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

Abstract

A special three-dimensional chaotic system was proposed in 2016, as a dual system of Chua system, which is satisfied a 12 •a 21 < 0. The dynamics characteristics are different from the Jerk system (a 12 •a 21 =0) and Chua system (a 12 •a 21 > 0). In this paper, a method for generating M×N×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. INDEX TERMS Grid multiple scroll attractors; homoclinic and heteroclinic orbits; Shilnikov bifurcation; Coexisting attractors.

Nonlinear circuits with parallel‐/series‐connected HP‐type memory elements and their characteristic analysis

“…Subsequently, M. A Zidan et al evaluated the state of the art in memristor‐based electronics and explored where the future of the field lies, 26 including on‐chip memory and storage, biologically inspired computing, and general‐purpose in‐memory computing. Moreover, chaotic oscillator, 27 dynamic systems, 11,28,29 application on neural networks, 1 long short‐term memory networks, 30 the high‐frequency device, 19 nano‐programmable logics, 31,32 the circuit implementation of nano‐scale TiO2 memristor, 32 and so on with memory elements have emerged.…”

“…MC and u(t) are state variables of the system, and a u , b u , c u , d u , e u , f u are the coefficients of MR, MC, and ML.Furthermore, let x = ρ, x 0 = ϕ, x 00 = u, if a voltage source is used as an input signal, by substituting(28) into(29), the unified form can be obtained as following:a u x 0 + b c u Á x 0 + d u ð…”

Coexisting Multiple Attractors in a Fourth Order Chua’s Circuit With Experimental Verifications by Analog and Digital Circuits

“…On the basis of this classical circuit, the various forms of generalized Chua's circuits and systems [13]- [26] build a deep recognition of chaos for the academia further. In these studies, the new features such as stable equilibrium points [14]- [17], multi-scroll attractors [18]- [21], multiple attractors [17], multiple scroll coexisting attractors [22], SC-CNN-based mode [23] and inductorless implementation approaches [24]- [26] have been revealed in succession. The fourth order Chua's circuit is also a form of generalized Chua's circuit, yet the related reports are not often noticed.…”

Continuation-Based Method to Find Periodic Windows in Bifurcation Diagrams With Applications to the Chua’s Circuit With a Cubic Nonlinearity

“…The existence of Shilnikov and topological chaos is proved in [23] and [24]. Bifurcations diagrams are constructed for the dual Chua's system in [25]. The Chua's circuit with a third order polynomial (cubic) nonlinearity is introduced in [26].…”