Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability
View this Special IssueResearch Article  Open Access
M. Fouodji Tsotsop, J. Kengne, G. Kenne, Z. Tabekoueng Njitacke, "Coexistence of Multiple Points, Limit Cycles, and Strange Attractors in a Simple Autonomous Hyperjerk Circuit with Hyperbolic Sine Function", Complexity, vol. 2020, Article ID 6182183, 24 pages, 2020. https://doi.org/10.1155/2020/6182183
Coexistence of Multiple Points, Limit Cycles, and Strange Attractors in a Simple Autonomous Hyperjerk Circuit with Hyperbolic Sine Function
Abstract
In this contribution, a new elegant hyperjerk system with three equilibria and hyperbolic sine nonlinearity is investigated. In contrast to other models of hyperjerk systems where either hidden or selfexcited attractors are obtained, the case reported in this work represents a unique one which displays the coexistence of selfexcited chaotic attractors and stable fixed points. The dynamic properties of the new system are explored in terms of equilibrium point analyses, symmetry and dissipation, and existence of attractors as well. Common analysis tools (i.e., bifurcation diagram, Lyapunov exponents, and phase portraits) are used to highlight some important phenomena such as perioddoubling bifurcation, chaos, periodic windows, and symmetric restoring crises. More interestingly, the system under consideration shows the coexistence of several types of stable states, including the coexistence of two, three, four, six, eight, and ten coexisting attractors. In addition, the system is shown to display antimonotonicity and offset boosting. Laboratory experimental measurements show a very good coherence with the theoretical predictions.
1. Introduction
Most authors have been interested in chaotic systems because of their sensitivity to the initial conditions and also to the variation of system parameters. Since the discovery of this phenomenon by Lorenz [1], many classical chaotic systems have emerged. We can mention the Rossler system [2], Chen system [3], Jafari system [4], Pham system [5], and Lü system just to name a few [6]. In the last few years, special attention has been given to “jerk systems” because of their simplicity and complex dynamics [7–12]. From a mathematical point of view, a generalization of the jerk dynamics is usually given in the following form:
When , we have , which is called “jerk system” [13]. For , (1) turns to “hyperjerk system” or “snap system” [14]. In the literature, several authors have studied the latter. Generally, these systems exhibit multistability phenomenon which are the coexistence of multiple attractors solely depending on the initial conditions. These attractors are generally classified into two categories, namely, selfexcited and hidden attractors [15–20]. Remember that selfexcited attractors exist in systems with unstable equilibrium points [21–23]. In contrast, hidden attractors are characterized by the systems with no equilibrium [24–29], either by a line or a curve of equilibrium points, or system with stable equilibrium points [24, 25]. In addition, hidden attractors have a basin of attraction which does not intersect with the neighborhoods of equilibria.
Interested by the selfexcited attractors, many authors applied different techniques to hyperjerk systems. Some of these authors introduced different types of nonlinearities. For instance, in 2006, Cklouverakis and Sprott [22] presented a numerical study of a simple subclass of hyperjerk systems and showed that the 4^{th} and 5^{th} order hyperjerk systems developed some simple chaotic behaviors. In 2015, Sundiarapandian and coworkers [23] presented a new hyperchaotic 4D hyperjerk system by adding a quadratic nonlinearity to the hyperjerk system of Chlouverakis–Sprott system. The authors present some qualitative and quantitative analyses of the new system. In 2017, Daltzis et al. [13] introduced a new hyperjerk system with two nonlinearities (absolute value and quintic term) and showed that the new system can develop hyperchaotic behaviors. Recently, Leutcho et al. [21] presented a new hyperjerk circuit with hyperbolic sine function and demonstrated that the novel proposed system is the unique one which is capable to exhibit the coexistence of nine periodic and chaotic attractors.
