竞争映射的动力学性态及其应用

We present the definitions of the order decomposition and the hypersurface in the competitive dynamical systems, and establish the general properties of the discrete-time competitive systems.We prove that every limit set lies in an invariant Lipschitz submanifold with codimension-one, and all nonwandering points lie in these Lipschitz submanifolds. Moreover, the topological.entropy of the competitive system is equal to the supremum of the topological entropies of the.restrictions of the system to these submanifolds. For planar competitive systems Sarkovskii Theorem and many other properties of one-dimensional dynamical systems still hold.For.n-dimensional dissipative irreducible Kolmogorov competitive mappings and type-K competitive autonomous systems, there exist at most countable many Lipschitz submanifolds with.codimension-one which attract all persistent orbits. We also provide the uniqueness condition of such a Lipschitz submanifold. We classify 3-dimensional L-V systems, give the fomula of the index of the type-K competitive mapping and give the nessecary and sufficient conditions.guaranteeing that all the orbits converge to the fixed point or the positive fixed point. For time-periodic reaction-diffusion equations, we give the weakest conditions guaranteeing that all the orbits converge to the periodic orbits of the system.

本项目研究抽象的离散竞争动力系统的长期性态。讨论周期点的吸引域、连结轨线及它们所组成的树结构；研究正极限集的性质、序结构及位置，着重研究它们位于什么样的李普希茨流形及这些流形的序结构、光滑性和余维；探索系统的复杂性，研究一般的动力系统到竞争系统的嵌入，对低维映射，证明沙尔可夫斯基型的定理，提供各种浑沌产生的判据。