Effects of Symmetric and Asymmetric Nonlinearity on the Dynamics of a Third-Order Autonomous Duffing–Holmes Oscillator

Unité de Recherche d’Automatique et Informatique Appliquée (URAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon Unité de Recherche de Matière Condensée, d’Electronique et de Traitement du Signal (URAMACETS), Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon

Multistability, namely, the occurrence of two or more attractors for a given set of system parameter values due to initial states (or/and noise sensitivity) [22][23][24], has explained a large number of events in biology [25,26] and chemistry [27], just to mention a few. Some important works based on the construction and study of multistable chaotic systems or chaotic systems with the coexistence of multiple attractors have recently been published in [28][29][30][31][32][33]. e work of Lai et al. [28] was based on the study of an extended Lü system; their results showed that the system has a pair of strange attractors, a pair of limit cycles, and a pair of point attractors for different initial conditions. e authors [29] have constructed an extremely simple chaotic system with an infinite number of coexisting chaotic attractors. Lai et al. [30] reported a new chaotic system they generated from the simplest chaotic memristor circuit by introducing a simple nonlinear feedback control input. e main characteristic of the new system is that it has an infinite number of coexisting equilibria and abundant attractors. e results of [31] report a new chaotic system without equilibrium with hidden attractors and coexisting attractors.
is special feature (multistability) can sometimes be justified by the fact that the system is symmetric [34]. In this line, Kengne and collaborators have highlighted the coexistence of four, six, and eight attractors from a cubic, hyperbolic tangent, and hyperbolic tangent-cubic nonlinearity-based chaotic jerk system, respectively [35][36][37]. e authors of [38] showed the presence of nine coexisting attractors in a hyperjerk chaotic system with hyperbolic sine nonlinearity. On the other hand, Negou and Kengne [34] recently introduced a jerk system with adjustable symmetry and nonlinearity. e system was obtained by replacing a hyperbolic sine nonlinearity in an existing jerk system with a smoothly adjustable nonlinearity. is modification demonstrated the relevance of adjustable symmetries in the emergence of striking multistability. e authors of [39][40][41] did similar works by considering adjustable symmetry and nonlinearities in the dynamics of a simple jerk system, snap system, and jerk circuit, respectively. As far as symmetry breaking is concerned, the results obtained highlighted issues, such as the presence of parallel bifurcation branches, hysteresis, and coexisting multiple asymmetric attractors in the mentioned systems and circuits. ese results are of particular practical interest given that perfect symmetry is not a practical reality.
Motivated by these results, we propose in this article a third-order chaotic system with adjustable symmetry, nonlinearity, and nontrivial equilibria as well.
e novel system is a generalized form of the third-order autonomous Duffing-Holmes oscillator introduced by Lindberg et al. [42]. e novelty is brought by adding a parametric quadratic term to the cubic nonlinear term of the former system, resulting in a nonlinear function of the form φ m (x) � x − mx 2 − x 3 . e quadratic term has been considered here because it models a possible imperfection that breaks the perfect symmetry of the original system. is modification can also be regarded as mathematical techniques to discover new nonlinear patterns. Historically, forced Duffing-Holmes system is one of the well-known simple nonautonomous second-order chaotic system [43,44]. In 2009, Lindberg et al. [42] suggested an autonomous third-order Duffing-Holmes type system, with a smoother spectrum. In spite of the numerous studies recently devoted to uncovering multistability in nonlinear systems, no report has shown that the autonomous thirdorder Duffing-Holmes system suggested by Lindberg et al. has such striking behavior (to the best of the authors' knowledge). With the generalized third-order autonomous Duffing-Holmes system proposed in this work, our aim is twofold. Firstly, we show that the autonomous Duffing-Holmes system introduced by Lindberg et al. (which is a particular case of the generalized one) exhibits multistability. Secondly, we investigate the effects of adjustable symmetry, nonlinearity, and nontrivial equilibria on the dynamics previously obtained (with the particular case). Moreover, amplitude control and adaptive synchronization of the system are performed for promoting possible applications in engineering. e remainder of this article is structured as follows. Section 2 is devoted to the description of the novel system and analytical examination of its basic properties. In Section 3, the system is numerically analyzed using appropriate tools of nonlinear dynamics. In Section 4, an electronic analog calculator of the system is designed and PSpice simulation results are also presented. Sections 5 and 6 are about amplitude control and adaptive synchronization, respectively. Finally, conclusions are drawn in Section 7.