Motivated by the above mentioned results, we present a new hyperjerk system with nonlinear position feedback involving a hyperbolic sine function. Our circuit is derived from the hyperjerk system proposed by Dalkirian and Sprott [7] by replacing the exponential nonlinearity by the hyperbolic sine function. The striking aspect of the proposed system is its ability to develop the coexistence of up to ten disconnected attractors including periodic, chaotic, and point attractors. The objectives of this work are as follows: (a) to present an analytical study of the proposed hyperjerk system; (b) to highlight regions in which we observe the coexistence of multiple attractors; (c) to point out some striking features like antimonotonicity and offset boosting; and (d) to verify the feasibility of the proposed model through an experimental study.
This research is organized as follows. Section 2 deals with the modeling process. The electronic consuration of the hyperjerk circuit is presented and the suitable mathematical model is derived to describe the dynamics of the novel hyperjerk, wherein some basic properties of the model are equally presented. In Section 3, the bifurcation structures of the system are investigated numerically. Also, in this section, some tools are used to show multistability observed in the novel system. Section 4 contains experimental study, and at the end of this section, it appears that coherence is observed between the theoretical and experimental analysis. Finally, Section 5 presents conclusion.
2. Description and Analysis of the Model
2.1. Circuit Description
It is important to know that the new circuit proposed here derives from the hyperjerk system proposed by Dalkirian and Sprott [7]. It is obtained by substituting the exponential nonlinearity by the hyperbolic sine function. Figure 1 represents the schematic diagram of the novel hyperjerk circuit. The circuit consists of four successive integrators associated to several feedback loops. In addition, the nonlinear feedback loop linked with the pair of semiconductor diodes is applied to the first integrator. The symmetrical nature [30] of the system is due to the antiparallel configuration of the diodes. In such type of configuration, the voltage across each diode is equal to the voltage of the resulting twoterminal device, while the current is the addition of the currents flowing through each diode. The symmetrical property of the nonlinearity is necessary for the occurrence of symmetric attractors [30]. We would like to recall that the pair of semiconductor diodes is the only nonlinear element responsible for the chaotic behavior displayed by the whole electronic circuit.
2.2. State Equations
The following assumptions will be adopted throughout our analysis. Firstly, we considered that capacitors and operational amplifiers are ideal with the latter operating in linear domains. Secondly, the currentvoltage characteristic (3) of the pair of semiconductor diodes is obtained from the Shockley diode equation [31, 32] as follows:where , , , , q, and are the intrinsic parameters of the diodes. By applying Kirchhoff’s laws to Figure 1 and considering the above assumptions, it can be shown that the voltages , , , and satisfy the following set of four coupled firstorder nonlinear differential equations:
Applying the following change of variables:we get the normalized circuit equations which are expressed by the following smooth nonlinear fourthorder differential equations easy for numerical integration:where the dot represents differentiation concerning the dimensionless time . Note that the nonlinear function only depends on the state variable in system (5). will be kept constant throughout the numerical analysis: . Therefore, during the bifurcation analysis of the 4D system, is considered like the control parameter (i.e., with respect to ). The values of electronic components used for both the numerical and experimental analyses are listed in Table 1. System (5) can be expressed equivalently in the general hyperjerk form as follows:

By observing equation (6), it can be noticed that our model belongs to the wider class of “elegant” hyperjerk dynamical systems defined in [14]. More interestingly, our model (5) represents one of the simplest autonomous 4D systems reported recently, displaying the coexistence of up to ten fixed points, periodic and chaotic attractors.
2.3. Symmetry, Dissipation, and Existence of Attractors
Equation (5) being invariant following the transformation , we can conclude that we will have a couple of solutions for a given parameter range. So, if is a solution of our system, then its symmetry will also be a solution. All these makes it possible to highlight the symmetrical nature of our system. In order to verify the dissipation property of our system, it is necessary to calculate the volume contraction rate (). For every point of space [33, 34], it is given by the following expression:
The above expression is negative and does not depend on the space coordinates of the system, and thus we can conclude that the introduced system is dissipative.
2.4. Fixed Point Analysis
By canceling the right side of equation (5), it is possible to determine the equilibrium points of the system which play a crucial role in the study of the system dynamics. The resolution of equation (8) permits to obtain different equilibrium points of the system.
