分段光滑Filippov系统的动力学研究

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. Successfully accomplishing the proposed research will make a significant contribution to the development of piecewise smooth Filippov systems in theories, methods and applications.

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。

本项目以分段光滑Filippov系统为研究对象，主要讨论由系统的不连续性导致的动力学行为。对于滑模动力学和奇异平衡点的理论研究，本项目去掉了Lyapunov函数的光滑性条件，得到了奇异平衡点的稳定和渐近稳定的结果；对于滑模动力学和奇异平衡点的应用研究，讨论了一类具有经济阈值策略的植物病虫害分段光滑Filippov模型的全局动力学，并得到了奇异平衡点有限时间收敛的结果；对于分段光滑Filippov系统的极限环研究，给出了分段线性Filippov系统极限环的存在性和稳定性的结果，并得到了至少存在两个以及恰好存在两个极限环的充分条件。通过本项目的研究，进一步揭示了不连续系统的本质特征，也给不连续系统的应用研究提供了理论基础。