Julia fractal based multi-scroll memristive chaotic system

Abstract

A memristor can be used in chaotic system as a nonlinear term, and thus enhancing the complexity of the chaotic system. Fractal theory is a leading and important branch of nonlinear science, and has been widely studied in many fields in the past few decades. The fractal and chaos are bound tightly and their relevant researches are well-established, but few of them focus on the research of the possibility of combining the fractal and the chaotic system. In order to obtain a multi scroll chaotic attractor, the fractal process is novelty introduced into the memristive chaotic system. In this paper, at first, a new memristive chaotic system is proposed. Then, the dynamic characteristics of the system are discussed from the aspects of symmetry, dissipation, stabilization of equilibrium points, power spectrum, Lyapunov exponent and fractional dimension. A mapping relationship based on classical Julia fractal is established. Through this mapping relationship, a multi-scroll memristive chaotic system based on the Julia fractal is obtained. Moreover, several deformed Julia fractal processes are applied to the memristive chaotic system, and abundant chaotic attractors are obtained. For example, the square term of the Julia fractal expression is multiplied by a coefficient, and according to the difference in coefficient, the resulting chaotic attractors have the same shape but different sizes. The exponent of the square term in the Julia fractal is changed into a variable, and the chaotic attractor of different scroll numbers is obtained according to the difference in power exponent. In addition, a rich multi-scroll chaotic attractor is obtained by using the fractal expression in the form of weighted sum polynomial. Finally, the influence of a complex constant in the fractal process on the system is discussed. The simulation results show that the combination of fractal process and chaotic system can obtain rich chaotic attractors, such as multi-scroll chaotic attractors. In general, compared with the single-scroll chaotic attractor, the multi-scroll chaotic attractor has a higher complexity and more adjustability. In addition, compared with other multi-scroll chaotic system, the proposed multi-scroll chaotic system is easy to adjust the number of the scrolls. To summarize, this work not only provides a new method of generating multi-scroll chaotic attractors, but also makes up for the lack of smoothness of the chaotic system caused by using functional methods.