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No-equilibrium system with chaotic behavior has attracted considerable attention recently because of its hidden attractor. We study a new four-dimensional system without equilibrium in this work. The new no-equilibrium system exhibits hyperchaos and coexisting attractors. Amplitude control feature of the system is also discovered. The commensurate fractional-order version of the proposed system is studied using numerical simulations. By tuning the commensurate fractional-order, the proposed system displays a wide variety of dynamical behaviors ranging from coexistence of quasiperiodic and chaotic attractors and bistable chaotic attractors to point attractor via transient chaos.

It is now well established from a variety of studies that a hyperchaotic system is specified by having at least two positive Lyapunov exponents [

It is interesting that Wang et al. found a hyperchaotic system without equilibrium [

The aim of this study is to introduce a new hyperchaotic system without equilibrium. The organization of the paper is as follows. The new no-equilibrium system is described in the next section. Section

Wei and Wang have introduced a special system which is different from the original Lorenz and Lorenz-like systems [

Motivated by the noticeable features of model (

Obviously, it is simple to find equilibrium points of system (

It is easy to see that no-equilibrium system (

It is interesting that the system without equilibrium exhibits hyperchaotic behavior for

Projections of hyperchaotic attractors in (a)

Dynamics of the system has been investigated by considering the effect of parameters on system’s behavior. Our simulations show that no-equilibrium system (

Bifurcation diagram of no-equilibrium system (

Three largest Lyapunov exponents of no-equilibrium system (

We have also investigated the multistability of the new no-equilibrium system by using the continuation diagram [

Continuations of new no-equilibrium system (

Coexistence of attractors in no-equilibrium system (

From the viewpoint of applications, the amplitude control of a chaotic signal is an important topic [

We introduce a single control parameter

Varying attractors of no-equilibrium system (

In order to control the amplitudes of variables

Controllable attractors of no-equilibrium system (

In this section, we focus on the effect of commensurate fractional derivation on the hyperchaotic system (

The phase portrait in the planes

For

The time series of

It is clearly seen in Figure

The phase portrait in the planes

From Figure

The phase portrait in the planes

The autocorrelation functions of

For

This paper introduces a new system, which has no equilibrium. However, different complex behaviors such as hyperchaos or coexistence of hidden attractors have been observed in such system. In addition, the new system without equilibrium is an amplitude-controllable system which is useful for practical applications. This study has found that commensurate fractional derivation affects the no-equilibrium system. Control and synchronization of such system should be studied in future works.

The authors declare that they have no conflicts of interest.

The authors thank Professor GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong, for suggesting many helpful references.