Recent Advances in Modeling, Analysis, and Synchronization of Chaotic Systems
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Dimensionality Reduction Reconstitution for Extreme Multistability in MemristorBased Colpitts System
Abstract
In this paper, a fourdimensional (4D) memristorbased Colpitts system is reaped by employing an ideal memristor to substitute the exponential nonlinear term of original threedimensional (3D) Colpitts oscillator model, from which the initialsdependent extreme multistability is exhibited by phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3D dimensionality reduction model is built using the state variable mapping (SVM) method, which allows the implicit initials of the 4D memristorbased Colpitts system to be changed into the corresponding explicitly initialsrelated system parameters of the 3D dimensionality reduction model. The initialsrelated equilibria of the 3D dimensionality reduction model are derived and their initialsrelated stabilities are discussed, upon which the dynamical mechanism is quantitatively explored. Furthermore, the initialsdependent extreme multistability is depicted by twoparameter plots and the coexistence of infinitely many attractors is demonstrated by phase portraits, which is confirmed by PSIM circuit simulations based on a physical circuit.
1. Introduction
Chua’s circuit [1] and Colpitts oscillator [2] are two important physical circuits used for generating chaos. In the Chua’s circuit, the unique nonlinear negative resistor is generally realized based on operational amplifier [3], which makes the oscillating frequency limited. By contrast, in the Colpitts oscillator, the nonlinear circuit element is implemented by a bipolar junction transistor [2], which allows the oscillating frequency to be adjusted from a few hertz up to the microwave region (gigahertz), depending on the technology. Due to the natural nonlinearities [4], memristors can be introduced into some existing circuits or systems to easily achieve chaotic oscillations. In the past few years, various memristorbased nonlinear oscillating circuits and systems were proposed, such as memristive HindmarshRose neuron model [5], memristive cellular nonlinear/neural network [6], memristive bandpass filter circuit [7], memristive spiking and bursting neuron circuit [8], memristive jerk circuit [9], memristive hypogenetic jerk system [10], memristive hyperjerk system [11], memristive TwinT oscillator [12], memristive Chua’s circuit [13], memristive canonical Chua’s circuit [14], memristive multiscroll Chua’s circuit [15], and memristive Chua’s hyperchaotic circuit [16]. However, relatively little attention has been received on the memristorbased Colpitts oscillator [17–19]. In addition, the memristive Colpitts oscillator implemented by replacing the bipolar junction transistor with memristor has not yet been reported. Because of the nanosized property, memristor is characterized by small size and low power consumption, leading to the fact that memristorbased Colpitts oscillator could have a good application prospect under some certain conditions.
The careful dynamical analyses of these constructed memristive systems show that the memristor initials do play a crucial role in dynamical characteristics of these systems [20, 21]. Particularly, memristive systems based on ideal memristors can produce the extreme multistability phenomenon of coexisting infinitely many attractors [22, 23]. Such a special phenomenon is commonly triggered in the systems with no equilibrium [24] or infinitely many equilibria [16, 25–28], entirely different from those generated from the offsetboostable flow by introducing an extra periodic signal [29–31]. In [17], a memristorbased Colpitts chaotic oscillator was proposed by introducing a nonideal extended memristor into original Colpitts oscillator [2]. This memristive system only had unique equilibrium and exhibited the rich parametersdependent dynamics. Like the reported ideal memristorbased systems, when an ideal memristor is introduced into original Colpitts oscillator, a natural question to ask is whether it will produce extreme multistability. Therefore, it is necessary to seek this special phenomenon in the ideal memristorbased Colpitts system.
The initialsdependent multistability [32, 33], or extreme multistability [34–36], pushes forward an immense influence on the study of dynamical characteristics in many nonlinear systems. Under the fixed system parameters, the solution trajectories of the systems can be represented by diverse stable states with the varied initials. Such a special phenomenon not only renders a nonlinear dynamical circuit or system to supply great flexibility for its potential uses in chaosbased information engineering applications [37], but also leads to new challenges for its control of the existing multiple stable states [32]. One might argue that this special phenomenon can hardly be achieved in practical engineering applications, as it is highly dependent on the initials. Moreover, due to the existence of zero eigenvalue at the equilibrium, it also presents new impediments for the traditional theoretical analysis of dynamical mechanisms. Interestingly enough, these problems can be solved by simplifying the mathematical models using proper state variables or applying reasonable approximation and simplification [38, 39].
