Physica D 240 (2011) 1395–1401 Contents lists available at ScienceDirect Physica D journal homepage: www.elsevier.com/locate/physd An averaging principle for stochastic dynamical systems with Lévy noise Yong Xu a,∗ , Jinqiao Duan b , Wei Xu a a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China b Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA article info Article history: Received 18 April 2010 Received in revised form 31 May 2011 Accepted 2 June 2011 Available online 12 June 2011 Communicated by G. Stepan Keywords: Averaging principle Stochastic differential equations Non-Gaussian Lévy noise Convergence to the averaged system abstract The purpose of this paper is to establish an averaging principle for stochastic differential equations with non-Gaussian Lévy noise. The solutions to stochastic systems with Lévy noise can be approximated by solutions to averaged stochastic differential equations in the sense of both convergence in mean square and convergence in probability. The convergence order is also estimated in terms of noise intensity. Two examples are presented to demonstrate the applications of the averaging principle, and a numerical simulation is carried out to establish the good agreement. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Asymptotical methods play an important role in investigat- ing nonlinear dynamical systems. In particular, the averaging methods provide a powerful tool for simplifying dynamical sys- tems, and obtain approximate solutions to differential equa- tions arising from mechanics, mathematics, physics, control and other areas. Some rigorous results on averaging principles date back to Krylov and Bogoliubov’s work [1] and the resulting KBM (Krylov–Bogoliubov–Mitropolsky) method [2–4]. Random fluctuations or noises appear in various engineering and social systems. Stochastic dynamics has mainly dealt with Gaussian noise. Some authors have started investigating averaging principles for systems with Poisson noise, a special non-Gaussian noise [5–7]. Averaging principles for stochastic systems were proposed by Stratonovich [8,9] in order to examine nonlinear os- cillation problems in the presence of random noise. For aver- aging principles for systems of stochastic differential equations, see [10–15], among others. The stochastic averaging methods have been found useful and effective for exploring stochastic differen- tial equations in problems in many fields [16–20]. It has become evident that random noises in practice are more likely to be non- Gaussian [21]. In [22], a rigorous theorem as regards an averag- ing principle for a class of stochastic differential equations with ∗ Corresponding author. E-mail addresses: hsux3@nwpu.edu.cn (Y. Xu), duan@iit.edu (J. Duan), weixu@nwpu.edu.cn (W. Xu). Poisson noise was obtained. Namely, the authors proved that un- der some conditions the solutions to averaged systems (still driven by Poisson noise) converge to the solutions of the original sys- tems (driven by Poisson noise) in a certain sense. In [23] similar results were provided for integral–differential equations with Pois- son noise. Note that Poisson noise is a special non-Gaussian Lévy noise [21,24]. Stochastic differential equations driven by Lévy noise have attracted great attention recently [25,26], but stochastic dynamics is still in its infancy. To the authors’ knowledge, averaging principles for stochastic differential equations with general non-Gaussian Lévy noise have not been considered. Therefore in this paper we consider these averaging principles. This paper is organized as follows. Section 2 introduces Lévy motion briefly. In Section 3, we recall some basic results for SDEs with Lévy noise and properties of solutions. In Section 4, we prove an averaging principle for standard stochastic differential equations with Lévy noise. Finally, in Section 5 we present two examples to demonstrate the averaging method. 2. Lévy motions A scalar Lévy motion L t is characterized by a drift parameter θ , a variance parameter d > 0 and a non-negative Borel measure ν . This measure ν is defined on (R, B(R)), is concentrated on R \{0}, and satisfies ∫ R\{0} (y 2 ∧ 1)ν(dy)< ∞, (1) 0167-2789/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2011.06.001