# Dimensionality Reduction Reconstitution for Extreme Multistability in Memristor-Based Colpitts System

Authors: Yunzhen Zhang; Zhong Liu; Mo Chen; Huagan Wu; Shengyao Chen; Bocheng Bao
Journal: Complexity
 (2019)

---

## Abstract

In this paper, a four-dimensional (4-D) memristor-based Colpitts system is reaped by employing an ideal memristor to substitute the exponential nonlinear term of original three-dimensional (3-D) Colpitts oscillator model, from which the initials-dependent extreme multistability is exhibited by phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3-D dimensionality reduction model is built using the state variable mapping (SVM) method, which allows the implicit initials of the 4-D memristor-based Colpitts system to be changed into the corresponding explicitly initials-related system parameters of the 3-D dimensionality reduction model. The initials-related equilibria of the 3-D dimensionality reduction model are derived and their initials-related stabilities are discussed, upon which the dynamical mechanism is quantitatively explored. Furthermore, the initials-dependent extreme multistability is depicted by two-parameter 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.


---

## Body

## 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 memristor-based nonlinear oscillating circuits and systems were proposed, such as memristive Hindmarsh-Rose neuron model [5], memristive cellular nonlinear/neural network [6], memristive band-pass filter circuit [7], memristive spiking and bursting neuron circuit [8], memristive jerk circuit [9], memristive hypogenetic jerk system [10], memristive hyper-jerk system [11], memristive Twin-T oscillator [12], memristive Chua’s circuit [13], memristive canonical Chua’s circuit [14], memristive multi-scroll Chua’s circuit [15], and memristive Chua’s hyperchaotic circuit [16]. However, relatively little attention has been received on the memristor-based 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 memristor-based 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 offset-boostable flow by introducing an extra periodic signal [29–31]. In [17], a memristor-based 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 parameters-dependent dynamics. Like the reported ideal memristor-based 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 memristor-based Colpitts system.The initials-dependent 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 chaos-based 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, flux-charge analysis method [13, 14, 22, 23] for the memristor-based dynamical circuits and state variable mapping (SVM) method [11, 40] for the memristor-based dynamical systems were proposed to achieve an equivalent dimensionality reduction model, leading to the fact that the circuit goes from high-order to low-order or the system goes from high-dimensional to low-dimensional. With these methods, the implicit initials in the original circuit or system can be changed into explicitly initials-related circuit/system parameters appearing in the dimensionality reduction model, and multiple stable states can be controlled by changing the initials-related circuit/system parameters [14], upon which the mechanism explanation for initials-dependent 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 memristor-based Chua’s circuits [13, 14, 22] and memristor-based hyperjerk system [11]. However, for memristor-based 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 four-dimensional (4-D) memristor-based Colpitts system is reaped by employing an ideal memristor [24] to substitute the exponential nonlinear term of the original three-dimensional (3-D) Colpitts oscillator model [2, 41]. The proposed memristive Colpitts system exhibits the initials-dependent extreme multistability. To focus on the revelation and reconstitution of this special phenomenon, an equivalent 3-D 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 4-D memristor-based Colpitts system are transformed into the explicitly initials-related system parameters of the 3-D dimensionality reduction model. Meanwhile, the initials-dependent extreme multistability in the 4-D memristor-based Colpitts system is reconstituted by the initials-related parameters-dependent dynamics in the dimensionality reduction model through traditional quantitative analyses.The rest of this paper is organized as follows. In Section2, a 4-D memristor-based Colpitts system is presented and the initials-dependent extreme multistability is revealed by phase portraits and two-dimensional (2-D) local basins of attraction. Thereafter, the equivalent 3-D dimensionality reduction model for the proposed memristive Colpitts system is built by the SVM method. In Section 3, to explore the dynamical mechanism, the initials-related equilibria of the 3-D dimensionality reduction model are derived and the initials-related stabilities are evaluated quantitatively. Furthermore, the initials-dependent extreme multistability is depicted by two-parameter 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. Memristor-Based Colpitts System and Dimensionality Reduction Modeling

