Communications in Nonlinear Science and Numerical Simulation
A chaotic system with only one stable equilibrium
Highlights
► This paper reports a surprising discovery of a simple chaotic system with only one stable equilibrium point. ► The Ši’lnikov homoclinic criterion is not applicable for the new system. ► The attracting basin of the stable equilibrium expands gradually as the parameter a increases.
Introduction
For three-dimensional (3D) autonomous hyperbolic type of chaotic systems, a commonly accepted criterion for proving the existence of chaos is due to Ši’lnikov [1], [2], [3], [4], which has a slight extension recently [5]. Chaos in the Ši’lnikov type of 3D autonomous quadratic dynamical systems may be classified into four subclasses [6]:
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chaos of the Ši’lnikov homoclinic-orbit type;
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chaos of the Ši’lnikov heteroclinic-orbit type;
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chaos of the hybrid type with both Ši’lnikov homoclinic and heteroclinic orbits;
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chaos of other types.
In this classification, a system is required to have a saddle-focus type of equilibrium, which belongs to the hyperbolic type at large.
Notice that although most chaotic systems are of hyperbolic type, there are still many others that are not so. For non-hyperbolic type of chaos, saddle-focus equilibrium typically does not exist in the systems, as can be seen from Table 1 which includes several non-hyperbolic chaotic systems found by Sprott [7], [8], [9], [10]. More recently, Yang and Chen also found a chaotic system with one saddle and two stable node-foci [11] and, moreover, an unusual 3D autonomous quadratic Lorenz-like chaotic system with only two stable node-foci [12]. In fact, similar examples can be easily found from the literature.
In this paper, we report a very surprising finding of a simple 3D autonomous chaotic system that has only one equilibrium and, furthermore, this equilibrium is a stable node-focus. For such a system, one almost surely would expect asymptotically convergent behaviors or, at best, would not anticipate chaos per se.
From Table 1, one may observe that the Sprott D and E systems also have only one equilibrium, but nevertheless this equilibrium is not stable. From this point of view, it is easy to understand and indeed easy to prove that the new system will not be topologically equivalent to the Sprott systems.
Section snippets
The mechanism of generating the new system
The mechanism of generating the new system is simple and intuitive.
To start with, let us first review some of the Sprott chaotic systems listed in Table 1, namely those with only one equilibrium. One can easily see that systems I, J, L, N and R all have only one saddle-focus equilibrium, while systems D and E both degenerate in the sense that their Jacobian eigenvalues at the equilibria consist of one conjugate pair of pure imaginary numbers and one real number. Clearly, the corresponding
The co-existence of stable equilibrium and chaotic motion
The new finding in this paper shows that the relation between the local stability of an equilibrium and the global complex dynamical behaviors of a system is subtle. Mathematically, the Hartman–Grobman theorem is about the local behavior of a dynamical system in the neighborhood of a hyperbolic equilibrium point. The new system discussed in this paper shows that although such a system has only one hyperbolic equilibrium point but they turn out to be chaotic globally.
Attracting basin of the equilibrium
When a < 0, the equilibrium is
Conclusion
This paper has reported the finding of a simple three-dimensional autonomous chaotic system which, very surprisingly, has only one stable node-focus equilibrium. The discovery of this new system is striking, because with a single stable equilibrium in a 3D autonomous quadratic system, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Yet, unexpectedly, this system is chaotic. Although the equilibrium is changed from an unstable saddle-focus to a stable
Acknowledgement
This research was supported by the National Natural Science Foundation of China under Grant 10832006 and the Hong Kong Research Grants Council under Grant CityU1114/11E.
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