System Description.
e autonomous Duffing-Holmes system derived by Lindberg et al. [42] is as follows: where b represents the damping coefficient, k is the feedback coefficient, and w f is the cutoff frequency of the filter (see Ref. [42] for more details). e system under investigation is obtained by adding the parametric quadratic term mx 2 to the second equation of the autonomous Duffing-Holmes system (1), that is: e quadratic nonlinearity coefficient m is introduced to modify system's nonlinearity, symmetry, and equilibria as it will be shown later. Note that system (1) can always be obtained by setting m � 0. ough this has been studied in [42], some complex behaviors such as coexistence of attractors were not revealed. With the generalized model (system (2)), we are going to revisit system (1) and investigate the case with m ≠ 0.

Dissipation and Existence of Attractors.
e presence of attractors in a system can be determined by calculating its divergence. e divergence of system (2) is given by 2 Complexity Given that w f and b are strictly positive, the divergence of system (2) is strictly negative whenever w f > b (case considered in this work).
is proves the existence of attractors because the volume elements contract after a unit of time. is contraction reduces a volume V 0 by a factor is means that each volume containing the trajectory of system (2) converges to zero when t tends to infinity at an exponential rate of (w f − b). erefore, all the orbits of system (2) are finally limited to a specific subset having a zero volume.

Symmetry.
Symmetry is an interesting feature of dynamic systems. It tells us if the solutions are unique or even.
is a solution of the system for a set of parameter values, then s(− x, − y, − z) is also a solution for the same values of parameters.
is symmetry will be responsible for the appearance of the solutions in pair by polarity inversion of the initial conditions [50] and thus leads to the coexistence of attractors in the state space. However, the aforementioned symmetry disappears for m ≠ 0.

Analysis of Fixed Points and Hopf Bifurcation.
When studying the dynamics of a system, it is always interesting to analyze the stability of its fixed points. By setting the lefthand side of equation (2) equal to zero ( _ x � _ y � _ z � 0), we find that system (2) has three fixed points: one trivial equilibrium p 1 (x 01 , y 01 , z 01 ) � (0, 0, 0) and two nontrivial equilibria p 2,3 (x 02,03 , y 02,03 , z 02,03 ) � ((− m ± ����� � m 2 + 4 √ /2), 0, 0). It should be noted that coordinates of nontrivial equilibria depend on the parameter m; it is therefore possible to adjust their positions by acting on the parameter m. e Jacobian matrix of system (2) associated with any of these fixed points is given by the following expression: e nature of equilibria can be found from the following characteristic equation (det(M J − λI 3 ) � 0): where I 3 denotes the 3 × 3 identity matrix. By applying Routh-Hurwitz stability criteria to characteristic equation (5), we obtain the following generalized stability conditions for the equilibrium points of system (2): For the trivial fixed point p 1 (x 01 , y 01 , z 01 ) � (0, 0, 0), we obtain the following stability conditions: while for the equilibrium points P 2,3 , we have the following stability conditions:  (7) and (8), for different values of the parameters w f , bet m.
Considering the change of variable λ � iw 0 (w 0 > 0) and substituting into (5), after separation of real part from imaginary part, we obtain relation (10) for the trivial point of equilibrium (P 1 ) when m � 0.
Differentiation of equation (5) with respect to k gives the following relation: Complexity 3 Equation (10) defines the frequency of stable oscillations as well as the critical value of k corresponding to the Hopf bifurcation of system (2). From relation (11), note that the transversality condition is always satisfied provided that w f is a strictly positive parameter.

Numerical Investigations
3.1. Methodology. In this section, we do a thorough analysis of the dynamics of system (2) using numerical tools such as bifurcation diagrams, maximum Lyapunov exponent, Table 1: Stability of the equilibrium points P 1 , P 2 , and P 3 obtained from equations (7) and (8) for different values of w f , b, k, and m.