Latterly, to solve the abovementioned problem, fluxcharge analysis method [13, 14, 22, 23] for the memristorbased dynamical circuits and state variable mapping (SVM) method [11, 40] for the memristorbased dynamical systems were proposed to achieve an equivalent dimensionality reduction model, leading to the fact that the circuit goes from highorder to loworder or the system goes from highdimensional to lowdimensional. With these methods, the implicit initials in the original circuit or system can be changed into explicitly initialsrelated circuit/system parameters appearing in the dimensionality reduction model, and multiple stable states can be controlled by changing the initialsrelated circuit/system parameters [14], upon which the mechanism explanation for initialsdependent dynamics can be realized. Moreover, dimensionality reduction modeling can reduce the complexity of quantitative analyses and numerical simulations, which is of theoretical significance and engineering application value.
The aforementioned analytic strategies have been preliminarily verified in several memristorbased Chua’s circuits [13, 14, 22] and memristorbased hyperjerk system [11]. However, for memristorbased Colpitts system, applicability and effectivity of the state variable mapping method still need comprehensive investigations and the concept of dimensionality reduction reconstitution is insistent to be clarified. Enlightened by the above ideas, a novel fourdimensional (4D) memristorbased Colpitts system is reaped by employing an ideal memristor [24] to substitute the exponential nonlinear term of the original threedimensional (3D) Colpitts oscillator model [2, 41]. The proposed memristive Colpitts system exhibits the initialsdependent extreme multistability. To focus on the revelation and reconstitution of this special phenomenon, an equivalent 3D dimensionality reduction model is obtained using the SVM method reported in [11] and several determined isolated equilibria are thereby yielded. Consequently, the implicit initials of the 4D memristorbased Colpitts system are transformed into the explicitly initialsrelated system parameters of the 3D dimensionality reduction model. Meanwhile, the initialsdependent extreme multistability in the 4D memristorbased Colpitts system is reconstituted by the initialsrelated parametersdependent dynamics in the dimensionality reduction model through traditional quantitative analyses.
The rest of this paper is organized as follows. In Section 2, a 4D memristorbased Colpitts system is presented and the initialsdependent extreme multistability is revealed by phase portraits and twodimensional (2D) local basins of attraction. Thereafter, the equivalent 3D dimensionality reduction model for the proposed memristive Colpitts system is built by the SVM method. In Section 3, to explore the dynamical mechanism, the initialsrelated equilibria of the 3D dimensionality reduction model are derived and the initialsrelated stabilities are evaluated quantitatively. Furthermore, the initialsdependent extreme multistability is depicted by twoparameter bifurcation plots and the coexistence of infinitely many attractors is demonstrated by phase portraits. In Section 4, with the circuit implementation of the dimensionality reduction model, PSIM circuit simulations are used to validate the numerical simulations. The conclusion is drawn in Section 5.
2. MemristorBased Colpitts System and Dimensionality Reduction Modeling
2.1. 4D MemristorBased Colpitts System and InitialsDependent Extreme Multistability
The constructing scheme is adopted through imitating the method narrated in [10]. For the input x and output y, an incoming ideal memristor with an inner state variable φ can be modeled asInspired by [24], the memductance W(φ) chosen here is quadratic in φ, which is characterized bywhere the parameters α and β are two positive constants. Note that the circuit module of W(φ) can be synthesized by referring to [24].
A classic 3D Colpitts oscillator model with an exponential nonlinear term was reported in [2, 41], which was described aswhere Q and g are positive real constants, k = 0.5, and the exponential nonlinear termis used to characterize the voltagecurrent relation of the bipolar junction transistor in the Colpitts oscillator. When the parameters appearing in (2a) are set as Q = 1.415 and g = 3.1623 [2], the 3D Colpitts oscillator model (2a) and (2b) is chaotic and displays a spiral attractor.
