Abstract
Symmetry is an important property found in a large number of nonlinear systems. The study of chaotic systems with symmetry is well documented. However, the literature is unfortunately very poor concerning the dynamics of such systems when their symmetry is altered or broken. In this paper, we investigate the dynamics of a simple jerk system with hyperbolic tangent nonlinearity (Kengne et al., Chaos Solitons, and Fractals, 2017) whose symmetry is broken by adding a constant term modeling an external excitation force. We demonstrate that the modified system experiences several unusual and striking nonlinear phenomena including coexisting bifurcation branches, hysteretic dynamics, coexisting asymmetric bubbles, critical transitions, and multiple (i.e., up to six) coexisting asymmetric attractors for some suitable ranges of system parameters. These features are highlighted by exploiting common nonlinear analysis tools such as graphs of largest Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The control of multistability is investigated by using the method of linear augmentation. We demonstrate that the multistable system can be converted to a monostable state by smoothly adjusting the coupling parameter. The theoretical results are confirmed by performing a series of PSpice simulations based on an electronic analogue of the system.
1. Introduction
Recently, a particular attention has been paid to the study of nonlinear and chaotic dynamic systems. This is due to the rapid development of increasingly powerful computers on the one hand and on the other hand to the many potential applications in several fields of science and engineering. These systems are capable of several forms of complexity such as chaos, hyperchaos, multirhythmicity, bifurcations, intermittency, hysteresis, and multistability [1–3]. Concerning the latter feature, it should be noted that a multistable dynamic system is capable of displaying two or more attractors for the same set of parameters. In this situation, each of the coexisting attractors is connected with an attraction basin that represents all the initial conditions leading to the underlined attractor [4]. Fixed points, limit cycles, toruses, and chaotic attractors can coexist in the same system, in various combinations depending on the choice of parameters. The term extreme multistability refers to the situation where an infinite number of attractors coexist [5–9]. Multistability is relevant from the view point of practical application as it may give rise to unexpected and even disastrous consequences [10]. If the phenomenon of multistability is encountered in dynamic systems with no symmetry property, it should be noted that symmetric systems are much more likely to develop this phenomenon [10]. Concerning the symmetry property, it should be mentioned that it is shared by many systems in several fields of science and engineering [2, 4]. In addition to multistability mentioned above, symmetric dynamic systems exhibit interesting behaviors such as period doubling, spontaneous symmetry breaking, merging crisis, hysteresis, and intermittency [1]. The study of symmetrical systems is well documented. However, to the best of the authors’ knowledge, the literature is very poor concerning the behavior of these systems when their symmetry is altered or broken. The symmetry break purposefully induced in a nonlinear dynamical system may be adjusted to discover many complex nonlinear phenomena (e.g., multirhythmicity, bursting, coexisting bubbles, hysteresis, critical transitions, and coexisting multiple asymmetric attractors) as previously discussed in several nonlinear systems [11–17].
