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A novel generation method of chaotic attractor is introduced in this paper. The underlying mechanism involves a simple three-dimensional time-varying system with simple time functions as control inputs. Moreover, it is demonstrated by simulation that various attractor patterns are generated conveniently by adjusting suitable system parameters. The largest Lyapunov exponent of the system has been obtained.

Chaos and Chaotic systems, which have been extensively investigated during the last four decades, have been found to be very useful in a variety of applications such as science, mathematics, engineering communities [

Many chaotic attractors in dynamical systems have been found numerically and experimentally, such as Lorenz attractor [

In general case, it is relatively easy to generate chaotic systems numerically, but it is usually very hard to analyze or verify the dynamical characteristics of nonsmooth systems, even for the switched systems with low dimensions [

Motivated by previous works of chaotic attractor generation, we have made further effort to generate more chaotic behaviors, by introducing time functions, that is, a time-varying system with various time functions is investigated. To our happiness, chaotic attractors are observed. And moreover, the shape of the created attractors can be changed easily by changing parameters, and various complex patterns can be obtained. The statistic behavior is also discussed, which reveals the regularities in the complex dynamics.

The rest of this paper is organized as follows. Section

There are two major researching methods of the chaotic attractor, analytical method and numerical analysis. The analytical methods had two general methods, the Melnikov method and the Kalashnikov method. However, in general case, we used the numerical method due to the difficulties of the analytical method.

Numerical method bases on theoretical achievements and computer simulation. Recently, there were a lot of those achievements, such as the Poincare cross-section, power spectrum, subsampling frequency, and continuous feedback control. In addition, the spectrum analysis, Lyapunov index, fractal dimension and topological entropy, and so forth, were common methods to describe the statistical properties of chaotic.

Some analysis results of chaotic attractor generation were given [

In this section, we studied the conditions of chaotic attractor generation. We first concern general linear system as follows:

We assume that

The range of motion of chaotic attractor can be defined, and there should be both of convergent motions and divergent motions of chaotic attractor in the certain region. In consideration of the above, it can be simulated in the case of the equilibrium being stable (where convergent) or the equilibrium is unstable (where divergent).

Therefore, a necessary condition of chaotic attractor generation for system (

At the equilibrium

From the analysis in Section

We assume that:

Considering the system trajectory nearby

And the characteristic polynomials of (

The characteristic polynomial in (

The system (

Chaotic attractor generated by system (

Chaotic attractor

In this section, we pay attention to the dynamical behaviors of the time-varying system (

At first, we do not change constant parameters from (

Chaotic attractor changed by

In the three cases, the largest Lyapunov exponents are

Then, we chose parameters from (

Different values of parameters

Different values of parameters

The largest Lyapunov exponents are given as follows:

From these numerical simulations, it is shown that the system (

This paper has given a novel chaotic attractor generation method. The generation of novel chaotic attractors via a simple three-dimensional time-varying system with various time functions has been introduced. The results once again support the long-accepted belief that properly designed simple systems can perform complex dynamical behaviors. Moreover, this system can produce various attractor patterns within a wide range of parameter values, and the statistic behavior which reveals the regularities in the complex dynamics is also discussed. In addition, the method has been developed in this article can also be applied to nonlinear dynamical systems and other fields. It is desirable that one could design more chaos generators by means of the method proposed in this paper.