Based on the 3D Colpitts oscillator model presented in (2a), a novel 4D memristorbased Colpitts system is reaped by employing the proposed memristor given in (1a) and (1b) to substitute the exponential nonlinear term described in (2b), whose mathematical model is formulated aswhere and two positive parameters a = 2g/Q, b = 1/Q are introduced for simplicity [41]. To focus on the revelation and reconstitution of extreme multistability, the parameters are determined as a = 5.2, b = 0.9, α = 0.5, and β = 0.1.
The ideal memristor (1a) and (1b) causes system (3) to possess line equilibrium therein, leading to the emergence of complex and sensitive initialsdependent extreme multistability with coexisting infinitely many attractors [22, 23]. To show the intriguing phenomenon, some intuitions about the extreme multistability of system (3) are exhibited by phase portraits, as shown in Table 1 and Figure 1, where the point attractor in Figure 1(e) is marked as fivepointed star. Apparently, a variety of disconnected attractors with different topologies, periodicities, and locations are coined in system (3) under different initials. Particularly, asymmetric chaotic doublescroll attractors (Figure 1(a)), symmetric chaotic doublescroll attractor, and chaotic spiral attractor (Figure 1(c)) can be observed in Figure 1, which are completely different from the chaotic spiral attractor reported in the original 3D Colpitts oscillator model (2a) and (2b) [2]. It is demonstrated that system (3) has more complex attractor structure.

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The phase portraits of coexisting infinitely many attractors in Figure 1 demonstrate that dynamical behaviors of system (3) are extremely depended on their initials. To inspect the dynamical behaviors distributed in the initial planes, 2D local basins of attraction in different initial planes are drawn, as shown in Figure 2, where only the periodicities of the state variable x_{1} are considered and the topologies and locations of the attractors are ignored here. The red regions marked by CH represent chaotic behaviors. The black and blue regions labeled by DE and P0 denote unbounded divergent and stable point behaviors respectively. Whereas the other color regions labeled by P1 ~ P4 stand for periodic behaviors with different periodicities. Therefore, the emergence of extreme multistability is disclosed, indicating the coexistence of infinitely many attractors in the 4D memristorbased Colpitts system.
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In addition, lots of unbounded divergent behaviors can be observed in Figure 2, which is rarely reported in a general memristive chaotic system [14], indicating that the proposed 4D memristorbased Colpitts system (3) is less robust to the initials.
2.2. Dimensionality Reduction Modeling
To explore dynamical mechanism of the initialsdependent extreme multistability emerged in system (3), an equivalent dimensionality reduction model for system (3) needs to be built [11, 13, 14, 22, 23]. Pursuant to the SVM method [11], integrating the four equations of (3) from 0 to τ, one getswhereRecalling the forth equation of (3), there exists . Thus the integral term in (4) is signified asThen system (4) can be rewritten asFrom (7), it is not difficult to find that the righthand sides of the first three equations do not depend on X_{4}, i.e., the forth equation of (7) is independent of the other three equations. Therefore, an equivalent 3D dimensionality reduction model can be described as
Similar to [11], there are correspondences between the state variables of systems (3) and (8) such thatBased on the relations in (9), the dynamical behaviors in (8) can be transformed back into those in (3).
Noteworthily, the implicit initials x_{i}(0) of the 4D memristorbased Colpitts system are mapped as explicitly initialsrelated system parameters δ_{i} appearing in the 3D dimensionality reduction model. What needs illustration is that, under the situation X_{1}(0) = X_{2}(0) = X_{3}(0) = 0, system (8) exhibits the completely same dynamical behaviors as the proposed system (3) [11]. To easily distinguish the different system parameters in system (8), we call a, b, α, β as the intrinsic system parameters and δ_{1}, δ_{2}, δ_{3}, δ_{4} as the extrinsic initialsrelated system parameters. It follows that the aforementioned 3D dimensionality reduction model can be utilized for quantitatively investigating the initialsdependent dynamics of the 4D memristorbased Colpitts system by changing the initialsrelated system parameters δ_{i}.
