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Control of Coexisting Attractors with Preselection of the Survived Attractor in Multistable Chua’s System: A Case Study
Although the control of multistability has already been reported, the one with preselection of the desired attractor is still uncovered in systems with more than two coexisting attractors. This work reports the control of coexisting attractors with preselection of the survived attractors in paradigmatic Chua’s system with smooth cubic nonlinearity. Techniques of linear augmentation combined to system invariant parameters like equilibrium points are used to choose the desired surviving attractors among the coexisting ones. Nonlinear dynamical tools including bifurcation diagrams, standard Lyapunov exponents, phase portraits, and cross section of initial conditions are exploited to reveal the selection scenarios of the survived attractor in the multistability control process of Chua’s system. The main crisis towards annihilation of multistability in Chua’s system when varying the coupling strength is interior crisis and border collision. Theoretical and numerical results obtained are further validated with PSpice analysis.
In the study of nonlinear dynamic systems, the simultaneous existence of attractors (finite or infinite), also known as multistability [1–13], extreme multistability [14–16], or megastability , is now in the forefront. Recall that the famous Chua’s circuit is among the widely studied electronic circuits capable to display chaos . When Professor Leon Chua introduced that circuit, it was intentionally built in such a way that three equilibria of the model were unstable. Based on the local stability of each point, the circuit was able to exhibit a double-scroll chaotic attractor . During the investigation of this oscillator, the main challenge was to design the nonlinear part called Chua’s diode. Using op-amps , diodes , transistors , current feedback op-amps , and inductor free CNN (cellular neural network) cells , many experimental results were reported on the realization of Chua’s diode. Finally, by exploiting two-stage op-amp-based negative impedance converters (NICs) in parallel, a usual implementation of Chua’s diode was proposed and accepted as a standard . Also in , Bao and collaborators, during their investigation, found that, with an unstable zero saddle point and two symmetric stable nonzero node-foci, improved Chua’s circuit can also generate a self-excited chaotic attractor. Very recently, exponential sampled-data control for T-S fuzzy systems with application to Chua’s circuit has been explored . Of particular interest, Chua had also presented the ability to display the coexistence of multiple attractors for the same set of system parameters but using different initial conditions . This coexistence of attractors is sometimes undesirable and needs to be avoided; hence, the investigation was carried out in this work.
Remark that these coexisting solutions in a given nonlinear dynamics system can be self-excited or hidden. Recall that an attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of an unstable fixed point. Otherwise, it is called a hidden attractor . On the contrary, hidden solutions have attraction basin which does not overlap with the neighborhood of an equilibrium point, and thus may be difficult to find numerically [11, 27–34]. However, the localization of the latter was made possible using an algorithm proposed by Leonov et al. . Such system types (with hidden attractors) can be potentially dangerous and very unpredictable (unrelated to butterfly effect). The multistability of the nonlinear system means that the system is able to exhibit different types of coexisting stable states and different forms of attraction basin for an identical set of system parameters but using different initial conditions [2–4, 8, 14, 36, 37]. Since the coexistence of attractors has been commonly used in image processing [16, 38], it becomes very urgent to control this phenomenon when, sometimes, periodic and chaotic orbits exist simultaneously. Up to date, the prominent methods reported in the relevant literature which enable to turn a multistable system to a monostable system are the noise selection , pseudo-forcing , short pulses , harmonic perturbation , intermittent feedback , temporal feedback , and linear augmentation [2, 3, 44–50]. Except for the temporal feedback and linear augmentation methods, in almost all other existing methods, the control is applied to one parameter of the system to remove on the attractors for all initial points. Thus, external control such as the temporal feedback or linear augmentation method would be preferred. Recall that, in many bistable dynamical systems, only one of the stable states is desired to track certain system performance. In this regard, Sharma et al.  presented control of some bistable systems with annihilation and selection of attractors using the linear augmentation scheme. This work has been carried out using two well-known paradigmatic systems that are the autonomous Chua oscillator and a neuronal system. Furthermore, the linear augmentation method has been successfully used in  to control the bistability property exhibited by the Lorenz–Rössler system. In 2015, the same research team  exploited the linear control scheme to stabilize a system to a fixed-point state even when the original system did not have any fixed point. Recently, in , the authors proposed the method of temporal feedback in autonomous as well as nonautonomous systems to target the coexisting attractor. The experimental realization of the introduced method was also addressed. However, all these results were based only on bistable systems.
