Antimonotonicity, Chaos and Multidirectional Scroll Attractor in Autonomous ODEs Chaotic System

Abstract

Three-dimensional autonomous ordinary differential equations (ODEs) are the simplest and most important chaotic systems in nonlinear dynamics. In fact, they have been applied in many fields. In this paper, a systematic methodology for analyzing complex behavior of the ODEs chaotic system, as one of the ODEs chaotic systems, the improved TCS which satisfies the condition a 12 a 21 = 0, is proposed. It is dissipative, chaos, symmetric, antimonotonicity and can generate multiple directional (M × N × L) scroll attractors. Then, bifurcation diagrams, Lyapunov exponents, time series, Poincare sections, and Hausdroff dimensions are analyzed by setting the parameters and initial value. More interestingly, antimonotonicity (named reverse period-doubling bifurcation) and coexisting bifurcations are also reported. Finally, the results of theoretical analyses may be verified by electric experimental.

Bounded Orbits and Multiple Scroll Coexisting Attractors in a Dual System of Chua System

“…2); (ii) with the same conditions but the shorter time, multi-scroll attractors can also be obtained, but the number of number is less than the analysis results, which is not a final state and dangerous. Once it is applied to the engineering, it will lead to the wrong effective or immeasurable losses; (iii) the different number scrolls could be presented and identified hysteretic dynamics (imply multiple stability), which have previously observed in some chaotic system [21,26]. When a=1, b=0.5, c=0.85, d=1.5, h=0.1, e=0.6, the coexisting scroll-1 attractors are given in Fig.8.…”

“…Thus, such a novel method for generating M×N×L grid scroll attractors and the existence proof on bounded orbits should be supplied, which are our main objective. Furthermore, the related research on multiple scroll coexisting attractors has been found in no literature before, except [21].…”

“…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. Kengne et al presented dynamics for Jerk or Jerk-like system with multiple attractors [11,12,19], Q. Lai and S. Chen proposed a polynomial function method for generating multiple strange attractors from the Sprott B system [31], just to name a few; (ii) all kinds of analysis methods have been described on unusual and striking dynamics, such as the, self-excited antimonotonicity [11,14,21] and hidden attractors [15], coexisting attractors [16][17][18][19][22][23][24], without equilibrium chaotic system [10,18,19], and multiple stability [20] and synchronization [8,33] as new research directions, are still in their infancy. Especially, in n-scroll chaotic system, the occurrence of two or more asymptotically stable or attracting basin have been analyzed, symmetry and broken symmetry have been reported.…”

“…For example, chaos prediction in nonlinear system subjected to subsonic flow and external load was demonstrated by D. Younesian and H. Norouzi [6], chaotic system synchronization and message recovery scheme was proposed by V. Sharma, B.B. Sharma, and R. Nath [8,33], the new chaotic systems and their electronic implementation were proposed [13,[21][22][23][24], image encryption based on some chaotic systems were explored [23][24][25], design and implementation based on the ARM-embedded and FPGA for secure video communication system were presented [37][38][39][40], and so on; (iv) the proof for existence on the complex scenario of bounded orbits [26][27][28][29] for chaotic and hyper-chaotic systems are in its infancy. It has became a serious attempt and gained a wide attention in academic circles.…”

“…There are many literatures on analysis and applications for some classical chaotic systems, i.e. Chua system, Jerk circuit, Van der Pol-Duffing circuit and Sprott B system [3][4][5]12,15,16,19,21], et al However, some essential works have been neglected, such as the dual system of Chua system, which is also a special type of a canonical form and proposed in 2016. This system satisfies the condition a 12 •a 21 < 0.…”