Abstract
A special three-dimensional chaotic system was proposed in 2016, as a dual system of Chua system, which is satisfied $a_{12}\cdot $ $a_{21} . The dynamics characteristics are different from the Jerk system ( $a_{12}\cdot $ $a_{21}=0$ ) and Chua system ( $a_{12}\cdot $ $a_{21}>0$ ). In this paper, a method for generating M $\times $ N $\times $ L grid multiple scroll attractors is presented for this system. Also, in order to ensure the rigor of the theoretical results, we prove existence of the complex scenario of bounded orbits, such as homoclinic and heteroclinic orbits, and illustrate concurrent created and annihilated of symmetric orbits. Then, Shilnikov bifurcation and the possible relationship between the birth and death of the scroll attractors are studied. Furthermore, two theorems are demonstrated for these bounded orbits. Finally, the Lyapunov exponents, bifurcation diagrams, and multiple scroll coexisting attractors are displayed, which are related to the parameters and initial condition.
Highlights
By and large, the research on chaos, various types of strange attractors, coexisting attractors, synchronization control and their applications have became the hotspots [15]. Their interesting and complex dynamical phenomena are applied in various fields, for example chaos prediction in nonlinear viscoelastic plates [6], nonlinear resonances and multi-stability in simple neural circuits [7], synchronization control [8, 33], grid multi-scroll chaotic attractors [4, 5, 9, 30], multi-wing hyper chaotic system [10, 30], electronic circuits [11,12,13, 22,23,24], image encryption application [25, 34], oscillator of memory elements [14, 35, 36,41], embedded implementation for digital systems [37,38,39,40] et al As well as we known, the chaotic application in electronic circuits began Chua circuit and generation of multiple scroll attractors started from Suykens and Vandewalle, beyond the double scroll [1]
In 2016, we proposed a dual system of Chua system, which can be thought a systematic methodology for creating multiple scroll chaotic attractors from a simple threedimensional system
In system (1), the attractors are only in one direction, the others directions cannot be realized; (v) as the dual model of Chua system and a new type of canonical form, the distribution of multiple scroll attractors should be throughout the whole real space
Summary
The research on chaos, various types of strange attractors, coexisting attractors, synchronization control and their applications have became the hotspots [15]. The main work and research on chaos theory, analysis methods and application fields are as follows: (i) chaotic systems and scroll attractors seem to have reached their maturity, i.e. some scholars and their co-workers proposed plenty of methods for designing the grid multi-scroll and multi-wing chaotic and hyper chaotic attractors [4, 5, 9, 29,30,31], L. Zhou and his fellows proposed a novel no-equilibrium hyper chaotic system [10], J. THE THREE-DIMENSIONAL CHAOTIC SYSTEM AND M×N×L GRID MULTIPLE SCROLL ATTRACTORS
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