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

A special three-dimensional chaotic system was proposed in 2016, as a dual system of Chua system, which is satisfied <inline-formula> <tex-math notation="LaTeX">$a_{12}\cdot $ </tex-math></inline-formula> <inline-formula> <tex-math notation="LaTeX">$a_{21}< 0$ </tex-math></inline-formula><italic>.</italic> The dynamics characteristics are different from the Jerk system (<inline-formula> <tex-math notation="LaTeX">$a_{12}\cdot $ </tex-math></inline-formula> <inline-formula> <tex-math notation="LaTeX">$a_{21}=0$ </tex-math></inline-formula>) and Chua system (<inline-formula> <tex-math notation="LaTeX">$a_{12}\cdot $ </tex-math></inline-formula> <inline-formula> <tex-math notation="LaTeX">$a_{21}>0$ </tex-math></inline-formula>). In this paper, a method for generating <italic>M</italic> <inline-formula> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> <italic>N</italic> <inline-formula> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> <italic>L</italic> 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.