System (8) is a 3D nonlinear system, whose initials can also influence the dynamical behaviors. Similar to [11, 42], under the fixed initialsrelated system parameters, system (8) only exhibits two kinds of oscillating states. Taking δ_{1} = 10^{−9} and δ_{2} = δ_{3} = δ_{4} = 0 as an illustration, a bounded chaotic behavior under the initials (0, 5, 0) and an unbounded behavior under (−9, 0, 0) are coexisted in the X_{1} − X_{3} plane, as shown in Figure 3(a). Furthermore, the local basin of attraction in the X_{1}(0) − X_{2}(0) initial plane with X_{3}(0) = 0 is depicted, as shown in Figure 3(b); it can be easily observed that there are only two oscillating states, namely, the bounded chaotic behavior (red) and unbounded divergent behavior (yellow), respectively. Consequently, the 3D dimensionality reduction model is less sensitive than the 4D memristorbased Colpitts system to the initials.
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3. Dynamical Mechanism Illustrations for Extreme Multistability
3.1. Equilibria and Stabilities Depending on the InitialsRelated System Parameters
By setting = = = 0 and solving for the equilibrium of system (8), one getsin which is solved byDefine P and Q asIn pursuance of the classical Cardan discriminant Δ = (Q/2)^{2} + (P/3)^{3} [14, 43], the roots of (11) are derived as
The detailed breakdowns of the equilibrium S are given in Table 2. With reference to these results, it can be found that system (8) has only one determined equilibrium when Δ > 0 and only two determined equilibria when Δ = 0. In contrast, system (8) has three determined equilibria when Δ < 0. Based on the characteristic polynomial at the determined equilibrium S of system (8), the stability analysis of system (8) can be effectively performed. By the RouthHurwitz criterion, if and only ifis satisfied, the determined equilibrium S is stable and a point attractor will be prevailed in its neighborhood. The intuition of Table 2 and (14) is that, under the fixed intrinsic system parameters, the equilibrium locations and stabilities are decided by the initialsrelated system parameters δ_{i} (i = 1, 2, 3, 4). Thus, the initialsdependent extreme multistability presented in 4D memristorbased Colpitts system can be deduced from the evolutions of the determined equilibrium in the 3D dimensionality reduction model [40, 42].

Take δ_{1} = 10^{−9} and δ_{2} = δ_{3} = 0 as an example. When the initialsrelated system parameter δ_{4} varies within , system (8) invariably has three equilibria S_{3,1}, S_{3,2}, and S_{3,3}; the coordinates of the determined equilibria are depicted in Figure 4(a). Stabilities of these three determined equilibria are evaluated by their eigenvalues and denoted with different colored lines, where the red dash, blue solid, and black dashdot lines denote the unstable saddlefocus (USF), stable nodefocus (SNF), and unstable nodefocus (UNF), respectively. More specifically, the USF denotes the equilibrium S has one negative real root and a pair of conjugated complex roots with positive real parts; the SNF indicates the equilibrium S is of one negative real root and a pair of conjugated complex roots with negative real parts; the UNF represents the equilibrium S has one positive real root and a pair of conjugated complex roots with negative real parts. The corresponding bifurcation diagram of the state variable X_{1} is presented in Figure 4(b), in which [X_{1}(0), X_{2}(0), X_{3}(0)] = are determined; the upper is the bifurcation diagram such that δ_{4} varies from −4 to 0, and the lower is bifurcation diagram such that δ_{4} varies from 0 to 4. It can be seen that the representing dynamics in Figure 4(b) matches with the stabilities of three determined equilibria stated in Figure 4(a).