In this work, we consider a simple jerk system with hyperbolic tangent nonlinearity [18] whose symmetry is broken by the introduction of an additive constant. We address the chaos generation mechanism, the formation of bubbles of bifurcation, and the coexistence of multiple attractors in both the symmetric and the asymmetric regimes of operation. For convenience, recall that jerk dynamic systems [19–23] refer to 3D ordinary differential equations (ODEs) in the form where the nonlinear vector function indicates the “jerk” (i.e., the time derivative of the acceleration). The hyperbolic tangent function is relevant in numerous problems such as nonideal operational amplifiers, activation function in neural network, magnetization in ferromagnetic systems, and solar wind-driven magnetosphere-ionosphere systems [24–28]. Multistability in simple jerk dynamic systems has recently drained tremendous research interest in varied fields of science and technology resulting in several publications. On this line, Kengne and colleagues reported the coexistence of four self-excited mutually symmetric attractors in a jerk system possessing a cubic nonlinearity [23] based on both numerical and experimental methods. This striking feature of multiple attractors is mainly due to the system’s symmetry and thus is also obtained with a hyperbolic sine [29], a hyperbolic tangent [18], a composite tanh-cubic nonlinearity [21], or a voltage controlled memristor [30], whose intrinsic current-voltage characteristics has the form of a pinched hysteresis loop. Despite the pertinence and the importance of the abovementioned results, we would like to stress that all cases of multistability discussed so far is restricted to symmetric jerk systems; also, multistability in jerk dynamic systems in case of a broken symmetry is very little studied. Motivated by previous results on jerk dynamical systems, this paper focuses on the effects engendered by symmetry break in a simple autonomous jerk system with hyperbolic tangent nonlinearity previously analyzed in [18]. Thus, the novel chaotic flow is smoothly tuned to behave either symmetrically or to develop no symmetry property using a single parameter. Importantly, the investigations clearly reveal that the modified system can experience coexisting bubbles of bifurcation, coexisting multiple (symmetric or asymmetric) attractors, and crises phenomena not found in the original symmetric system [18]. Despite the fact that the addition of a constant term may be viewed as a purely mathematical technique to induce new nonlinear phenomena, one of the key motivations is that symmetries are rarely exact in real physical systems, and some symmetry-breaking imperfections are always present [31–33].
The structure of the paper is as follows. Section 2 describes the evolution equations of the modified jerk system with hyperbolic tangent and analyses possible symmetries. Analytical conditions for the occurrence of Hopf-type bifurcations are established, and the stability of the equilibrium points is investigated with respect to parameters. In Section 3, numerical results concerning the bifurcation behaviour of the model, the coexistence of numerous attractors, and the coexistence of bubbles of bifurcation are presented. The control of multistability based on the linear augmentation scheme is described in Section 4. Section 5 is concerned with the experimental study of the modified system. A convenient electronic circuit (i.e., the analogue simulator) is designed for investigating the extremely complex dynamics of the system. PSpice simulation experiment supports the results of the theoretical study. Finally, Section 6 presents the conclusions of the whole work.
2. Description and Analysis of the Model
2.1. The Model
The state equation of the autonomous jerk system, which is considered in this work, is expressed by the following third-order nonlinear system (ODEs) with a single nonlinear function:where , , and denote (real) positive control parameters. Notice that the nonlinearity is smooth and involves only one state variable (i.e., ). Here, is the symmetry control parameter of the model. Specifically, for, system (1) exhibits a perfect symmetry and reduces to the case previously studied by Kengne and coworkers [18]. The case corresponds to an asymmetric model for which more complex nonlinear phenomena arise (that cannot be explained by using the symmetry arguments) including, for instance, the presence of multiple coexisting asymmetric attractors, coexisting bifurcation branches, and crisis events (see Section 4). The graphical representations of the nonlinear function are provided in Figure 1 for several discrete values of parameter. Interestingly, we would like to quote that the hyperbolic tangent nonlinearity has also been considered in many problems related to neural networks and Chua’s system as well [24–28]. The inclusion of this nonlinear term in model (1) engenders the extremely complex and striking bifurcation patterns developed by the whole system. The most “elegant” form [4] of system (1) is achieved by expressing it as a jerk equation:
Finally, it should be remarked that the state variable appears solely in the third equation and, consequently, represents an offset boosted variable [34, 35].
2.2. Symmetry and Dissipation
As previously indicated above, it can easily be seen that systems (1a) and (1b) are symmetric with respect to the origin of the coordinates for the special case. For this singular case, systems (1a) and (1b) represent an inversion invariant nonlinear dynamic system, provided that it remains unchanged when performing the coordinate substitution: . As a result, all attractors of systems (1a) and (1b) occur either as individual symmetric attractors or as mutually symmetric couples. This property is the key ingredient to justify the presence of multiple coexisting stable states appearing for some suitable sets of system parameters. More importantly, a suitable exploitation of the symmetry property of the evolution equation simplifies considerably the numerical experiment (e.g., the stability analysis, the calculation of phase space trajectory, and the basins of attraction as well).