Equilibrium points
Parameter values Eigenvalues Stability

Complexity
Poincaré section, phase portraits, time series, cross section of the basin of attraction, and so on. Bifurcation diagrams are obtained from the local maxima of one of the state variables for different values of the control parameters. Maximum Lyapunov exponents are computed using Wolf's algorithm [51]. In fact, the Maximum Lyapunov exponent makes it possible to estimate the rate of convergence or divergence between two close trajectories. us, if the maximum Lyapunov exponent is (i) positive, the system is considered as chaotic because for a small perturbation, its trajectory diverges (this implies the occurrence of a strange attractor in phase space); (ii) negative, the system is considered stable because for all perturbations, it returns to its equilibrium point; (iii) null, the system is considered periodic or quasiperiodic.
All the numerical integrations are based on the fourthorder Runge-Kutta algorithm with an integration step h � 5.10 − 3 , for better precision.

Considering b as a Control
Parameter with m � 0. In this section, we consider b as a control parameter and m is equal to zero. e dynamics obtained is that represented by the bifurcation diagrams of Figure 1(a), carried out in the ascending direction for 0.22 ≤ b ≤ 0.32, w f � 0.5, and k � 1.6.
is figure presents a very interesting phenomenon which is the coexistence of two symmetric bifurcation diagrams. is coexistence is highlighted through the superposition of two diagrams (red and blue) obtained by symmetrically changing the initial conditions ((0.25, 0, 0)⇔(− 0.25, 0, 0)). ey are in good agreement with the corresponding graphs of Lyapunov exponents as shown in Figure 2(a). Figure 3 shows some 3D phase portraits obtained for different values of the control parameter with initial conditions as indicated. Figures 4(a)-4(d) show a symmetric chaotic attractor, its time series, its frequency spectrum, and its Poincaré section (these representations are consistent with the chaotic behavior), respectively, for b � 0.32. All these confirm the symmetric nature of the dynamic of system (2).

Considering b as a Control
Parameter with m ≠ 0. In this section, we still consider b as a control parameter but are plotted for m equal to 0.05, 0.10, and 0.25, respectively. Let us recall that red color is used for (x 0 , y 0 , z 0 ) � (0.25, 0, 0) and blue color is used for (x 0 , y 0 , z 0 ) � (− 0.25, 0, 0). Comparing Figures 1 Figure 1(a), we can observe that bifurcation diagrams in red are no more symmetric to their counterpart (in blue). ere is a gap (discontinuity) in the blue bifurcation diagrams (Figures 1(b)-1(d)), which increases as m is increased. is observation is caused by a shift in equilibrium points (see Table 2) and implies the existence of coexisting attractors. e coexistence of a period-2 limit cycle with a period-3 limit cycle ( Figure 5(c)) and a chaotic attractor with a periodic attractor ( Figure 5(d)), for instance, is revealed in Figure 5. Figures 6(a)-6(d) represent an asymmetric chaotic attractor, its time series, its frequency spectrum, and its Poincaré section (these representations are specific to chaotic attractors), respectively, for b � 0.32. e aforementioned results indicated an increase in system's complexity when m ≠ 0.

Considering k as a Control
Parameter with m � 0. Considering k as a control parameter with m equal to zero, the dynamics of system (2) is governed by the superimposed bifurcation diagrams and Lyapunov exponents of e different values of the initial conditions (x 0 , y 0 , z 0 ) are as follows: for the coexistence of the four attractors-period-1 ( ± 0.12, 0, 0) and chaotic spiral ( ± 0.39, 0, 0); for the coexistence of the six attractors-n-finite periods ( ± 0.28, 0, 0), period-6( ± 0.06, 0, 0), and period-1 ( ± 0.108, 0, 0). ese multistabilities are well illustrated on the cross sections of the basins represented in Figures 12(a)-12(d), where we can observe several regions distinguished by different colors that correspond to each attractor.

Considering k as a Control
Parameter with m ≠ 0. Keeping k(2 ≤ k ≤ 3) as the control parameter, with m ≠ 0(m � 0.10, 0.125), w f � 0.5, and b � 0.41, we obtain the asymmetric bifurcation diagrams of Figures 7(b) and 7(c), which govern the evolution of system (2) for m � 0.10 and 0.125, respectively. ese diagrams are obtained for the initial conditions ( ± 0.75, 0, 0). e asymmetry observed in these diagrams is due to the destruction of the system's symmetry. is is caused by a nonsymmetric displacement of nontrivial equilibrium points with respect to the values of m (see Table 2). is can also be observed on the corresponding Lyapunov exponents diagrams of

Two Parameters and Lyapunov Exponents.
e changes that occur in chaotic dynamic systems may not be only related to small differences of initial conditions but also rely on the variation of parameters to which the system is sensitive. Figure 13 shows an interesting way to represent the behavior of system (2) according to its parameters Complexity (b, k, and w f ) through the maximum Lyapunov exponents (λ max ) [52,53]. In this figure, we can appreciate behavioral changes marked by transitions, from periodic zones (where λ max < 0) in blue to chaotic zones (where λ max > 0) in red (see Figures 13(a) and 13(b)).
is way of presenting the dynamics of the system (2) thus makes it possible to prove the existence of chaotic solution for a combination of the values of its parameters.