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Since the trajectory of system (8) starts from the original point, its evolution route is mainly elicited by the stability of the equilibrium neighboring to the original point and somewhat affected by the other equilibria. The bifurcation behaviors are symmetric for the negative and positive δ_{4} in the region I, but are asymmetric in the regions II and III. More narrowly, in the region I, when δ_{4} varies within , the three equilibria S_{3,1}, S_{3,2}, and S_{3,3} are all unstable, such that the system orbit may be randomly pushed toward one of these three unstable equilibria. And system (8) starts from the chaotic state and goes into the periodic state via reverse perioddoubling bifurcation route. In the region of I and region II, S_{3,2} becomes a stable equilibrium, and are still unstable equilibria. The dynamical behaviors of system (8) are mainly determined by the stable equilibrium , leading to the occurrence of point attractors. In the region III, the three equilibria are all unstable and the system orbit will randomly push toward one of these unstable equilibria, resulting in the generation of limit cycle, chaotic attractor, or unbounded orbit. In the region , system (8) displays the almost symmetric dynamical behaviors as those in . Accordingly, the stability distributions of these three determined equilibria related to the initialsrelated system parameter δ_{4} lead to the emergence of complex dynamical behaviors in system (8).
3.2. Extreme Multistability Reconstitution
Observed from Figure 4(b), we know that system (8) can display rich dynamical behaviors hinging on the initialsrelated system parameters δ_{1}, δ_{2}, δ_{3}, and δ_{4}. For intuitively manifesting the coexistence of infinitely many attractors, twoparameter bifurcation plots in different initialsrelated parameter planes are plotted, as shown in Figure 5. Here the twoparameter bifurcation plots are depicted by examining the periodicities of the state variable X_{1}, which are different from the parameterspace plots given in [44]. Similar to the color regions shown in Figure 2, the red region labeled by CH represents chaos, the black region by DE indicates divergence, the blue region by P0 denotes stable point, and the other color regions by P1 ~ P4 stand for periodic limit cycles with different periodicities. Comparing the numerical results in Figure 5 with those in Figure 2, the similarity of dynamical behaviors can be observed and the fact that the 3D dimensionality reduction model can be utilized for quantitatively investigating the initialsdependent dynamics of the 4D memristorbased Colpitts system by changing the initialsrelated system parameters is further validated. As the original state variables in the system (3) are the derivatives of the new state variables in system (8) and the computational errors always exist in numerical simulations [45], there are some slight differences between the numerical results in Figures 2 and 5. Therefore, it can be concluded that the 3D dimensionality reduction model is the equivalent representation of the 4D memristorbased Colpitts system.
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When the initialsrelated system parameters δ_{1} = δ_{2} = 0, the coexistence of infinitely many attractors in the δ_{3} − δ_{4} parameter plane can be observed in Figure 5(a). In the regions and of δ_{4}, the system can generate asymmetric chaotic doublescroll attractors. In contrast, the intuition of the region of δ_{4} is that the system can generate symmetric chaotic doublescroll attractors. Furthermore, when δ_{3} = δ_{4} = 0, δ_{1} = δ_{3} = 0, and δ_{2} = δ_{3} = 0, Figures 5(b), 5(c), and 5(d) reveal the coexistence of infinitely many attractors in different initialsrelated system parameter planes, respectively, and the emerged dynamical distributions are completely different from those shown in Figure 5(a).
Corresponding to the part of different color areas in Figure 5, different types of coexisting attractors are listed in Table 3. Automatically, referring to Figure 1, for eleven sets of different initialsrelated system parameters (δ_{1}, δ_{2}, δ_{3}, δ_{4}) in different color areas of Figure 5, the phase portraits of coexisting attractors in the X_{1} − X_{2} plane are numerically simulated, as displayed in Figure 6, where the point attractor in Figure 6(e) is marked as fivepointed star homogeneously. Obviously, Figure 6 shows exactly the same dynamical characteristics as Figure 1. It can be seen that many different kinds of disconnected attractors, such as chaotic attractors with different topologies, limit cycles with different topologies and periodicities, stable point, and unbounded orbit, can be observed in system (8). As a result, the initialsrelated parametersdependent dynamics featured by Figure 6 intuitively verify the initialsdependent extreme multistability in the proposed 4D memristorbased Colpitts system.