More generally, systems (1a) and (1b) are nonsymmetric for any value of. However, we notice the invariance of systems (1a) and (1b) following the coordinate transformation . Consequently, we restrict our analysis for positive values of parameter all over this work. The dynamics for negative values of can be deduced from the latter transformation.
The divergence of the vector field (1a) and (1b) is computed as follows:
It follows that, for any point in phase space, the divergence is always negative. Accordingly, system (1) is dissipative and consequently can develop attractors [1–3].
2.3. Fixed Point Analysis
The study of equibria always represents the first issue to be addressed when performing the investigation of a nonlinear dynamic system. Their study yields preliminary insights into the dynamics of complete system [1–4]. By equating to zero all the derivatives in system (5), we found that (see Figure 1), for , there exist three different rest points , where verifies the following transcendental equation:
In view of the graph presented in Figure 1, we notice that, for , the system has two fixed points, while a single equilibrium point exists in case . Throughout the rest of this work, we restrict our analysis for values of where the system exhibits three fixed points. It should be remarked that, for, the system displays three symmetric fixed points amongst which the origin [18]. Using the set of parameters defined above, the roots equation (4) have been numerically obtained for two discrete values of (i.e., and ) by using the “fzero” build in function of Matlab. Recall that the “fzero” function is a MATLAB subroutine to search for the zeros of a single variable real-value function. As sample results, we have obtained the following fixed points: (i) and for ; (ii) , , and for . We notice that the positions of the equilibrium points in state space are defined solely by the value of parameter. Evaluated at any given fixed point , the Jacobian matrix of system (1) takes the following form:
We obtain the related eigenvalues by searching for the zeros of the characteristic polynomial:where refers to the identity matrix. From the graphs in Figure 1, we notice that the equilibrium point has its x-coordinate with magnitude smaller than unity. Also, the related characteristic polynomial possesses coefficients with different signs, and thus, is always unstable according to the Routh stability theorem. The stability of the pair of fixed points ( and ) changes with the values of parameters and. From both the Routh–Hurwitz stability criterion [1, 2] and the Hopf bifurcation theorem, we derived the following results about the stability of the above pair of equilibrium points. Each of the fixed points remains stable only for values of . If the parameter is brought beyond the critical value, the fixed point becomes unstable. We now investigate the Hopf bifurcation related to the fixed point () when is considered as the bifurcation control parameter. The following analytical conditions have been derived:
As a result, systems (1a) and (1b) exhibit a Hopf bifurcation from when , and a limit cycle will develop around the point . Equation (7b) defines the frequency of oscillations (), while the transversality condition is expressed by equation (7c). As sample numerical results, both equilibrium points and undergo a Hopf bifurcation at for and . In contrast, for and , the Hopf bifurcation values are and , respectively, for and . From the study presented above, we conclude that both three equilibriums are unstable in any regime (periodic or chaotic oscillations) of operation, and systems (1a) and (1b) exhibit self-excited attractors accordingly [36,37].
3. Numerical Study
3.1. Scenario to Chaos
To highlight the influence of system parameters on the dynamics of the system, we keep and use and as control parameters. Figure 2 provides the bifurcation diagrams of the coordinate against and related plots of largest Lyapunov exponent [38] for two different values of (i.e., and ). These diagrams are obtained by scanning the parameter downward without resetting the values of initial conditions, starting the system from the initial state , respectively. We know that, for (see Figure 2), the system is symmetric as well as related dynamical structures (i.e., equilibrium points, attractors, and basins of attractions). For this particular case, it can be seen from the diagram of Figure 2 (left panel) that there exist two symmetric bifurcation branches (blue and magenta), exhibiting a period doubling sequence to chaos for decreasing . These branches merge at approximately via the well-known symmetry recovering the crisis process. At this point, two mutually symmetric mono-scroll chaotic attractors (corresponding to the blue and magenta branches) combine to form a double-scroll strange attractor (see Figure 3). Completely different routes are found in the nonsymmetric system (i.e., ). In fact, for a nonzero value of parameter (e.g., ), it can be captured from Figure 2 (right panel) that a pair of asymmetric limit cycles with different periodicity experiences each its own route of period doubling cascade to chaos when the control parameter is decreased in small steps. Accordingly, blue and magenta bifurcation branches display a horizontal shift which increases with parameter . For example, in the bifurcation plot of Figure 2 (right panel), the first period doubling takes place at for the blue branch and for the red one. Here, the merging process never occurs. Instead, the series of period doublings of coexisting asymmetric cycles yields an asymmetric double-scroll strange attractor (see Figure 4). Moreover, most fascinating properties of the asymmetric system is the sudden disappearance (via a critical transition) of one of the bifurcation branches (i.e., the magenta branch; see Figure 2) when decreasing the control parameter for any nonzero value of . Past this crisis event, the system experiences a single attractor that metamorphoses to an asymmetric double-scroll chaotic attractor as is further decreased.
