Chaos Entanglement: Leading Unstable Linear Systems to Chaos

Chaos entanglement is a new approach to connect linear systems to chaos. The basic principle is to entangle two or multiple linear systems by nonlinear coupling functions to form an artificial chaotic system/network such that each of them evolves in a chaotic manner. However, it is only applicable for stable linear systems, not for unstable ones because of the divergence property. In this study, a bound function is introduced to bound the unstable linear systems and then chaos entanglement is realized in this scenario. Firstly, a new 6-scroll attractor, entangling three identical unstable linear systems by sine function, is presented as an example. The dynamical analysis shows that all entangled subsystems are bounded and their equilibrium points are unstable saddle points when chaos entanglement is achieved. Also, numerical computation exhibits that this new attractor possesses one positive Lyapunov exponent, which implies chaos. Furthermore, a 4 x 4 x 4-grid attractor is generalized by introducing a more complex bound function. Hybrid entanglement is obtained when entangling a two-dimensional stable linear subsystem and a one-dimensional unstable linear subsystem. Specifically, it is verified that it is possible to produce chaos by entangling unstable linear subsystems through linear coupling functions - a special approach referred to as linear entanglement. A pair of 2-scroll chaotic attractors are established by linear entanglement. Our results indicate that chaos entanglement is a powerful approach to generate chaotic dynamics and could be utilized as a guideline to effectively create desired chaotic systems for engineering applications.