Abstract
The investigations of hidden attractors are mainly in continuous-time dynamic systems, and there are a few investigations of hidden attractors in discrete-time dynamic systems. The classical chaotic attractors of the Logistic map, Tent map, Henon map, Arnold’s cat map, and other widely-known chaotic attractors are those excited from unstable fixed points. In this paper, the hidden dynamics of a new two-dimensional map inspired by Arnold’s cat map is investigated, and the existence of fixed points and their stabilities are studied in detail.
Highlights
The investigations of the chaotic system were greatly encouraged by the discovery of the Lorenz system [1]
The Lorenz system is one of the most wildly-studied continuous-time dynamic systems, and other classical continuous-time dynamic systems include the Rössler system, Chua system, Chen system, Lü system, and Sprott system [2,3,4,5,6]. Most attractors of those classical continuous-time dynamic systems are excited from unstable equilibria
A lot of systems improved when classical continuous-time dynamic systems were proposed [8,9,10,11,12,13,14,15,16,17,18,19]. These investigations of hidden attractors can be classified by the number and stability of equilibria, such as no equilibrium, finite stable equilibria, and infinite stable equilibria
Summary
The investigations of the chaotic system were greatly encouraged by the discovery of the Lorenz system [1]. A lot of systems improved when classical continuous-time dynamic systems were proposed [8,9,10,11,12,13,14,15,16,17,18,19]. In line with continuous-time dynamic systems, these hidden attractors in the classical chaotic maps are excited from unstable fixed points. Due to the restriction of the form of the Arnold’s cat map, the new two-dimensional chaotic attractor can only appear in the case of no fixed point. The complexity of output time series is tested by approximate entropy in Section 5, and Section 6 summarizes the conclusions of this paper
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