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3.2. Coexistence of Multiple Attractors
The coexistence of multiple stable states [10] is one of the most attractive properties the jerk system considered in this work. This intriguing feature has been deeply investigated for the symmetric system (i.e., ) in the reference work [18]. In this section, we investigate the impact of the excitation term on the mechanism governing the generation of multiple solutions. In this regard, several bifurcation plots are produced following appropriate numerical techniques [18] in order to track parameter domains corresponding to the presence of multiple coexisting stable states. As sample results, Figure 5 shows the bifurcation diagrams of local maxima of variable, obtained when varying parameter in the range for four discrete values of . Details of numerical procedures employed to produce these plots are described in Table 1. These diagrams display the intricate phenomenon of parallel branches and hysteresis which justify the occurrence of multiple attractors for several parameter sets. More importantly, notice that the merging process does occur for the symmetric system (i.e., ). In contrast, one of the branches undergoes a critical transition and collapses for nonzero values of parameter when the control parameter is slowly decreased. The numerical techniques used to obtain those diagrams are provided in Table 1. A close examination of Figure 5 reveals that various combinations of coexisting attractors can be obtained when suitably selecting the system parameters. For example, Figure 6 presents the coexistence of two different chaotic attractors (a, b) computed for using two different values of initial conditions. The corresponding cross section of the basins of attraction is provided in (c) using the same colors as the relevant attractors. In this figure, the red zone indicates unbounded dynamics. In the same line, Figure 7 depicts three different asymmetric coexisting attractors and corresponding cross sections of the basin of attraction. More interestingly, by choosing appropriately the values of system parameters and initial conditions, four asymmetric coexisting attractors can be found as exemplified in Figure 8. A more intriguing situation is depicted in Figure 9 where up to five different asymmetric attractors coexist. The corresponding bifurcation like sequence of local maxima of the coordinate versus initial condition is provided in the graph of Figure 9(f). The cases reported above relate to the asymmetric mode of operation, and a different configuration of coexisting attractors occurs in the symmetric regime where coexisting attractors now appear only in mutually symmetric pairs (see Figure 10).
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From the above investigations, we notice that the occurrence of multiple attractors is possible both in the unforced (i.e., ) and the forced () regimes. The latter situation is being much more challenging for analysis provided that the occurrence of multiple solutions cannot be explained based on symmetry arguments. At this point, we would like to stress that the occurrence of multiple attractors has reported numerous other problems from diverse fields of science and technology. This feature can be advantageously exploited in engineering applications such as image encryption and random signal generation as well. However, in cases where only a single stable attractor is desired, some control strategies may be developed. Detailed analysis concerning this point is out of the scope of this paper. Accordingly, interested readers may obtain precious information from the review work of [10].