Very recently, Fonzin Fozin et al.  investigated the annihilation of the coexistence of multiple stable states in a self-excited memristive hyperchaotic oscillator based on the linear augmentation method. Exploiting nonlinear analysis tools such as bifurcation diagrams, Lyapunov exponent spectrum, phase portraits, basins of attraction, and relative basin sizes, the authors show that when increasing the control parameter, the bifurcation routes followed by each of the three coexisting attractors were progressively merged in order to give a unique diagram. The results of the authors show that, for higher values of the control parameter, the multistable system with up to three coexisting attractors becomes a monostable one with only one surviving attractor. The same result was found when the same research team was addressing the control of multistability (involving three disconnected attractors) in simplified canonical Chua’s oscillator with smooth hyperbolic sine nonlinearity using the linear augmentation scheme [3, 44]. Much recently, Tabekoueng Njitacke et al.  investigated the coexistence of firing patterns and their control in two neurons coupled through an asymmetric electrical synapse. Their numerical results show the effectiveness of the control strategy through annihilation of the periodic coexisting firing pattern. They found that, for higher values of the coupling strength, only a chaotic firing pattern survives. From these results, it can be seen that intensive works have been done on the multistability control of the nonlinear oscillators based on the linear augmentation method. Remark that all these successful results on the control of multistability were performed so far only on systems with unique equilibrium point [2, 3, 44, 50]. This unique equilibrium point excludes the possibility to target/select a desired attractor during the multistability control process. Henceforth, we propose in this work the following:(a)To exploit the linear augmentation method to track and select one attractor among the four coexisting attractors using each of the three equilibrium points of the model(b)To design an analog electronic circuit of controlled Chua’s oscillator to further support the numerical investigation
The layout of the paper is as follows: in Section 2, we recall some basic properties of Chua’s oscillator with a smooth nonlinearity. In Section 3, numerical tracking of the coexisting attractors using usual nonlinear dynamics tools is addressed. In Section 4, the linear augmentation scheme is exploited to track each of the three coexisting attractors which survive around each of the three equilibriums points. The circuit realization of the investigated model is provided in Section 5. In Section 6, some conclusions are summarized.
2. Description of Chua’s Oscillator with Smooth Nonlinearity
Chua’s oscillator with smooth cubic nonlinearity in which multistability is controlled in this work is given by dimensionless equation (1) as follows [5, 51]:where the nonlinear function is defined as
The parameter values used in this work are the ones of the model studied in  and are set as , , , and with and being tunable. As it can be seen in equation (1), the model remains identical under the substitution . The stable state generated by the model will appear in symmetric pairs in to re-establish the real symmetry of the model. If not, the stable states produced will remain symmetric if the real symmetry of the attractors has already been re-established. This approach has been widely exploited recently to find coexisting attractors in symmetrical systems such as jerk [6, 8, 52–56], hyperchaotic and chaotic Chua’s oscillators [3, 44, 57–59], Hopfield neural networks [10, 36, 60, 61], and Duffing oscillator , just to name a few. In addition, it is easy to show that the model processes three equilibrium points given by the following expression: , and , in which is given by .
3. Selection of Coexisting Attractors: A Numerical Approach
3.1. Computational Method
In this section, we will use traditional nonlinear analysis tools such as bifurcation diagrams, graph of maximum Lyapunov exponent, phase portraits, two-parameter diagrams, standard Lyapunov stability diagrams, and attraction basins to hunt down windows in which controlled Chua’s oscillator with a smooth nonlinearity exhibits either hysteretic dynamics of parallel bifurcation branches. These various tools are computed using the Runge–Kutta formula in Turbo Pascal software with variables and constants chosen in the extended precision mode. In this contribution, we use a constant time grid of , and investigations are carried out for a very long time. In this way, the transient behavior is suppressed. Some bifurcation diagrams in this work are computed either by increasing the control parameter stating from different initial conditions or using an upward and backward continuation technique. These methods are the best for finding windows in which the model displays the coexistence of bifurcations. Graph of Lyapunov exponent is calculated using the algorithm proposed by Wolf et al. . Two-parameter diagrams and standard Lyapunov stability diagrams are computed by varying simultaneously two system parameters with making up of suitable colorful diagrams. These previous diagrams, as well as the basins of attraction, are obtained by numerically computing the maximum Lyapunov exponent on a grid of 350 × 350 values of the chosen space parameters.
