On dynamics analysis of a new chaotic attractor

doi:10.1016/j.physleta.2008.07.032

Physics Letters A

Volume 372, Issue 36, 1 September 2008, Pages 5773-5777

https://doi.org/10.1016/j.physleta.2008.07.032 Get rights and content

Abstract

In this Letter, a new chaotic system is discussed. Some basic dynamical properties, such as Lyapunov exponents, Poincaré mapping, fractal dimension, bifurcation diagram, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed in this Letter is a new chaotic system and deserves a further detailed investigation.

Introduction

Since Lorenz found the first chaotic attractor in a smooth three-dimensional autonomous system, considerable research interests have been made in searching for the new chaotic attractors. In 1976, Rössler found a three-dimensional autonomous smooth chaotic system. Later, more and more attractors were found. Some new chaotic systems were recently coined, such as Chen system, Lü system, Liu system, the generalized Lorenz system family, and the hyperbolic type of the generalized Lorenz canonical form [1], [2].

A new chaotic system is discussed in this Letter, its coined process is similar to the Lü attractor coined. It is a three-dimensional autonomous system, according to the detailed numerical simulation as well as the theoretical analysis, the chaotic attractor obtained from this new system is also the butterfly-shaped attractor. The chaotic system is a new attractor which is similar to the Lorenz chaotic attractor, but it is not topological equivalent with the Lorenz chaotic system [3], [4]. This Letter is devoted to a more detailed analysis of this new chaotic attractor.

Section snippets

Finding the new chaotic system

In this section, we briefly describe how the anti-control procedure suggested in [4], [5] leads to the finding of the new chaotic attractor.

Start with the controlled Lorenz system [3]: ${\begin{matrix} \dot{x} = a (y - x), \\ \dot{y} = b x - x z - y + u, \\ \dot{z} = x y + c z, \end{matrix}$ where a, b, c are constants, currently not in the range of chaos, and u is a linear feedback controller in the following form $u = l_{1} x + l_{2} y + l_{3} z,$ in which $l_{1}$ , $l_{2}$ , $l_{3}$ are constant gains to be determined. The controlled system Jacobian evaluated at $(x_{0}, y_{0}, z_{0})$ is given by $J_{x_{0}, y_{0}, z_{0}} = (\begin{matrix} - a & a & 0 \\ b + l_{1} \end{matrix}$

Some basic properties of the new system

In this section, we will investigate some basic properties of the new system (6).

Conclusions

In this Letter, we have constructed a new chaotic system (6). This new chaos attractor is different from the Lorenz attractor. In the new chaotic system, there are abundant and complex dynamical behaviors. The topological structure of the new system should be completely and thoroughly investigated, and the forming mechanism of the system (6) needs further studying and exploring. It is expecting that more detailed theory analyses and simulation investigations will be provided in a forthcoming

Acknowledgement

The authors are grateful to the National Natural Science Foundation of Chain (60503027).

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