Discovering unknown aspects of non-equilibrium systems with hidden strange attractors is an attractive research topic. A novel quadratic hyperjerk system is introduced in this paper. It is noteworthy that this non-equilibrium system can generate hidden chaotic attractors. The essential properties of such systems are investigated by means of equilibrium points, phase portrait, bifurcation diagram, and Lyapunov exponents. In addition, a fractional-order differential equation of this new system is presented. Moreover, an electronic circuit is also designed and implemented to verify the feasibility of the theoretical model.
After the emergence of chaos as a field of study, various chaotic systems have been reported in the literature. Most of them are 3D differential equations with a limited number of equilibria , , . Therefore Shilnikov criteria can be used to prove the chaos in those systems , . It has always been of interest to find chaotic systems with special features like multi-scroll attractors , , , , multistability , , , , and extreme multistability , , .
Leonov and Kuznetsov categorized dynamical systems into two groups: dynamical systems with self-excited attractors, and dynamical systems with hidden attractors , , , . If the basin of attraction for an attractor involves (even in its border) at least one equilibrium, then that attractor is self-excited , , ; otherwise, it is hidden. Hidden attractors have been observed in many real dynamical systems , . Design, localization, and control of hidden attractors have been of great interest in recent years , , , , .
In 1994, the first non-equilibrium chaotic flow was found . After about 20 years, some other chaotic systems with non-equilibrium were introduced , , , , . It can be easily concluded that the chaotic attractor in such systems is hidden , , , . They have attracted the interest of the scientific community rapidly because chaotic systems with hidden attractors allow unexpected responses to perturbations in dynamical systems . Hidden attractors also may be seen in 4D systems. As noted earlier, chaotic behaviours can emerge in (at least) 3D differential equations. The first 4D chaotic system was found by Rossler , which was the first step in designing a 4D chaotic system.
There is another common type of chaotic systems, which is jerk systems. It is the lowest derivative for which an ordinary differential equation (ODE) with smooth, continuous functions can give chaos . A dynamical system that is described by an nth-order ODE with n>3 is a hyper jerk system, which can be shown to have an explicit form where the (jerk) function J is a simple quadratic or cubic polynomial. Such systems are surprisingly general and prototypical examples of complex dynamical systems in a high-dimensional phase space , .
In this paper, a new hyper jerk 4D chaotic system with hidden attractors is introduced. This system is the first example of 4D hyper jerk systems with no equilibrium. In the next section, we present the dynamical analysis of this system. Bifurcation and Lyapunov exponent analysis are studied in Section 3. In Section 4, fractional-order and time-delayed versions of this new 4D system are proposed. To use this new chaotic system in engineering applications, in Section 5 we report a circuital implementation of the theoretical model. Finally, we conclude the paper in Section 6.
2 Dynamics of a New 4D Hyper Jerk Chaotic Flow
Consider the following new four-dimensional hyper jerk chaotic flow:
If a×e>0, this system has no equilibrium. Here x, y, z, and w are state variables, while a, b, c, d, and e are positive parameters. Interestingly, when a=2, b=5.05, c=2.5, d=0.7, and e=2 and the initial condition [x(0), y(0), z(0), w(0)]=(0, −5, 0, 0) is selected, the new system (1) has a chaotic solution. In this case, the Lyapunov exponents of the system (2) are λ1=0.0772, λ2=0, λ3=−0.8044, and λ4=−1.8293. Different projections of the chaotic attractor are illustrated in Figure 1.
3 Bifurcation Analysis
In this part, the dynamical behaviors of system (1) with respect to changing parameters d are investigated (see Fig. 2). In Figure 2a, the bifurcation diagram of the system is shown, and in Figure 2b the Lyapunov exponents are shown. It can be seen that changing the parameter d shows a familiar period doubling route to chaos. Figure 3 shows the basin of attraction for the hidden attractor. It is hidden because there is no equilibrium in this new chaotic hyper jerk system.
4 Fractional Order Hidden Jerk System (FOHS)
Recently, many researchers have discussed fractional-order calculus and its applications , . Fractional-order nonlinear systems with different control approaches have been investigated in , , . Fractional-order memristors based on non-equilibrium chaotic and hyper chaotic systems have been proposed in , . A novel fractional-order non-equilibrium chaotic system has been investigated in , and a fractional-order hyper chaotic system without equilibrium points has been investigated in . A memristor-based fractional-order system with a capacitor and an inductor is discussed in . Numerical analysis and methods for simulating fractional-order nonlinear system are proposed in , and MATLAB solutions for fractional-order chaotic systems are discussed in .