3.3. Coexisting Bubbles of Bifurcation
Another interesting and striking event revealed monitoring the parameters of system (1) is the phenomenon of antimonotonicity [39–44]. In fact, the period doubling transition to chaos followed by the reverse bifurcation scenario is found when varying the control parameter for several values of in case of a zero forcing term (i.e., ). Sample results are depicted in Figure 11 which presented five bifurcation plots of the coordinate versus for five discrete values of . In each case of the diagrams in Figure 11, there are two symmetric bifurcation diagrams due to the symmetry of the model. From Figure 11, we note that a period-1 bubble is obtained for . As decreases the sequencebubble bubbles bubble bubblesfull, Feigenbaum tree takes place. This behavior corresponds to the symmetric system () and much more complex nonlinear dynamics arise in the presence of a nonzero excitation force (i.e., ), as exemplified in Figure 12. This latter figure depicts the bifurcation plots of the coordinate against parameter obtained for several discrete values of while maintaining and . In contrast to the situation presented in Figure 11, lower and upper bifurcation branches are nonsymmetric, depicting a horizontal shift and exhibiting different periodicities. This striking behavior (engendered by symmetry break) is rarely reported and thus represents an enriching contribution to the behavior of these types of systems.
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4. Control of Multistability
Recently, a control method referred to as linear augmentation is described which is suitable to control the dynamics of a nonlinear system without perturbing the system’s parameters [45–47]. In this strategy, a nonlinear dynamic system is coupled with a linear one. The motivations of this coupling scheme are twofold: (a) to stabilize the steady state in a given nonlinear oscillator; (b) to adjust the number of coexisting attractors for a multistable system. Accordingly, the dynamics of the jerk system with hyperbolic tangent nonlinearity coupled to a linear system is described by the following fourth-order system:
Here, represents the decay parameter of the linear system, denotes the coupling strength, and is the control parameter, which serves to locate the position of the equilibrium points. For, both oscillators evolve independently, and the linear system exhibits a fixed-point motion. For a nonzero coupling coefficient (i.e.,), there is a mutual influence between the nonlinear oscillator and the linear system resulting in the symmetry breaking of the whole system even for . The fixed points of the coupled system are yielded by the following nonlinear algebraic system:
System (9) clearly shows that the fixed points are asymmetrically located in state space, and their number strongly depends on the values of the linear system parameters as well as the coupling strength. Considering the case where the system develops six distinct periodic and chaotic attractors, we examine the range of coupling parameter corresponding to a monostable behavior of the coupled system. To this end, the parameters are fixed as in the caption of Figure 13. The latter figure shows the bifurcation diagrams illustrating the transition from a multistable state (see Figure 10) to monostability when smoothly varying the coupling strength in the range. Regions A, B, C, and D correspond to the coexistence of five, four, three, and two attractors, respectively, while a single attractor is observed in region E. Six sets of data are superimposed. These data are obtained by scanning the parameter upward starting from each of the six coexisting attractors without resetting the initial conditions. We present in Figures 15 and 16 sample phase portraits of the system (corresponding cross sections of the basins of attraction), highlighting the transition of the system to a monostable state.
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5. PSpice Simulations
It is predicted from the above results that the jerk system with a single hyperbolic tangent function can undergo extremely varied dynamic behaviors. The design and implementation of a convenient electrical circuit (i.e., the analogue simulator) for the experimental study of the model are presented in this section. PSpice simulation [48] investigations are carried out to check the results of analytical and numerical analyses. The possibility of monitoring capacitors initial voltages and evaluating the corresponding impact on the behavior of the whole circuit represents one of the main advantages of using of PSpice. Interestingly, evidence of several coexisting stable solutions [49–54] in the system may easily be demonstrated both in the symmetric and the asymmetric modes of operation. Moreover, the hardware realization of theoretical chaotic mathematical models is convenient for engineering utilization including, for instance, random signal generation, chaos-based communications, and image encryption.