3.2. Parallel Bifurcation Branches and Coexistence of Multiple Attractors
In the study of nonlinear dynamical systems, the coexistence of attractors for the same set of system parameters but using different initial conditions is known as multistability. Two curious and striking manifestations of such behavior are extreme or hidden extreme multistability [63, 64] and megastability phenomenon . This phenomenon of multistability has already been found in several nonlinear systems including the memristor-based oscillator [8, 16, 57], jerk/hyperjerk systems [55, 56, 65], and hyperchaotic Chua’s oscillator [3, 4], just to name a few.
Figure 1 represents a bifurcation obtained when varying the control parameter in the range . Two sets of data with their corresponding graph of maximum Lyapunov exponent are superimposed. A large window of coexisting bifurcations which are related with parallel bifurcation branches is presented. As a result, this superposition of the bifurcations is the coexistence of multiple attractors for the same sets of system parameters. For example, when , Chua’s circuit displays the coexistence of a symmetric pair of period-4 limit cycles (black and blue) and a symmetric pair of chaotic attractors (green and red), using different initial conditions, as depicted on the three-dimensional (3D) projection of the attractors in Figure 2(a). The set of initial conditions which enable to obtain each of the previous attractors is provided in Figure 2(b).
For this same value of the control parameter, the equilibrium points of the oscillator and their stability are checked as follows: with eigenvalues given by and and with eigenvalues given by and . Since the eigenvalues of the investigated model around the fixed points are unstable for the set of the parameter used for the study and its basin of attraction intersects with any open neighborhood of an unstable ﬁxed point, we conclude that the coexisting attractors found are self-excited . When , Chua’s circuit displays the coexistence of a symmetric pair of period-2 limit cycles (black and blue) and a symmetric pair of chaotic attractors (green and red), using different initial conditions, as depicted on the three-dimensional (3D) projection of the attractors in Figure 3(a). The basin of attraction associated to each coexisting attractor is provided in Figure 3(b). The equilibrium points of the model as well as the eigenvalues for that discrete value of the control parameter are given as with eigenvalues given by and and with eigenvalues given by and , which further supported the unstable nature of the equilibria. From this attraction basin of Figures 2(b) and 3(b), it can be observed that each attractor has its set of initial conditions which intercepts with the one of its direct neighbor. For each basin of attraction, zones of unbounded motion are marked with yellow color, whereas the pair of coexisting attractors is painted in red and green, while the pair of periodic attractors is painted in blue and black.
4. Control of Coexisting Attractors Using the Linear Augmentation Method
4.1. Description of the Control Method
From the already published works [45–49], the theory of the linear augmentation control method consists of coupling the nonlinear system displaying multistable behavior with a linear system as depicted by equation (3). Remark that the choice of scalar control was guided by recent results of control and synchronization on chaotic systems [66, 67]. Indeed, it has been demonstrated that control and synchronization of the chaotic system using scalars offer great flexibility than vectors. In fact, the drawback of using vector than scalar for control or synchronization is that the full states of the systems are involved in the process. When these full states of the systems are used, the energy and resource consumption are high.
In this equation, represents a standard form of a nonlinear dynamical system, is an m-dimensional vector of the system variable, and is the vector field on which it is associated. Parameter represents the connection weight which enables to link the nonlinear system and the linear one. Vector stands for the dynamics of the linear system , where represents its decay parameter. When the controller is off, i.e., , the linear system tends to zero with an exponential following a decay rate .
represents another key parameter of controlled Chua’s oscillator which will be used to track the wished stable state. It is generally selected at the neighborhood of the equilibria of the uncontrolled system . These stable states, which can originate from unstable equilibria, are checked by certain sets of invariants, i.e., coexisting attractors and existing fixed points. The fixed points are found either in the center of the coexisting attractors or lying on the boundary separating the basins of attraction of the stable state. Then, by considering vector identical with one of the unstable fixed points, one can obtain the death of some of the coexisting stable states through merging crises when increasing coupling/connection weight between the coupled oscillators. For superior values of the connection weight, only one surviving attractor is obtained which enables the system for chosen parameter sets to turn from multistable to a monostable one. The controlled scheme presented above is now used for Chua’s oscillator. Coupling is applied along the variable with the coupling strength as depicted in the following equation:
The equilibrium points of controlled Chua’s oscillator are obtained by solving the following equation:
After some algebraic manipulations, we obtain the expression of the equilibrium points as follows:where is obtained by solving the following equation:
Considering and , equation (7) becomes
The roots of (8) can be derived using the Cardano–Tartaglia method. According to the Cardano discriminant [68, 69], if , there is a real root and two complex roots. Since the fixed point cannot be a complex number, one equilibrium point exists.where .