Time-delayed differential equations play an important role in most engineering applications , . Stability analysis of differential equations with delays has also been discussed in . Synchronization of such time-delayed systems is a major complex problem, and therefore many synchronization schemes are discussed in the literature , . A time-delayed chaotic system was obtained from a logistic chaotic map . A parameter identification problem for a general time-delayed chaotic system is considered and analysed in . A novel time-delayed chaotic system with hidden attractors is discussed in .
4.1 Designing Fractional-Order Hidden Jerk System (FOHS)
In this section we derive the fractional-order hidden jerk system (FOHS). There are three commonly used definition of the fractional-order differential operator, namely Grunwald-Letnikov, Riemann-Liouville, and Caputo , . The dimensionless mathematical model of the FOHS can be derived as follows:
where qx, qy, qz, and qw are the fractional orders of the FOHS system. The three main approaches derived to solve fractional-order chaotic systems are the frequency-domain method , the Adomian decomposition method (ADM) , and the Adams-Bashforth-Moulton (ABM) algorithm . The frequency-domain method is not always reliable in detecting the chaotic behaviour in nonlinear systems . On the other hand, ABM and ADM are more accurate and convenient to analyse dynamical behaviours of a nonlinear system. Compared to ABM, ADM yields more accurate results and needs less computing and memory resources . Hence, the proposed FOHS system is implemeted in a field-programmable gate array (FPGA) by applying the ADM scheme. As the ADM algorithm converges fast , the first six terms are used to get the solution of the FOHS system as in real cases, and it is impossible to find an accurate value of x when t takes larger values . Hence we have to design a time discretization method. That is to say, for a time interval ti (initial time) to tf (final time), we divide the interval into (tn, tn+1) and we get the value of x(n+1) at time tn+1 by applying x(n) at time tn+1 using the relation x(n+1)=F(x(n)) .
where are the Adomian polynomials with i=1, 2, 3, 4, and As can be seen from the FOHS (2), there are four nonlinear terms x2, xy, xz, and xw. The Adomian polynomials for the linear terms can be calculated as and the first six Adomian polynomials for the nonlinear terms can be derived as shown in Table 1.
|Nonlinear term||Adomian polynomials|
Using (3) and (4), the FOHS is numerically solved. Figure 4 shows the 2D phase portraits of the FOHS with h=0.001, q=0.995, the parameters a=2, b=5.05, c=2.5, d=0.7, e=2, and the initial conditions [0, −5, 0, 0].
4.2 Dynamic Analysis of the Fractional-Order Hidden Jerk System
4.2.1 Bifurcation with Fractional Order
Dynamical behaviors of the FOHJS are illustrated in Figure 5. We first derive the bifurcation plots of the FOHJS with parameters b and e, and the plots are shown in Figures 5a and b. The FOHJS shows chaotic oscillations for 5≤b≤5.1 and 1.5≤e≤2 with routine period doubling route to chaos. For b<5 and e>2, the system goes unbounded and loses chaotic oscillations. The next most important analysis of interest when investigating a fractional-order system is the bifurcation with fractional order. As can be seen from Figure 5c, bifurcation of the FOHS system for change in fractional order shows that the system oscillations remain chaotic if qi>0.992 and the system shows hyper chaotic behaviour for 0.993≤q≤0.998, and the largest positive Lyapunov exponents (L1=0.3166, L2=0.08217) of the FOHS system appear when q=0.998, as can be seen from Figure 5d against its largest integer-order Lyapunov exponents (L1=0.2991, L2=0.07634). Figure 5e and f show the 2D phase portraits in the XY-plane for various fractional orders.
4.2.2 Time-Delayed Hidden Jerk System (TDHJS)
In this section we derive the time-delayed, dimensionless model of the hidden jerk system (HS) by introducing multiple time delays in the second state variable (y). TDHJS can be defined as
where a=2, b=5.05, c=2.5, d=0.7, and e=2. Similar to HS, the TDHJS shows chaotic oscillations for the initial conditions [0, −5, 0, 0], and the time delays are τ1=0.05 and τ2=0.04. Figure 6 shows the 2D phase portraits of TDHJS.
4.2.3 Dynamic Properties of TDHJS
184.108.40.206 Lyapunov Exponents
There are various algorithms proposed based on chaos synchronization for the estimation of the Lyapunov exponent of time-delayed dynamical systems , . In this paper we adopt the technique employing the synchronization of identical systems coupled by a linear negative feedback mechanism  for finding the exact Lyapunov exponents of TDHJS. The calculated Lyapunov exponents are L1=0.1684, L2=0, L3=−0.9974, and L4=−2.078.
