exact solutions of one-dimensional nonlinear Schrödinger equation. By analyzing the properties of these solutions, the authors proposed Bose-Einstein condensates for studying the effective range of one-dimensional Gross-Pitaevskii equation, and found that the number of atoms in bright solitons maintains dynamic stability; In [13], the authors studied the dynamic generation of fractionated semi quantum vortices in sodium atom Bose-Einstein condensates, and found that an obvious periodic modulated spin density wave spatial structure is always embedded in the square semi quantum vortex lattice; In [14], the authors studied the local nonlinear material waves of quasi two-dimensional Bose-Einstein condensates. The results show that all Bose-Einstein condensates can have any number of local nonlinear material waves with discrete energy, which are the mathematical exact orthogonal solutions of Gross-Pitaevskii equation.
There are many effective methods to study the existence of periodic solutions of nonlinear differential equations. For conservative system, the system has a variational structure, so the existence of periodic solutions can be transformed into a functional extreme value problem. Thus, the variational method (minimax principle) can effectively solve such problems (See [15][16][17]). In particular, for the secondorder differential equation on the plane, the existence of periodic solutions can also be studied by the area-preserving Poincaré mapping or the fixed points of successor mapping. For example, the Poincaré-Birkhoff torsion theorem is used to prove the existence of infinitely many periodic solutions of Duffing equation (See [18][19][20]). This kind of method has formed one systematic theory for the second-order plane system. For non-conservative system, due to the variational structure is destroyed, many research methods of periodic solutions are limited in application, especially for high-dimensional non-conservative differential system, the research tools of periodic solutions still need to be developed continuously.
The theory of topological degree has been widely used in the existence of periodic solutions. For example, Fabry et al. in [21], combined the topological degree theory with the theorem of the upper and lower solutions in semi-ordered Banachspace, and obtained the following multi-period solution results of second-order differential equation: where F is continuous and 2 -periodic with respect to t. We assume the following conditions hold.
(H1) There exist numbers R 1 > 0 and s 1 such that for all t ∈ ℝ and for all x ≤ −R 1 .
(H2) F satisfies the condition Berstein-Negumo condition, by which is meant that, for each R ∈ ℝ + , there exists a continuous function Theorem 1.1 [21,30] Assume that the conditions (H1) and (H2) hold. Moreover, we also suppose that: (2) for s = s 0 , Eq. (1.3) has at least one 2 -periodic solution; The application of topological degree theory [22] to the study with the stability of periodic solutions is mainly due to professor Ortega, who extended the index of fixed point to the topological index of periodic solutions [23][24][25][26], and then establishing the relationship between the topological index of periodic solutions and its stability, the relevant proof can also be found in [27]; In [28], Ortega also improved the topological degree index theorem of periodic solutions by degree theory and central manifold theorem. An important application of this result is the discussion on the where c > 0, g ∶ ℝ × ℝ ↦ ℝ is a continuous function and T-periodic with respect to t, and first-order smooth with respect to u. Let u = x 1 ,u = x 2 , system (1.6) becomes the following system which satisfies the following condition Based on an application of Theorem 1.1 to second order Liĕnard differential x) = +∞, uniformly in t [21], the references [27] gave some results on the existence and stability of periodic solutions of Eq. (1.6). These results are the basis for the study of non-autonomous contact Hamiltonian systems in this paper, that is, the following two lemmas: Lemma 1.1 [29] Assume that u is an isolated T-periodic solution of system (1.6) and there's always for all t. Then u is uniformly asymptotically stable (or unstable) if and only if T (u) = 1( T (u) = −1).

Remark 1.2 The constant
2 T 2 + c 2 4 in this condition has been proved irreplaceable.
Lemma 1.2 [29] Assume that the following conditions hold.
(1.6) u + ċu + g(t, u) = 0, Then there is a finite number s 0 such that (1) for s > s 0 , Eq. (1.6) has only two different T-periodic solutions, one is asymptotically stable and the other is unstable; (2) for s = s 0 , Eq. (1.6) has only one T-periodic solution, which is not asymptotically stable; (3) for s < s 0 , each solution of Eq. (1.6) is unbounded.
According to Lemma 1.1, we obtain a result of non-autonomous contact Hamiltonian system (1.1) in following Theorem.