Note that the point is a trivial equilibrium point, while and are the solutions of the transcendental equation:
By fixing and maintaining at the same previous value, we obtain the other nontrivial equilibrium points . The stability of the system in the neighborhood of those equilibrium points is studied by resolving the characteristic equation (12), resulting from the below equation:where represents the identity matrix and the Jacobian matrix defined as follows:
The characteristic equation obtained is with
By applying the Lyapunov stability theory, it is shown that at the equilibrium point , , and thus the equilibrium is unstable since the characteristic equation has coefficients with different signs. In contrast, the stability of the nonzero equilibrium points () depends on the control parameters and . By applying the Routh–Hurwitz criterion [32, 33], we have shown that for and , the nontrivial equilibrium points are unstable, but for , they are stable. These critical values are obtained from the calculation of the Hopf bifurcation conditions:with
Equation (13) provides the frequency of stable oscillations as well as the critical values of corresponding to the Hopf bifurcation of the system. From Table 2, it follows that in the regime of (periodic or chaotic) oscillations, the three equilibria are unstable, and thus the system generates selfexcited oscillations. For the following parameters , , , , and , the equilibrium point remains unstable for all values of control parameter . Moreover, for some values of bifurcation parameter , the nontrivial equilibria have pure imaginary roots, and thus the system presents the Hopf bifurcation. In order to verify the existence of the Hopf bifurcation in the system, eigenvalue locus is plotted. It shows the existence of Hopf bifurcation in the system which is characterized by the intersection of the eigenvalue locus with the imaginary axis. By observing Figures 2(a) and 2(b), we can certify that the new hyperjerk system presents Hopf bifurcation.

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3. Numerical Computation
3.1. Numerical Techniques
System (5) is resolved numerically in order to highlight the rich variety of bifurcation that can be observed in a new hyperjerk system. The dynamic properties of the model were numerically simulated in Turbo Pascal using the fourthorder Runge–Kutta method with a constant time step size of , and parameters are taken in extended precision mode. The transient phase is canceled by integrating the system for a long time. The bifurcation diagram and the Lyapunov exponent are the traditional tools that measure the dependence of the system on the initial conditions as well as the sequence that leads to chaos in the system. The algorithms of Wolf and his collaborators [35] are used for calculating the Lyapunov exponents.
3.2. Bifurcation, Chaos in a Novel Hyperjerk Circuit
Different scenarios exhibited by the proposed hyperjerk system are obtained by plotting the bifurcation diagrams. The bifurcation diagram of Figure 3(a) is obtained by plotting the local maxima of the variable according to the bifurcation parameter , the other parameters being fixed at , , , , and . It can be noted that it is a perioddoubling route to chaos because the transition from period1 attractor to doubleband chaos is as follows: period1 period2 period4 singleband chaos period5 singleband chaos doubleband chaos. Figure 4 clearly shows the above transition. It is obtained by progressively varying the control parameter. The exact nature of the attractors mentioned above is defined by the graphs of the four largest Lyapunov exponents shown in Figure 3(b). We can observe in Figure 3(b) that periodic attractors are characterized by , , , and , while chaotic attractors have the following characteristics: , , , and . A perfect coherence is observed between the bifurcation diagram and the corresponding graphs of the four largest Lyapunov exponents. In order to show the complexity of the new hyperjerk circuit, the chaotic attractor has been projected on several planes (Figures 5(a)–5(f)), as well as the Poincaré section (Figure 5(g)). We can observe that the doubleband chaos completely changes when moving from one plane to another. For the value of the bifurcation parameter , the coexistence of four periodic and chaotic attractors is observed in the novel proposed system (see Figure 6). In order to illustrate the Hopf bifurcation previously proved by theoretical calculations, the bifurcation diagram of Figure 7 has been represented. Stable state is characterized by a fixed point with , while oscillatory state is characterized by .
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3.3. Multistability
In this section, we demonstrate the variety of dynamical regimes in the new 4D system. We show that depending on the values of the system parameters, the system exhibits very rich dynamics and bifurcation scenarios. A multistable system is a system with various coexisting stable states (chaotic, point, and periodic state) under the same system parameters, with different initial conditions. In recent years, the phenomenon of multistability phenomenon has been reported in many nonlinear dynamic systems [13, 36–46].
