Existence of Periodic Solutions in Distribution for Stochastic Newtonian Systems
- PDF / 2,078,552 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 11 Downloads / 213 Views
Existence of Periodic Solutions in Distribution for Stochastic Newtonian Systems Xiaomeng Jiang1,2 · Yong Li1,3 · Xue Yang1,3 Received: 13 December 2019 / Accepted: 30 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Periodic phenomena such as oscillation have been studied for many years. In this paper, we verify the stochastic version of Levinson’s conjecture, which confirmed the existence of stochastic periodic solutions for second order Newtonian systems with dissipativeness. First, we provide a stochastic Duffing’s equation to display our result. Then, we apply Wong– Zakai approximation method and Lyapunov’s method to stochastic second order Newtonian systems driven by Brownian motions. With the help of Horn’s fixed point theorem, we prove that this kind of systems is stochastic dissipative and admits periodic solutions in distribution. Keywords Levinson’s conjecture · Stochastic Newtonian systems · Wong–Zakai approximations · Periodic solutions in distribution · Lyapunov’s method
1 Introduction In mechanics, a classical class of second order differential equations is applied to model oscillation. Several typical examples are listed below. 1. Duffing’s equation: 2π x¨ + d1 x˙ + d2 x + d3 x 3 = d4 cos t , (1) T
Communicated by Jorge Kurchan.
B
Xiaomeng Jiang [email protected] Yong Li [email protected] Xue Yang [email protected]
1
College of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
2
College of Computer Science and Technology, Jilin Unversity, Changchun 130012, People’s Republic of China
3
School of Mathematics and statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, People’s Republic of China
123
X. Jiang et al.
where T > 0, d1 , d2 , d3 , d4 are constants. Duffing’s equation can model, for instance, the motion of spring pendulum. When d1 = d3 = 0, this equation describes a simple harmonic motion. 2. Liénard’s equation: x¨ + l1 (x)x˙ + l2 (x) = e(t), (2) where l1 , l2 ∈ C(R, R). Moreover, l1 is an even function. l2 is an odd function. e ∈ C(R+ , R) is T -periodic. This equation can be applied in radio or vacuum tube technology to model oscillating circuits. 3. Rayleigh’s equation: x¨ + r1 (x) ˙ + r2 (x) = e(t), (3) where r1 , r2 : R → R are continuous and e is the same as in Eq. (2). This equation can be applied in electromagnetic oscillation circuit and model the changes of current effected by periodic force. These equations can be classified as the following Newtonian system x¨ + A(t, x)x˙ + ∇V (x) = e(t).
(4)
Here, A and e are T -periodic in time t. According to physical principle, the energy of this system would decrease when A > 0 and hence, every Newtonian system with friction is dissipative. According to physical principles, system (4) must admit T -periodic solutions. Namely, there at least exists a solution x such that, x(t + T ) = x(t) for all t ≥ 0.
(5)
In other words, effected by friction which could reduce the energy of the system, the o
Data Loading...