Circuit Design and PSpice Simulations
e theoretical and numerical predictions previously obtained are verified in this section using an electronic analog of system (2). e circuit diagram of the proposed electronic analog computer is shown in Figure 14. It involves three integrators, an inverter, a subtractor, and two multipliers which are at the origin of the nonlinear terms.
Using Kirchhoff's laws, equations governing the circuit can be obtained as with the following considerations: x � X, Equation (12) becomes  Complexity Systems (14) and (2)  Let us carry out analog simulations of the circuit's dynamics in PSpice. For these simulations, the following configurations of the transient analysis are adopted. Print step: 200 ns; final step: 500 ms; no-print delay: 480 ms; step ceiling: 4µs. For electronic components, C 1 � C 2 � C 3 � 10 nF and R � 10 kΩ while R b , R k , R m , and R w f are tunable. e different portraits presented in Figure 14 correspond to the transitions to chaos of the symmetric system obtained by fixing R m � 1 MΩ, R w f � 20 kΩ, and R k � 6.25 kΩ and by varying R b (see Table 3) for the initial conditions (X(0), Y(0), Z(0)) � ( ± 0.25 V, 0 V, 0 V). We can observe similarities between phase portraits of Figure 15 with those of Figure 3 obtained from the bifurcation diagram of the symmetric system (Figure 1(a)). Figure 16 shows the transitions of the asymmetric system to chaos obtained for R m � 200 kΩ, R w f � 20 kΩ, R k � 6.25 kΩ by varying R b (see Table 3) with the initial conditions     Figure 5 obtained by numerical integrations at different points of the bifurcation diagram of the asymmetric system (Figure 1(b)). e coexistence of the four and six attractors (see Figures 17 and 18) was obtained for R k � 3.746 kΩ and R k � 3.745 kΩ, respectively (see Table 3

Total Amplitude Control (TAC)
Amplitude control of chaotic signals is important in engineering applications where the amplitude is desired for signal generation and transmission [54][55][56][57][58]. It makes it possible to get the attractor large or small by changing one or more variables in a range, without changing the dynamic and topological properties of the attractor [59]. e amplitude control can be total (TAC), in this case all variables are controlled in a linked way. When all the variables are controlled separately, we speak of composite amplitude control (CAC). In the case where only the amplitude of one  or of n − 1(n ≥ 3) variables is controlled, it is a partial amplitude control (PAC) [56,59].
Referring to [56,[59][60][61], it is possible to perform a total amplitude control (TAC) of system (2) by introducing an amplitude control function f(r) on the cubic term, which leads to the following system: e function f(r) makes it possible to control the amplitudes of the variables x, y, and z according to Proof. Let: u � x/ ���� f(r), v � y/ ���� f(r) and w � z/ ���� f(r). e resulting system (17) considering m � 0 is identical to system (2): So, the function f(r) controls all the amplitudes according to 1/ ���� f(r). Consider f(r) � 1/r(r > 0), with r being the total amplitude control parameter. By acting on the amplitude control parameter r(r > 0), the variables are modified while leaving the dynamics of the attractor unchanged. Figure 19(a) shows the reductions in the amplitudes of the x, y, and z variables of the dual-band chaotic attractor when increasing the control parameter r. Figure 19(b) shows the largest Lyapunov exponent diagram, which remains unchanged when the control parameter varies, which justifies the invariance of the properties of the attractor during amplitude control. e phase portraits of this dual-band chaotic attractor are shown in Figure 20   amplitude control (TAC) because the amplitudes of the three variables are reduced without altering the dynamic and topological properties of the attractor. is control can be achieved by introducing a potentiometer in the electronic circuit. is is convenient or beneficial in the context of secure communication as well as in various fields of information processing in engineering [54][55][56][57][58]. □ 6. Adaptive Synchronization e importance of synchronizing chaotic and/or hyperchaotic systems could be its application in areas of engineering such as secure communication [62][63][64][65][66]. To date, there are several synchronization schemes that have already been developed with notable applications [67][68][69][70][71]. Among these, one of the best known methods is the adaptive method, which we will use in this section.
Consider the master and slave subsystems described by expressions (18) and (19), respectively: where (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) represent the state variables of the master and slave, respectively. u i (i � 1, 2, 3) are the nonlinear controllers to choose so that the subsystems (18) and (19) synchronize. e synchronization errors between (18) and (19) are given by   Figure 7 showing the impact of the value of m on the symmetry of system (2) considering k as a control parameter.   108, 0, 0).
where e 2 � y 2 − y 1 , For reasons of simplicity, consider the expressions of the controllers described by with b, w f , and k being the estimated parameters of b, w f , and k, respectively. k i (i � 1, 2, 3) is the positive feedback gain.
By substituting (22) in (19) and (20), we obtain the following expressions for the slave subsystem (23) and the synchronization errors (24):  Figure 14: Circuit realization of the three-dimensional autonomous Duffing-Holmes like oscillator described by system (2) where U i (i � 1, 2, 3, 4, 5) are TL084CN and U j (j � 6, 7) are AD633JN.  16 Complexity   Complexity where , e synchronization problem amounts to impose that the synchronization error e(τ) ⟶ 0 (where e � [e 1 , e 2 , e 3 ] T ) when τ ⟶ ∞ (asymptotically stable around its point of equilibrium) [72]. For this, let us consider the following positive definite function as Lyapunov candidate function: V e 1 , e 2 , e 3 , e b , e w f , e k � 1 2 e 2 1 + e 2 2 + e 2 3 + e 2 b + e 2 w f + e 2 k .  Complexity   Figure 19: Bifurcation diagrams (a) and largest Lyapunov exponent (b), illustrating the change in amplitude of the variables x, y, and z (in red, blue, and black, respectively) as a function of the parameter r, without alteration of the properties of the system. e parameters b, w f , k, and m are set to 0.3, 0.5, 1.6, and 0.0, respectively.