Remark 1.3 T (u) represents the topological degree index of the periodic solution
The contact Hamiltonian system (1.1) is extended to the following contact Hamiltonian system and its corresponding differential system as follows: q, m) is T-periodic with respect to t. Obviously, system (1.13) includes system (1.2), and here we obtain a result of non-autonomous contact Hamiltonian system (1.12) according to Lemma 1.2, that is,    (2) for d = s 0 , system (1.13) has only one T-periodic solution, which is not asymptotically stable; (3) for d < s 0 , every solution of system (1.13) is unbounded.

Theorem 1.4 Assume that the following conditions hold.
Then there exists a number s 0 with finite and (1) for d < s 0 , system (1.13) has only two different T-periodic solutions, one is asymptotically stable and the other is unstable; (2) for d = s 0 , system (1.13) has only one T-periodic solution, which is not asymptotically stable; (3) for d > s 0 , every solution of system(1.13) is unbounded. Theorem 1.2 shows that the stability of the periodic solution of system (1.2) in three-dimensional space depends on the topological degree index of the isolated periodic solution of its corresponding system (1.11) in two-dimensional space. If the nonlinear system (1.11) is complex and difficult to calculate the topological degree index, which can be transformed into a simpler differential system by homotopy transformation, and then the topological degree index of periodic solution can be calculated [22]; In the conclusions of Theorems 1.3 and 1.4, a finite (1.16) lim |q|→+∞ 1 m g(t, q, m) = +∞, (uniformly in t). number s 0 (independent of the value d) is obtained. Although s 0 is difficult to be calculated, but it is like a critical value which divides the dynamics of the contact Hamiltonian system (1.12) satisfying the conditions into three kinds in three-dimensional space. In a sense, the dynamic properties of the conditional contact Hamiltonian system (1.12) in three-dimensional space are completely characterized. From the discussion above, it can be seen that the T-periodic solution (u(t),̇u(t), s 0 (t)) of system (1.2) is uniformly asymptotically stable (or unstable) if and only if T (u) = 1( T (u) = −1) . This completes the proof of Theorem 1.2.

Remark 2.1
Take q = u(t) , we obtain It is easy to verify that p =̇u(t) is asymptotically stable, and the stability (or asymptotic stability) of periodic solution (u(t),̇u(t)) of system (1.11) depends on the stability (or asymptotic stability) of u(t).

Experimental System and Measurement Results of Theorem 1.3
In the following, we give a practical example to verify our main results, that is (H(p, q, s, t) is T-periodic with respect to t); The corresponding differential equations of the contact Hamiltonian system above are as follows:   = p m , p = −m � qe cos 2 (k 1 t)+b 1 + e −q (sin 2 (k 2 t) + b 2 ) � + md − pc, s = p 2 2m − m � 1 2 q 2 e cos 2 (k 1 t)+b 1 − e −q � sin 2 (k 2 t) + b 2 � � + mdq − cs.

Conclusion
In this paper, using the topological index theorem of periodic solutions and the multiplicity result of periodic solutions of forced nonlinear second ordinary differential equations, the periodic dynamic behaviors of a class of non-autonomous contact Hamiltonian systems with time variables are studied. Under the appropriate conditions, the number and stability of periodic solutions of this class of systems are completely analyzed; Because of the topological index theory of periodic solution is limited to two-dimensional space and its application in Liĕnard equation (c(t) in Liĕnard equation ü + c(t)̇u + g(t, u) = 0 is required to be a constant), which limits the study of more general non-autonomous contact Hamiltonian systems. We also expect to study the periodic dynamics of the contact Hamiltonian system H(p, q, s, t) = p 2 2m + G(t, q, m) − mdq + c(t)s with variable sign damping coefficient and higher dimensional contact Hamiltonian system.