We, for the first time, investigate the basic behaviours of a chaotic switching fractional system via both theoretical and numerical ways. To deeply understand the mechanism of the chaos generation, we also analyse the parameterization of the switching fractional system and the dynamics of the system's trajectory. Then we try to write down some detailed rules for designing chaotic or chaos-like systems by switching fractional systems, which can be used in the future application. At last, for the first time, we proposed a new switching fractional system, which can generate three attractors with the positive largest Lyapunov exponent.

Over the last two decades, since chaos has been demonstrated that it can be useful and well controlled [

On the other hand, fractional calculus is a mathematical branch which has more than 300 years of history but just been interested recently in physics [

Generating a new chaotic or chaos-like system is always on the core of the chaos research with great theoretical and applied meanings. In addition a new chaotic system usually gets more complex dynamic behaviours being less recognized by people who does not catch up with the nonlinear dynamics. Thus, a new chaos system may be popular used in chaotic secure communication and encryption. Not like lots of the existing switching systems literatures [

The paper is organized as follows. Section

There exits three main definitions of fractional-order derivatives. They are Grünwald-Letnikov fractional derivatives (G-L):

And since they can transform to each other, our use of fractional derivatives can be free for all of these three definitions.

For the requirement in the next part, we list several theories and lemmas. All of them come from [

We first define

(i)

(ii)

(iii)

More details of

If

If

If

Using Lemma

Consider the following initial value problem for a nonhomogeneous fraction differential equation under nonzero initial conditions:

First, we discuss the chaos generated by the switching fractional system

This kind of chaos was proposed by the literature [

Here we design the switching rule as follows: when

And we take

Then we want to discuss the dynamic behaviours of the chaos more carefully. The switching function (

We first rewrite

And we can obtain the eigenvalues:

Since the eigenvalues are different, we obtain a transformation matrix:

Because of (

By using Lemma

Taking (

Using (

taking (

So we see that the trajectory of

For

Then we can quickly write

So we see that the trajectory of

Now we can do some numerical simulations and research in quantities of relationship between the parameters and the dynamic behaviours of the chaos. We take all the parameters under the restricted conditions we proposed above, so that the switching fractional system (

The chaotic attractor generated by the switching fractional system (

The trajectory of the switching fractional system (

We first focus on the parameters

Phase portraits of the switching fractional system (

Phase portraits of the switching fractional system (

Then we focus on the parameters

Phase portraits of the switching fractional system (

For

Phase portraits of the switching fractional system (

Phase portraits of the switching fractional system (

Then we focus on the parameters

Phase portraits of the switching fractional system (

faster. This conclusion is established in all conditions, which satisfy

Here we epitomize some more detailed rules of relationship of the dynamic behaviours of the switching fractional systems and the parameters of the switching fractional systems. Take the switching fractional system (

So the rules are that we should balance all the parameters when we apply the rules obtained in the literature [

Under the rules above, we here propose a new chaotic or chaos-like switching fractional system

We take the switching rule as follows: when

We take

We take the parameters:

The chaotic attractor generated by the switching fractional system (

We take the parameters:

The chaotic attractor generated by the switching fractional system (

In this paper, we for the first time study the basic behaviours of a chaotic switching fractional system both theoretically and numerically. We also analyse the parameterization of the switching fractional system and the dynamics of the system’s trajectory. Then we try to write down some more detailed rules for designing chaotic or chaos-like systems by switching fractional systems based on the basic rules given in the literature [

This research is supported by the National Natural Science Foundation of China (nos. 61173183, 60973152, and 60573172), the Superior University Doctor Subject Special Scientific Research Foundation of China (no. 20070141014), and the Natural Science Foundation of Liaoning Province (no. 20082165).