Conclusion
is work focused on the effects of symmetric and asymmetric nonlinearity on the dynamics of a third-order chaotic system, namely, the Duffing-Holmes autonomous oscillator. is has been investigated by supplementing a parametric quadratic term (mx 2 ) to the cubic nonlinear term (− x 3 ) of the existing third-order autonomous Duffing-Holmes  system [42]. An electronic circuit analog to the proposed system has also been designed. Using the Routh-Hurwitz criteria, stability conditions of the fixed points have been established and the existence of a Hopf bifurcation is obtained. e dynamics of the system has been studied by considering two control parameters (b and k) for discrete values of the parameter m. e parameter m made it possible to adjust the symmetry of the system by modifying nontrivial equilibrium points. With b as control parameter, the system exhibits a coexistence of symmetric bifurcation diagrams, confirmed by superposition of the Lyapunov exponent diagrams, phase portraits, and others when m � 0. When m ≠ 0, symmetry is destroyed. With k as control parameter, dynamics of the system is still symmetric in the case of m � 0. is has been confirmed by the superposition of bifurcation diagrams obtained from different initial conditions. With this, the coexistence of four and six attractors in a set of symmetric pairs has been revealed at certain points of the bifurcation diagrams. e cross sections of the basins of attraction have been represented in these different points of coexistence. For the case of m ≠ 0 (with k as control parameter), the system exhibits asymmetry behavior confirmed by bifurcation diagrams obtained from different initial conditions. ese symmetric breakdowns (asymmetry) are related to the movements of the nontrivial equilibria, caused by the values of the parameter m. e representations of the dynamics of the system according to its parameters and the Lyapunov exponents revealed the presence of periodic and chaotic regions. e results from the analog simulations in PSpice show good agreement with the numerical results. e results obtained under total amplitude control (TAC) and synchronization analyses conducted using the adaptive method prove that this new system is suitable for applications in various fields of engineering such as encryption of images [73], secure communication [74], and random bit generator [75]. e model reported in this work is a typical example of a 3D system with three rest points and a nonlinear function with a region of negative slope. Also, we conjecture that the results obtained in this work may also be found in the Chua circuit, Shinriki oscillator, and jerk system (with cubic, hyperbolic sine, or hyperbolic tangent) just to cite a few, hence the pertinence and relevance of our study.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.