3.3.1. Coexistence of Attractors with respect to Bifurcation Parameter
By changing the system parameters and considering as bifurcation parameter, we observe a completely different behavior. In addition, a very interesting phenomenon which is the coexistence of multiple attractors appears in the new 4D hyperjerk. For this phenomenon to be illustrated, the bifurcation diagrams of Figure 8 are plotted using the following method:(i)The blue diagram is obtained by simultaneously increasing the value of the control parameter as well as the initial condition . At each iteration, we assign to the new value of the control parameter c.(ii)The red diagram is obtained by incrementing from its minimum value 2.34 to its maximum value 2.985, with a carefully chosen step. Note that the solutions of the system at each iteration are considered as the initial condition of the next iteration.(iii)The cyan diagram respects the previous procedure, with the initial condition , whereas the black diagram follows the same procedure as previously described but the only difference is the decrease of the control parameter . The initial condition is .(iv)The magenta diagram is obtained by increasing the control parameter from 2.398 to 2.985, followed by decreasing the bifurcation parameter from 2.398 to 2.34.
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We can observe in Figure 8 several windows of coexisting attractors. For more details about the methods used to plot the bifurcation diagrams of Figure 8, see Table 3. The enlarged bifurcation diagram of Figure 9 shows the hysteretic domain, plotted in the range , and the techniques used to plot the diagrams are also presented in Table 3. Figure 9 shows the coexistence of six and eight different limit cycles, chaotic and point attractors. Some sample phase portraits showing the coexistence of six and eight attractors are presented in Figures 10 and 11, respectively. Some basins of attractions showing the initial conditions domains of the coexisting attractors are presented in Figure 12. The coexistence of four attractors is clearly denoted (a pair of periodic attractors (black and yellow) and a pair of chaotic attractors (blue and green)). Note that there is a perfect symmetry between the different cross sections of the competing attractors.
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The initial conditions of the coexistence of ten attractors exhibited by the proposed hyperjerk system are presented in Figure 13. The bifurcation like sequence of Figure 13 shows the variation of in terms of the control parameter , and the other initial conditions are set to zero (). Note that chaotic attractors are characterized by an unlimited number of points, while periodic attractors are characterized by a finite number of points. In the same line, the basin of attraction shows the different domains of convergence of similar attractors. The phase portrait of Figure 14 illustrates the coexistence of ten attractors. The initials conditions of the coexisting attractors are given in Table 4.
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3.3.2. Coexistence of Attractors with respect to Bifurcation Parameters and
In order to investigate the sensitivity of the new hyperjerk system in terms of the bifurcation parameter , the other parameters are fixed as follows: , , , , and . We found that the novel hyperjerk system can exhibit striking bifurcation sequences when varying the control parameter in the range . With reference to Figure 11, the bifurcation diagram in black and the one in blue are obtained by increasing and decreasing the values of the parameter , while the one in red is obtained by fixing the initial conditions at . A window of hysteretic dynamics can be identified in the range . The enlarged bifurcation diagram of Figure 12 clearly illustrates the domain of the coexistence of multiple attractors observed in the new hyperjerk system according to the bifurcation parameter . Different methods used to plot these bifurcation diagrams are presented in Table 3. Up to six different periodic, chaotic, and point attractors can be obtained by only changing the initial conditions. For instance, sample phase portraits of the coexistence of six distinct attractors are presented in Figure 13.
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During the mathematical analyses, it has been shown that the Hopf bifurcation was depending on the control parameter . The bifurcation diagram of Figure 13(a) clearly illustrates this phenomenon characterized by the stable state followed by the unstable state. Moreover, this control parameter also highlights the coexistence of multiple attractors exhibited by the new 4D system. By considering the following sets of the parameters: , , , , , and , we discover that the new 4D system displays the coexistence of four distinct chaotic and point attractors. The phase portraits of the Figure 14 and their corresponding cross section of the basin of attraction clearly show the coexistence phenomenon and also give the initial condition domain of each attractor. The green and black domains represent the initial conditions regions of the pair of chaotic attractors, while the yellow domain represents the initial condition regions of the pair of point attractors.
