分段光滑Filippov系统的动力学研究

项目名称： 分段光滑Filippov系统的动力学研究

项目编号： No.11301551

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 王佳伏

作者单位： 中南林业科技大学

项目金额： 22万元

中文摘要： 分段光滑Filippov 系统大量出现在控制工程、生物学、物理学和经济学等领域的动力学建模中。分段光滑Filippov系统的"不连续性"导致动力系统的许多理论和方法不再适用，且往往引起奇异动力学行为的出现，这给其研究带来了困难和挑战。本项目主要研究分段光滑Filippov系统的由"不连续性"引起的滑模运动、奇异平衡点和奇异周期闭轨三类常见且具有重要应用价值的动力学行为：（1）利用Filippov理论和微分包含理论，讨论滑模运动的存在性，滑模域以及在滑模域上的动力学性质；（2）借助集值分析的不动点定理和非光滑分析的广义Lyapunov方法，研究奇异平衡点的存在性、稳定性和有限时间收敛性；（3）在分岔方法和奇异摄动理论下，利用Poincaré映射研究奇异周期闭轨的存在性和稳定性。通过本项目的研究，发展不连续系统的研究方法，完善分段光滑Filippov系统的理论，为其应用研究提供理论基础。

中文关键词： Filippov系统；不连续系统；稳定性；平衡点；极限环

英文摘要： Piecewise smooth Filippov systems appear numerously in dynamic modeling in many fields such as control engineering, biology, physics and economics。 The "discountinuity" in piecewise smooth Filippov systems makes many theories and methods not work, and leads to the appearance of singular dynamical behaviors,which means that there are difficulty and challenge in the study of piecewise smooth Filippov system. The main goal of this proposed research is to study three important classes of dynamic behaviors, namely sliding motion,singular equilibria and singular peiriodic trejectories, which induced by the "discountinuity" of piecewise smooth Filippov systems. Specifically, we focus on the following three aspects: (1)Employing Filippov theory and differential inclusion theory to explore the existence of sliding motions and to find their sliding domains, and also investigate the dynamics of sliding motions on the sliding domains; (2) Applying fixed point theorems in set-valued analysis and the generalized Lyapunov method in non-smooth analysis to establish the existence, stability and convergence in finite time of singular equilibria; (3) Using the bifurcation theory and singular perturbation theory, together with the Poincare map, to derive conditions for the existence and stability of singular periodic trajectories.

英文关键词： Filippov system；Discontinuous system；Stability；Equilibrium；Limit cycle