A No-Equilibrium Hyperchaotic System and Its Fractional-Order Form
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 . Hyperchaotic systems have been applied in various areas due to their higher level of complexity with respect to chaotic systems [2–6]. There has been an increasing amount of literature on hyperchaos [7, 8]. Authors have introduced and studied different hyperchaotic systems such as switched hyperchaotic system , four-wing hyperchaotic attractor , hyperchaotic Chua’s circuits , and hyperchaotic Lü attractor . It is noted that there are countable numbers of equilibrium points in such reported hyperchaotic systems.
It is interesting that Wang et al. found a hyperchaotic system without equilibrium , which is different from normal hyperchaotic systems. Following up the first discovery of Wang et al., other hyperchaotic systems without equilibrium were presented [14–17]. Recent studies have attempted to explore special features of no-equilibrium hyperchaotic systems. Wang et al. investigated multiwing nonequilibrium attractors in simplified hyperchaotic systems . Bao et al. constructed a memristive hyperchaotic system which does not display any equilibrium . This memristive system generated coexisting hidden attractors. Moreover, fractional-order systems without equilibria were discovered in [20, 21]. It is worth noting that attractors in such no-equilibrium hyperchaotic system are “hidden" from the viewpoint of computations [22, 23]. More recent attention has focused on hidden attractors due to their importance in both theoretical issues and engineering applications [24–27].
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 3 presents dynamics of the new system without equilibrium. Fractional derivation effect on the proposed system is investigated in Section 4. Our conclusions are drawn in the last section.
2. The Model of the New System without Equilibrium
Wei and Wang have introduced a special system which is different from the original Lorenz and Lorenz-like systems . Wei-Wang system is given byin which , , and are state variables while , , , and are parameters. There are two equilibrium points in system (1) for and : and . Moreover, Wei-Wang system (1) is chaotic when both equilibrium points are stable .
Motivated by the noticeable features of model (1), we construct a novel four-dimensional system by introducing an additional fourth state as follows:in which , , , and are four state variables. In system (2), positive parameters are , , , , , , and . System (2) is invariant under the transformation . It means that the system has rotational symmetry around the -axis.
It is easy to see that no-equilibrium system (2) is dissipative with an exponential contraction ratebecause ofHowever, it is noted that the dissipativity in the sense of Levinson should be studied further [29, 30].
It is interesting that the system without equilibrium exhibits hyperchaotic behavior for , , , and and initial conditions as illustrated in Figure 1. The Lyapunov exponents of the no-equilibrium system are , , , and . We have applied Wolf’s method  to calculate the Lyapunov exponents. The time of computation is 10,000 and the initial conditions are . It is noted that the values of Lyapunov exponents are not the same for any initial point on invariant set . Computations of Lyapunov exponents should be considered seriously [33–35].
3. Dynamical Behavior of the System without Equilibrium
Dynamics of the system has been investigated by considering the effect of parameters on system’s behavior. Our simulations show that no-equilibrium system (2) displays rich dynamics for the parameter . We have studied the dynamics of no-equilibrium system (2) by varying the value of parameter . Bifurcation diagram and Lyapunov exponents of no-equilibrium system (2) are presented in Figures 2 and 3, respectively. For , the system without equilibrium only generates periodical behavior. However, no-equilibrium system (2) displays different attractors such as periodic, quasiperiodic, chaotic, and hyperchaotic attractors for .
We have also investigated the multistability of the new no-equilibrium system by using the continuation diagram . As can be seen in Figure 4, a window of multistability dynamics is identified in the range of from 1.905 to 2.215. For example, Figure 5 illustrates the coexisting attractor in no-equilibrium system (2) for .
From the viewpoint of applications, the amplitude control of a chaotic signal is an important topic [37–40]. Interestingly, no-equilibrium system (2) is an amplitude-controllable system. Here we illustrate this special feature of no-equilibrium system (2).
We introduce a single control parameter into system (2):The control parameter is used to boost the amplitude of the variable . Therefore, we can change the signal easily, for example, from a bipolar signal to unipolar one as illustrated in Figure 6.
In order to control the amplitudes of variables and , the coefficient is included into system (2) as follows:Here we can consider the coefficient as an amplitude controller. Attractor of no-equilibrium system (2) is controlled as shown in Figure 7.
4. Fractional Derivation Effect on the System without Equilibrium
In this section, we focus on the effect of commensurate fractional derivation on the hyperchaotic system (2) when , , , and (see Figure 1). The fractional-order form of system (2) is obtained by replacing the integer-order derivatives by fractional-order ones. The mathematical description of the fractional-order version of system (2) is expressed aswhere is the derivative order satisfying . The fractional-order form of system (2) has no-equilibrium points; therefore the effect of fractional derivation on the hyperchaotic system (8) can only be numerically investigated. Here, the Adams-Bashforth-Moulton predictor-corrector scheme is used [41, 42]. This method is based on the Caputo definition of the fractional-order derivative, given by where and is the Gamma function. We perform the numerical simulations of fractional-order system (8) for different fractional-order (). We present in Figure 8 the phase portraits in the planes , , and obtained for two specific values of commensurate fractional-order .
For , the fractional-order system (8) displays a point attractor as shown in Figure 8(a). When the fractional derivative order increases, the fractional-order system (8) presents a transient chaos at (see Figure 8(b)). The transient chaos is confirmed in Figure 9 which depicts the time series of the corresponding phase portraits of fractional-order system (8) shown in Figure 8(b).
It is clearly seen in Figure 9 that the trajectories of fractional-order system (8) display chaotic behavior for . For , they converge to an equilibrium point. For , the phase portraits in the planes , , and are plotted in Figure 10.
From Figure 10, one can note that the fractional-order system (8) exhibits bistable double-scroll chaotic attractors at . The chaotic behavior found in Figure 10 is confirmed in Figure 12 which presents the autocorrelation function of the outputs , , , and of the corresponding phase portraits of fractional-order system (8) depicted in Figure 10. The coexistence between double-scroll chaotic and quasiperiodic attractors is shown in Figure 11 which presents the phase portraits in the planes , , and for .
For and using the initial conditions , double-scroll chaotic attractor is obtained in the fractional-order system (8), while for and using the initial conditions , the fractional-order system (8) exhibits quasiperiodic attractor. The chaotic and quasiperiodic behaviors are confirmed in Figure 12 which presents the autocorrelation function of the outputs , , , and of the corresponding phase portraits of fractional-order system (8) depicted in Figure 11. In order to know the dynamical behavior found in Figures 10 and 11, we calculate autocorrelation function of the outputs , , , and . In Figure 12, we present the autocorrelation functionwhere , is the mean value of the amplitude along the trajectory, and is the time shift. The coefficient is bounded in the range and it stays high for periodic, quasiperiodic, and chaotic cases and decays to zero in the case of the hyperchaotic attractor .
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.
Conflicts of Interest
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.
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