5.1. Design of the Experimental Circuit
The circuit diagram of the proposed electronic simulator is shown in Figures 16(a) and 16(b). The hyperbolic tangent nonlinearity module [53, 54] whose detailed schematic diagram is depicted in Figure 16(b) consists of resistors, a dual-transistor pair, a pair of operational amplifiers, and a dc current source. A detailed analysis of the hyperbolic tangent circuit can be found in [53, 54]. Operational amplifiers and related circuitry (in Figure 16(a)) implement the basic operations of addition, subtraction, and integration. By choosing a suitable time scaling, the simulator outputs can directly be displayed on the screen of a double trace oscilloscope by feeding the output voltage of to the X input and the output voltage of to the Y input. With the hypothesis of ideal operational amplifiers operating in their linear regime, upon applying Kirchhoff current and voltage laws to the circuit diagram in Figure 16(a), it can be established that the voltages , , and satisfy the set of three coupled first-order nonlinear differential equations:
Choosing the following rescale of time and variables: , system (10) is identical to system (1) with the following definition of parameters:
From the above equations, it follows that the dynamics of system (1(a) and 1(b)) can be simulated at any desired frequency (within the bandwidth of op. amplifiers) by choosing the value of the three capacitors.
5.2. PSpice Simulation Results
The behavior of the circuit shown in Figure 16 is studied in PSpice by employing the values of parameters provided in Table 2 in order to check the theoretically predicted results of Section 3, in particular the period doubling route to chaos and the presence of coexisting bifurcation branches. (i.e., equivalently parameter) is chosen as main bifurcation control resistor. The electronic components’ values (, , , and ; see Table 2) are selected so as to match the dimensionless values ( and ) of Section 3 in view of allowing the comparison between theory and PSpice results. When varying progressive resistor, we observe the same sequence of bifurcations described in PSpice. Sample results showing the projections of the double-scroll attractor emanating from the merging crisis of coexisting asymmetric mono-scroll chaotic attractors for are provided in Figure 17 along with corresponding theoretical ones. Similarly, Figure 18 depicts various projections of the asymmetric double-scroll chaotic attractor obtained in PSpice (right panel) and the corresponding theoretically obtained ones (left panel) when the control resistors are fixed as , , , and . The bifurcation sequences observed in PSpice perfectly agree with those of theoretical studies carried out in Section 3. On the contrary, using the electronic component values fixed as , , , and , we have observed the coexistence of four different asymmetric attractors, namely, a pair of asymmetric period-3 cycles, a period-2 cycle, and a chaotic attractor when starting the system from four different initial conditions (see Figure 19). The latter situation is identical to the case reported in Figure 8 during the theoretical analysis. We have avoided the inclusion of other cases of multistability obtained in PSpice for the sake of brevity. We would like to point out the existence of some small shifts in the values of the control resistor in PSpice in comparison to the theoretically predicted values. Such discrepancies are mainly due to the unavoidable simplifications adopted during the modeling step of the analogue simulator (e.g., ideal bipolar junction transistor model and ideal op. amplifier model, in comparison with more realistic/complex models implemented in PSpice).
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6. Concluding Remarks
In summary, this paper has explored the dynamics of a simple chaotic jerk system with hyperbolic tangent nonlinearity whose symmetry is destroyed by the adding a constant term acting as an external excitation force. We have shown that the modified system exhibits several unusual and interesting nonlinear patterns such as coexisting bifurcation branches, hysteretic behaviors, coexisting symmetric and asymmetric bubbles, critical phenomena, and multiple (i.e., two, three, four, five, or six) coexisting asymmetric attractors for some appropriately chosen sets of its parameters. These features were illustrated by exploiting common nonlinear analysis tools such as graphs of largest Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The control of multistability based on the linear augmentation scheme is exploited to tune the system from the state of six coexisting attractors to monostability. An appropriate electronic analogue of the system was designed and simulated in PSpice. The theoretical results show a very good agreement with the PSpice simulation investigations.
The model considered in this work can be regarded as prototypal autonomous 3D system with three rest points and an odd symmetry. Also, we conjecture that the dynamics induced by symmetry break observed in this work may also be found when using the jerk equation with other types of nonlinearities (e.g., cubic, quintic, hyperbolic sine, and piece-wise quadratic). Moreover, the extension of the analysis presented in this paper to cases of other chaotic oscillators such as Chua’s, Shinriki, autonomous van der Pol-Duffing, and hyperjerk circuits is under consideration.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.