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4. PSIM Circuit Simulations
The 3D dimensionality reduction model described by (8) is equivalently implemented in an analog circuit form, as manifested in Figure 7, where the gains of two multipliers M_{1} and M_{2} are set as 1. According to basic circuit theory, the circuit state equations are formulated in a general form aswhere v_{1}, v_{2}, and v_{3} represent the state variables and RC is the integrating time constant. The initialsrelated system parameters δ_{1}, δ_{2}, δ_{3,} and δ_{4} are implemented by additional DC voltage sources or directly linking to the ground. Note that the R_{6} in Figure 7 is only a negative feedback resistor to ensure the dissipativity of the physical circuit without selfexcited oscillation, which is used to implement the selffeedback term in Equation (15). More details about the circuit design principle can refer to the operational amplifier stability in [46, 47].
To better confirm the extreme multistability generated from the equivalent circuit in Figure 7, PSIM circuit simulations are considered to confirm the phase portraits of coexisting attractors given in Figure 6. The circuit parameters shown in Figure 7 are taken as R = 10 kΩ, R_{1} = R/(aβδ_{4}^{2} − aα), R_{2} = R/a = 1.9231 kΩ, R_{3} = R/aβδ_{4}, R_{4} = 3R/aβ = 57.6923 kΩ, R_{5} = 2aR = 104 kΩ, R_{6} = R/b = 11.1111 kΩ, and C = 100 nF. The initials [v_{1}(0), v_{2}(0), v_{3}(0)] are assigned as (0 V, 0 V, 0 V) and the initialsrelated system parameters (δ_{1}, δ_{2}, δ_{3}, δ_{4}) are assigned as the same values by referring to those in Figure 6. PSIM intercepted phase plane plots in the v_{1} − v_{2} plane are shown in Figure 8. Ignoring the computational errors in PSIM simulations, PSIM simulated results in Figure 8 verify the complex phenomenon revealed in Figure 6 and illustrate that extreme multistability does exist in the proposed 4D memristorbased Colpitts system. What needs to be specified is that the initials of the 4D memristorbased Colpitts system are in the explicit form in the physical circuit of the 3D dimensionality reduction model described by (8), which can be used to easily achieve the controllable strategy for extreme multistability in the 4D memristorbased Colpitts system [14, 23].
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5. Conclusion
In this paper, a dimensionality reduction reconstitution scheme for extreme multistability in memristorbased Colpitts system was introduced. By employing an ideal memristor to substitute the exponential nonlinear term of original 3D Colpitts oscillator model, a novel 4D memristorbased Colpitts system was obtained. The initialsdependent extreme multistability of the proposed system was exhibited via phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3D dimensionality reduction model was constructed using SVM method. As a consequence, the implicit initials of the 4D memristorbased Colpitts system were transformed into the explicitly initialsrelated system parameters of the 3D dimensionality reduction model. Meanwhile, the dynamical mechanism was quantitatively explored by deriving the initialsrelated equilibria and discussing the equilibrium stabilities in the 3D dimensionality reduction model. Furthermore, the initialsdependent extreme multistability was verified by twoparameter bifurcation plots and the coexistence of infinitely many attractors was demonstrated by phase portraits and confirmed by PSIM circuit simulations based on a physical circuit. To sum up, this work has multiple advantages: the proposed 4D memristorbased Colpitts system has great practical importance, as it has much smaller size, lower power consumption, and more complex attractor structure; the dimensionality reduction model greatly reduces the computational overhead, as the system goes from the 4D to 3D; the traditional quantitative analyses can be used for exploring the extreme multistability phenomenon, because the implicit initials of the 4D memristorbased Colpitts system are transformed into the explicitly initialsrelated system parameters of the 3D dimensionality reduction model; the physical control and mechanism explanation for extreme multistability are realized through dimensionality reduction reconstitution.
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 they have no conflicts of interest.
Acknowledgments
This research issue was supported by the grants from the National Natural Science Foundations of China under Grant Nos. 61671245, 51777016, 51607013, and 61601062.
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