Xiong Wang

A chaotic system with only one stable equilibrium

Abstract If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the equilibrium is changed from an unstable saddle-focus to a stable node-focus, therefore the familiar Si’lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.

Constructing a chaotic system with any number of equilibria

In the chaotic Lorenz system, Chen system and Rössler system, their equilibria are unstable and the number of the equilibria are no more than three. This paper shows how to construct some simple chaotic systems that can have any preassigned number of equilibria. First, a chaotic system with no equilibrium is presented and discussed. Then a methodology is presented by adding symmetry to a new chaotic system with only one stable equilibrium, to show that chaotic systems with any preassigned number of equilibria can be generated. By adjusting the only parameter in these systems, one can further control the stability of their equilibria. This result reveals an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of a chaotic system.

Constructing a Novel No-Equilibrium Chaotic System

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincare map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

COEXISTENCE OF POINT, PERIODIC AND STRANGE ATTRACTORS

For a dynamical system described by a set of autonomous ordinary differential equations, an attractor can be a point, a periodic cycle, or even a strange attractor. Recently, a new chaotic system with only one stable equilibrium was described, which locally converges to the stable equilibrium but is globally chaotic. This paper further shows that for certain parameters, besides the point attractor and chaotic attractor, this system also has a coexisting stable limit cycle, demonstrating that this new system is truly complicated and interesting.

A Novel No-Equilibrium Chaotic System with Multiwing Butterfly Attractors

Discovering unknown features of no-equilibrium systems with hidden strange attractors is an attractive research topic. This paper presents a novel no-equilibrium chaotic system that is constructed by using a state feedback controller. Interestingly, the new system can exhibit multiwing butterfly attractors. Moreover, a new chaotic system with an infinite number of equilibrium points, which can generate multiscroll attractors, is also proposed by applying the introduced methodology.

A Chaotic System with Different Shapes of Equilibria

Although many chaotic systems have been introduced in the literature, a few of them possess uncountably infinite equilibrium points. The aim of our short work is to widen the current knowledge of t...

Is that Really Hidden? The Presence of Complex Fixed-Points in Chaotic Flows with No Equilibria

In this letter we investigate the role of complex fixed-points in finding hidden attractors in chaotic flows with no equilibria. If these attractors could be found by starting the trajectory in the neighborhood of complex fixed-points, maybe it would be better not to call them hidden.

A Chaotic System with Different Families of Hidden Attractors

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

A SIMPLE YET COMPLEX ONE-PARAMETER FAMILY OF GENERALIZED LORENZ-LIKE SYSTEMS

This paper reports the finding of a simple one-parameter family of three-dimensional quadratic autonomous chaotic systems. By tuning the only parameter, this system can continuously generate a variety of cascading Lorenz-like attractors, which appears to be richer than the unified chaotic system that contains the Lorenz and the Chen systems as its two extremes. Although this new family of chaotic systems has very rich and complex dynamics, it has a very simple algebraic structure with only two quadratic terms (same as the Lorenz and the Chen systems) and all nonzero coefficients in the linear part being -1 except one -0.1 (thus, simpler than the Lorenz and Chen systems). Surprisingly, although this new system belongs to the family of Lorenz-type systems in some existing classifications such as the generalized Lorenz canonical form, it can generate not only Lorenz-like attractors but also Chen-like attractors. This suggests that there may exist some other unknown yet more essential algebraic characteristics for classifying general three-dimensional quadratic autonomous chaotic systems.

From Wang–Chen System with Only One Stable Equilibrium to a New Chaotic System Without Equilibrium

Wang–Chen system with only one stable equilibrium as well as the coexistence of hidden attractors has attracted increasing interest due to its striking features. In this work, the effect of state feedback on Wang–Chen system is investigated by introducing a further state variable. It is worth noting that a new chaotic system without equilibrium is obtained. We believe that the system is an interesting example to illustrate the conversion of hidden attractors with one stable equilibrium to hidden attractors without equilibrium.