三维超混沌映射拓扑马蹄寻找算法及应用

拓扑马蹄理论是严格研究混沌的重要理论,然而却很少用在超混沌的研究中.主要原因是超混沌系统不仅相空间维数比普通混沌高,而且存在的拉伸方向数也较多,导致拓扑马蹄的寻找难度很大.为此,本文针对三维超混沌映射,提出一种实用的拓扑马蹄寻找算法.超混沌系统通常有较大的负Lyapunov指数,其吸引子会靠向某一曲面.基于这种特性,本文首先沿着系统收缩方向进行降维,得出二维平面投影系统;接着在新系统中搜索二维拉伸的投影马蹄;最后利用投影马蹄升维构造出原三维系统拓扑马蹄.为了验证算法的有效性,本文以经典Lorenz超混沌系统和著名Saito超混沌电路为例,利用数值计算,在它们的Poincare映射中找出了具有二维拉伸的三维拓扑马蹄.

Algorithm for finding horseshoes in three-dimensional hyperchaotic maps and its application

Topological horseshoe theory is fundamental for studying chaos rigorously, which, however, has rarely applied to hyperchaos. The reason is that it is too hard to find a topological horseshoe in a hyperchaotic system, due to the high dimension of the system and the multiple expansion directions in the state space. Therefore, in this paper a practical algorithm for three-dimensional (3D) hyperchaotic maps is proposed. Usually, a hyperchaotic system has a large negative Lyapunov exponent, its attractor is often contracted closely to a certain surface. Based on this feature, the algorithm first deducts the dimension along the direction of contraction to obtain a 2D projective system; then it detects a projective horseshoe with 2D expansion; finally, it constructs a 3D horseshoe for the original system. In order to verify the validity of the algorithm, it is applied to the classic hyperchaotic Lorenz system and the famous Saito hyperchaotic circuit, and their horseshoes with 2D expansion are successfully found from the Poincaré mapping.