Abstract
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors.
Highlights
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required
Since the Jacobian matrix of a chaotic system is closely related to Lyapunov exponents, the Chen–Lai algorithm can be used as the principle to construct new discrete chaotic maps
Through the existence and stability analysis of fixed points, we prove that such a class of quadratic polynomial maps cannot have hidden chaotic attractors
Summary
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. Since the general construction method of traditional chaotic systems cannot effectively analyze the existence and stability of fixed points, the exhaustive search is still the main way to find new chaotic systems with hidden chaotic attractors. For the lack of a systematic theory on fixed points, the existing general method of constructing new traditional chaotic maps and the analysis of chaotic hidden attractors cannot be combined organically. According to the period-three theorem, we propose a general method to construct quadratic polynomial chaotic maps, and systematically analyze the existence of hidden attractors in such a class of quadratic polynomial chaotic maps. The general method proposed in this paper overcomes the defect of the low efficiency of finding new chaotic maps by exhaustive search and poor amplitude control ability by the existing general construction method of traditional chaotic systems.
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