If , equation (8) would have two real roots. Finally, if , there are three real roots in equation (8), which manifests that controlled Chua’s oscillator has three equilibrium points and can be obtained from equations (9)–(11). The rest of the circuit parameters are the same as those used in Figure 3. The stability of controlled Chua’s oscillator is provided in Table 1. As it can be seen from Table 1, for some discrete values of the controller coefficient, controlled Chua’s oscillator, under the consideration, conserves its stability, hence its ability to generate self-excited attractors because of the unstable nature of its equilibria. It is found that when the coupling strength is null, there are three values of , among which the origin and a symmetric pair . When the coupling strength is increased, remains uniform, while and are attracted by decreasing (resp. increasing) the value of the equilibrium points and .
4.2. Control of Multistability in Chua’s Oscillator
An illustrative example of the linear control method is shown in Figure 4 using the two-parameter diagram and corresponding standard Lyapunov stability diagram in the parameter space . The integration method adopted here is identical with the one used in the previous section. Cyan color is tied to periodic oscillations, while magenta color is tied with chaotic motions. Good accordance is observed between two-parameter diagrams (left) and corresponding standard Lyapunov stability diagrams (right). The diagrams are obtained by sweeping upward (Figure 4(a)) and downward (Figure 4(b)) both control parameters. From a general point of view, it can be seen that, on both diagrams, four regions, namely, (R1), (R2), (R3), and (R4), can be observed. Regions (R1), (R2), and (R3) correspond to the set of parameters for which the model displays hysteretic dynamics which gives birth to the phenomenon of coexistence of multiple stable states, whereas (R4) represents the set of parameters in which the model displays monostable dynamics, in other words, absence of multistability. These diagrams are very important since they enable the engineer to have a general overview on the dynamics of the multistable oscillator when the bifurcation parameter and the controller coupling strength are both varying.
Since coupling is introduced along the “” variable, we fix which is one among the two nontrivial equilibrium points. When increasing the control parameter in the range as it can be seen in Figure 5(a), four sets of data are superimposed in the bifurcation diagram. Each set of data (marked by red, black, blue, and green colors) corresponds to the route followed by each attractor during the control mechanism. As depicted in Figure 5(a), three crises enable all the plotted routes to merge along the one in black for higher values of the coupling strength. In region (D1) of Figure 6 and for very small values of (i.e., ), four attractors coexist including two chaotic attractors (red color and green color) with two periodic attractors (black color and blue color).
At the upper boundary of (D1), the diagram in red (chaotic one) undergoes a merging crisis (first crisis) and blends with the diagram in blue. In region (D2), because of the previous merging crisis, there are only three distinct diagrams that follow their bifurcation sequences (see region (D2)). For a discrete value , we have the coexistence of three disconnected attractors, involving a period-2, period-3, and period-4 limit cycle as presented in Figure 6(a). The demarcation region of each coexisting attractor in region (D2) is provided in Figure 6(b). As it can be observed from Figure 6(b), the basin in blue has already absorbed the one in red, while the basin in black has started to absorb the one in green. At the upper boundary of (D2), a crisis (second crisis) enables the diagram in green displaying period-3 limit cycle to merge with the diagram in black. In region (D3), we observe the superposition of two diagrams including a periodic and chaotic one.
In this region for a discrete value , Chua’s oscillator displays coexistence of a period-2 limit cycle with an asymmetric chaotic attractor (see Figure 7(a)). The basin of attraction associated with each coexisting attractor is computed and plotted in Figure 7(b). From this basin of attraction, it can be observed that the basin in black has already absorbed the one in green. At the upper boundary of (D3), a crisis (third crisis) enables the diagram in blue displaying chaotic behavior to merge with the diagram in black. In region (D4), when the critical value , all the diagrams have already merged with the black one, and the control goal is achieved as depicted in region (D4) when “” is fixed as . We can say that the route followed by the black diagram (see Figure 7(b)) is a magnetized route that attracts towards it all the other routes as the control parameter is increased.
