具有奇异退化异宿轨的四维Lorenz型超混沌系统研究

Hyperchaos has attracted the great interest of scholars from various fields and areas, and is applied enormously. Due to its complex dynamical behavior, the research on mechanism of hyperchaos is almost nil. Based on the studies of singular degenerate heteroclinic cycle in three-dimensional chaotic systems, we will further study the existence of singular degenerate heteroclinic cycle in four-dimensional hyperchaotic Lorenz-type systems, thereby research on the mechanism of hyperchaos. First, instituting some expressions which can approach the heterocinic solution with higher precision. Second, proving the existence of singular degenerate heteroclinic cycle in some four-dimensional hyperchaotic Lorenz-type systems. Third, Based on the researches of the singular degenerate heteroclinic cycle, we will further reveal the complex dynamics of four-dimensional hyperchaotic Lorenz-type systems and investigate the mechanism of hyperchaos, thereby consummate the hyperchaotic theory system. Hyperchaos exists in nature widely, it has direct or indirect effects on people’s lives. Therefore, the research of four-dimensional hyperchaotic systems bears both theoretical and practical significance.

超混沌有着广泛的应用前景，其应用研究目前已引起普遍的关注，但是关于超混沌产生机理方面的研究则几乎为空白。本项目致力于研究具有奇异退化异宿轨的四维Lorenz型超混沌系统，并藉此来探索超混沌的产生机理。首先，建立更高精度的异宿轨级数逼近方法，给出异宿轨的更加精确的级数近似表达式；其次，证明四维Lorenz型超混沌系统中奇异退化异宿轨的存在性；最后，在以上关于奇异退化异宿轨的研究基础之上，进一步揭示四维Lorenz型超混沌系统的复杂动力学行为、探讨超混沌的产生机理，以期更加完善超混沌的理论体系。超混沌现象普遍存在于自然界中，直接或间接地影响着人们的生活，因此对四维Lorenz型超混沌系统的研究，具有十分重要的理论及实际意义。

本项目主要研究了四维Lorenz型超混沌系统的复杂动力学行为。针对具有有限个平衡点的系统，研究了Hopf及Zero-Zero-Hopf等局部分岔行为，揭示系统局部性质对其全局动力学的影响，以及研究了系统吸引子在全局范围内的共存现象。针对具有平衡点曲线的超混沌系统，首先，基于平衡点的局部稳定性的分析，通过数值地方法证明了奇异退化异宿轨的存在性，并有无穷多奇异退化异宿轨共存于同一相空间中；其次，大量的吸引子共存现象揭示了具有奇异退化异宿轨的超混沌系统的多稳定性；最后，在全局动力学方面，通过Poincare紧致化方法，研究了该类超混沌系统在无穷远处的动力学行为。