To understand the chaotic behaviour of TDHJS and the impact of time delays (τ1, τ2) and parameters (aτ, bτ, cτ, dτ, eτ), we derive the bifurcation plots and investigate the various regions of bifurcation. Figure 7 shows the bifurcation of TDHJS with time delays, and the time delays τ1 and τ2 show chaotic oscillations for 0.03≤τ1≤0.0785; for τ1>0.785, the TDHJS system goes unbounded. For 0.001≤τ1≤0.06, TDHJS shows chaotic oscillations, and for τ2>0.06 the system spirals out and goes into periodic orbits.
In addition, the bifurcation diagrams of the TDHJS system with different parameters are reported in Figure 8. To investigate the bifurcation of the TDHJS system in the parameter space, the time delays are fixed at τ1=0.05, τ2=0.04. First we investigate the bifurcation of TDHJS with the parameter aτ, with all the other parameter fixed to their respective values. TDHJS shows a chaotic region for 1.9≤aτ≤2.8 and shows period-2 oscillations for aτ<1.7 and period-3 oscillations for aτ>2.85 with period-doubling route to chaos. Figure 8a shows the bifurcation of TDHJS with the parameter aτ and Figure 8b shows the bifurcation of TDHJS with the parameter bτ. The plots show two distinct chaotic regions for 5≤bτ≤5.09 and 5.1≤bτ≤5.2 separated by a symmetry-breaking and symmetry-restoring region for 5.09≤bτ≤5.1 with period-4 oscillations for bτ≥5.25 and enters and exits the chaotic regions by period-doubling and period-halving, respectively. The third bifurcation analysis is for the parameter cτ, as seen from Figure 8c. The bifurcation plot of cτ shows a chaotic region for 2.4≤cτ≤2.6, period-2 oscillations for cτ<2.25, and period-4 oscillations for 2.25≤cτ≤2.32 with the routine period doubling route to chaos. For cτ>2.6, the TDHJS system goes unbounded. Figure 8d shows the bifurcation of TDHJS with dτ, which shows a chaotic region for 0.55≤dτ≤0.875 with period-doubling and period-halving entry and exit to the chaotic region. The fifth bifurcation analysis is for the parameter eτ, which shows chaotic region for 0.8≤eτ≤3, as can be seen from Figure 8e.
5 The Electronic Circuit Implementation of 4D New Chaotic System
In this section, the new chaotic system in (1) was implemented as an electronic circuit like in Figure 11. The electronic circuit of the new chaotic system was executed in the electronics lab. The circuit includes basic electronic materials like resistors, multipliers, and capacitors.
5.1 Scaling of the 4D New Chaotic System
To keep the amplitude values in the linear range of operational amplifiers, the chaotic system should be scaled. The x, y, z, and w signal values in the interval (−40, 60) are shown in Figure 9. The z and w values are not in the interval (−15, 15) for electronic circuit implementation. Thus, the z and w values need to be scaled for real-time applications. For the scaling process, let Then setting the original state variables x, y, z, w instead of the variables X, Y, Z, W, the scaled system first becomes
According to new values (X, Y, Z, W), we regenerate of all the values as
With the new values, the scaled system is as shown below.
Finally, new scaled chaotic system is given by
In Figure 10, the new time series of the scaled system with a=2, b=5.05, c=2.5, d=0.7, and e=2 is shown. The amplitudes of z and w are decreased according to Figure 1. After these processes, we can execute the electronic circuit application.
5.2 Circuit Implementation of the New Scaled Chaotic System
A circuit is designed as shown in Figure 11 for the scaled chaotic system. The experimental electronic circuit of the scaled chaotic system was designed for parameters a=2, b=5.05, c=2.5, d=0.7, and e=2 with initial conditions x(0)=0, y(0)=−5, z(0)=0, and w(0)=0.
We select C1=C2=C3=C4=1 nF, R1=400 kΩ, R2=200 kΩ, R3=R4=R6=R7=R9=R13=R14=100 kΩ, R5=160 kΩ, R8=15 MΩ, R10=39.6 kΩ, R11=40 kΩ, and R12=57 kΩ; the corresponding phase portraits on the oscilloscope are shown in Figure 12.
A novel quadratic four-dimensional hyperjerk system with hidden chaotic attractors was proposed in this paper. Dynamical properties of this new system were investigated by means of equilibrium points, phase portrait, bifurcation diagram, and Lyapunov exponents through numerical simulation and circuit implementation. In addition, a fractional-order differential equation of this new system, along with analysis of its dynamical properties, was presented.
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