As it can be observed in Figure 5(b), only two crises enable the control of the multistability around the origin . At the upper boundary of (D1), the first crisis enables the diagram in green to merge with the diagram in black. At the same time, the diagram in red merges with the one in blue. As presented in (D2), only a symmetrical pair of attractors coexists. At the upper boundary of (D2), the two coexisting diagrams undergo a symmetric restoring crisis and give birth to a unique diagram as depicted in region (D3).
Now, when tacking the other nontrivial equilibrium point (), the surviving attractor, when monitoring the control parameter, is obtained after three crises. These crises enable all the diagrams to finally merge with the diagram in blue. The details of this control process through merging of the diagrams are not provided for the sake of brevity. However, in Figure 8, we have provided some basins of attraction to illustrate merging of the basins of attraction. In Figure 8(a), the black diagram of Figure 5(c) has already merged with the green one. For that same value, the diagram in blue starts to absorb the one in red. In Figure 8(b), the diagram in blue has already completely merged with the red one, and only two coexisting attractors remain.
For , we have provided the unique attractors which have survived through the control scheme exhibited by Figures 5(a)–5(c) in Figure 9(a) and their corresponding basin of attraction in Figure 9(b). It is found that when the control, the techniques described in Figure 5(a), is exploited, the attractor in black is selected. When the control, the techniques described in Figure 5(c), is used, the attractor in blue is selected, and the control method displays by Figure 5(b) is used, the attractor in magenta is selected. These results clearly demonstrate that it is possible to target attractors based on steady points of the uncoupled system. Also, it is worth to emphasize here that the result of the multistability control of up to four coexisting attractors in Chua’s system addressed in this work has never been presented before and thus merit to be shared.
5. The Circuit Implementation
In this part of our work, the aim is to be able to set up an analog circuit that will allow us to make a comparison between the theoretical/numerical results obtained previously and the experimental results [8, 10, 27, 61, 70, 71]. The circuit diagram that allows us to perform various simulations in PSpice software is presented in Figure 10. The circuit of controlled Chua’s oscillator is designed using four capacitors , and , several resistors, eight op-amp TL082CD, a constant DC source, and two multipliers, which can be implemented practically using AD633JN versions of the AD633 four-quadrant voltage multiplier chips used to implement the nonlinear terms of our model. The signal () at the output is related to those at inputs , , , , and (). The circuit equation using Kirchhoff’s electrical circuit laws can be obtained as
When the controller is OFF ( or ), the circuit implementation of controlled Chua’s oscillator displays the phenomenon of the coexistence of up to four disconnected attractors as depicted in Figure 11. Thus, this result enables to support the fact that the previous obtained results on the coexistence of attractors in Chua’s oscillator were not artifacts. When the controller is ON for , selection of the attractor in controlled Chua’s oscillator occurs. For example, when and , the attractor in Figure 12(a) is selected. When and , the attractor in Figure 12(b) is selected. Finally, when and , the pair of the symmetric attractors of Figures 12(a) and 12(b) merges and gives the double-band chaotic attractor of Figure 12(c). Besides each selected attractor, its corresponding frequency spectrum is provided to further support the nature of the attractor.
This paper focused on selection of the coexisting attractor in multistable Chua’s oscillator with a smooth nonlinearity. The choice of paradigmatic Chua’s oscillator system within this work is based on the fact that it possesses three equilibrium points and thus opens the possibility to target or select a specific coexisting attractor located around unstable equilibria. Remark that the previous studies which focused on the multistability control were done on the system having three, four, and five coexisting attractors and only one equilibrium point [2, 3, 44, 50]. Based on linear augmentation, the unique equilibrium point excludes the possibility to target a coexisting attractor. This is why based on usual nonlinear techniques exploiting bifurcation diagrams, standard Lyapunov exponents, phase portraits, and cross-section basin of initial conditions, we show the possibility to control multistable Chua’s oscillator (with three equilibria) towards three monostable states depending on the choice of the equilibrium point used during the linear augmentation. These results clearly demonstrate that it is possible to select a coexisting attractor based on steady points of the uncoupled system such as predicted by Sharma et al. , where only bistable systems were investigated.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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