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3.4. Offset Boosting Scenario
Another property of system (3) is the possibility to develop an offset boosting effect. In our model, appears only in the fourth line of equation (3), and thus this variable is a bootable variable [47–52]. Assuming the transformation where is a constant, equation (5) can be rewritten accordingly as
Figure 19 clearly presents offset boosting of the doubleband chaotic attractor. The following values of parameter: (blue), (red) and (green) are used to plot them in x_{1}x_{4} and x_{1}x_{3} planes.
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3.5. Antimonotonicity
By decreasing the value of the control parameter , we can observe the formation and destruction of periodic orbits via reverse perioddoubling bifurcation sequences. This interesting phenomenon has been reported in the literature. It is reported in various nonlinear systems such as Duffing oscillator [12], Chua circuit [53], and secondorder nonlinear nonautonomous circuit [54, 55]. This phenomenon was reported for the first time in the hyperjerk system by Leutcho et al. [21]. The creation of periodic seas in the parameter space is the necessary requirement for a nonlinear system to experience forward and reverse perioddoubling cascade [21]. Sample illustrations are represented in Figure 20, where some bifurcation diagrams are shown. These diagrams are obtained for each discrete value of the control parameter . In Figure 20, note that for , we have period2 bubble and for a slight adjustment of the control parameter , period4 bubble is observed for , whereas for , we have a period8 bubble. In the same order, chaotic bubbles are formed for . The increase of control parameter causes the creation of other bubbles, and it finally results in an infinite tree (like chaos).
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4. Experimental Study
The objective of this section is to confirm the above theoretical results by realizing a laboratory experimental study. For this purpose to be achieved, several approaches have been proposed in the literature to implement chaotic circuit (by using many types of offtheshelf electronic components [56] or fieldprogrammable gate array (FPGA) technology [57–60] or fieldprogrammableanalogarray (FPAA) technology [61, 62] just to name a few). Only offtheshelf electronic components (i.e., resistors, capacitors, pair of semiconductor diodes (), and TL084 operational amplifiers types with a power supply of ±15 VDC) are used to realize the schematic diagram of Figure 1. The following values of electronic circuit components are used during the experimental process: , , , and (for the other parameters, see Table 1 case A). The complete sequence of phase portraits plotted in plan is obtained by adjusting the control resistor in the range . It can be seen in Figure 21 a good coherence between the numerical results (left side) and the experimental ones (right side). By changing the values of electronic components: ,, , , , and (for the other parameters, see Table 1 case B), the coexistence of attractors emerges. Figure 22 clearly illustrates the coexistence of fixed points, period2 attractor, and chaotic attractor. Those attractors appear randomly by switching on and off the power supply. We can conclude that the mathematical model proposed in this work perfectly describes the real behavior of the novel hyperjerk circuit.
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5. Conclusion
This work has proposed and investigated a new chaotic hyperjerk circuit with three equilibrium points having hyperbolic sine nonlinearity. The chaotic behavior observed in the system is due to the nonlinear component formed by two antiparallel diodes. Classical nonlinear analysis tools have been used to study the complete dynamics of the system. The bifurcation analysis of the new circuit shows that the chaotic doubleband attractor arises from the perioddoubling scenario followed by the symmetry recovering crisis event. In addition, some properties of the system such as antimonotonicity and offset boosting have been revealed. In particular, various regions in the parameter space in which the system develops the coexistence of up to ten disconnected attractors consisting of stable fixed points, limit cycles, and strange attractors have been reported. The coexistence of periodic, chaotic, and stable fixed points discovered in this work has not yet been reported in a hyperjerk system (at least as simple as the case discussed) and thus merits dissemination. To validate the theoretical study presented in this work, the new chaotic hyperjerk circuit has been realized and used for the investigations. Experimental results agree well with those obtained during the numerical experiment, thus confirming the feasibility of the proposed model. Owing to its extreme simplicity coupled with extremely rich dynamics, the new hyperjerk circuit introduced in this work has potential utility for information encryption as well as for other chaosbased applications [62].
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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