### 2.1. 4-D Memristor-Based Colpitts System and Initials-Dependent Extreme Multistability

The constructing scheme is adopted through imitating the method narrated in [10]. For the inputx and outputy, an incoming ideal memristor with an inner state variableφ can be modeled as(1a)y=Wφxφ˙=xInspired by [24], the memductanceW(φ) chosen here is quadratic inφ, which is characterized by(1b)Wφ=α-βφ2where the parameters α and β are two positive constants. Note that the circuit module ofW(φ) can be synthesized by referring to [24].A classic 3-D Colpitts oscillator model with an exponential nonlinear term was reported in [2, 41], which was described as(2a)x˙1=gQ1-kx3-nx2x˙2=gQkx3x˙3=-Qk1-kgx1+x2-1Qx3whereQ andg are positive real constants,k =0.5, and the exponential nonlinear term(2b)nx2=e-x2-1is used to characterize the voltage-current relation of the bipolar junction transistor in the Colpitts oscillator. When the parameters appearing in (2a) are set asQ = 1.415 andg= 3.1623 [2], the 3-D Colpitts oscillator model (2a) and (2b) is chaotic and displays a spiral attractor.Based on the 3-D Colpitts oscillator model presented in (2a), a novel 4-D memristor-based 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 as(3)x˙1=ax3-aWx4x2x˙2=ax3x˙3=-0.5x1+x2a-bx3x˙4=x2where W(x4)=α-βx42 and two positive parametersa =2g/Q,b =1/Qare introduced for simplicity [41]. To focus on the revelation and reconstitution of extreme multistability, the parameters are determined asa = 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 initials-dependent 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 five-pointed star. Apparently, a variety of disconnected attractors with different topologies, periodicities, and locations are coined in system (3) under different initials. Particularly, asymmetric chaotic double-scroll attractors (Figure 1(a)), symmetric chaotic double-scroll 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 3-D Colpitts oscillator model (2a) and (2b) [2]. It is demonstrated that system (3) has more complex attractor structure.Table 1
Attractor types with different initials of system (3).
Initials Attractor types Phase portraits (10−9, 0, 0, ±3.6) Asymmetric chaotic double-scroll attractors Figure1(a) (10−9, 0, 0, ±3.3) Period-2 limit cycles Figure1(b) (−1, 2, 0, 2.55) Spiral chaotic attractor Figure1(c) (red) (10−9, 0, 0, 0) Symmetric chaotic double-scroll attractor Figure1(c) (blue) (10−9, −1.5, 0, −2) Period-2 limit cycle Figure1(d) (red) (10−9, 0, 0, −0.8) Period-3 limit cycle Figure1(d) (blue) (−1, 0, 0, −3.4) Period-1 limit cycle Figure1(e) (red) (−1, 0, 0, −3.2) Point attractor Figure1(e) (blue) (−1, 2, 0, 2.2) Unbounded orbit Figure1(f)Figure 1
Phase portraits of coexisting infinitely many attractors in thex3 −x4 plane for different initials (x1(0),x2(0),x3(0),x4(0)). (a) Asymmetric chaotic double-scroll attractors for (10−9, 0, 0, ±3.6). (b) Period-2 limit cycles for (10−9, 0, 0, ±3.3). (c) Chaotic attractors with different topologies for (10−9, 0, 0, 0) and (−1, 2, 0, 2.55). (d) Period-2 and period-3 limit cycles for (10−9, −1.5, 0, −2) and (10−9, 0, 0, −0.8). (e) Period-1 limit cycle and point attractor for (−1, 0, 0, −3.2) and (−1, 0, 0, −3.4). (f) Unbounded orbit for (−1, 2, 0, 2.2).
(a) (b) (c) (d) (e) (f)The phase portraits of coexisting infinitely many attractors in Figure1 demonstrate that dynamical behaviors of system (3) are extremely depended on their initials. To inspect the dynamical behaviors distributed in the initial planes, 2-D local basins of attraction in different initial planes are drawn, as shown in Figure 2, where only the periodicities of the state variablex1 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 4-D memristor-based Colpitts system.Figure 2
Two-dimensional local basins of attraction of system (3) in different initial planes for four sets of the initials (x1(0),x2(0),x3(0),x4(0)). (a) Thex3(0) –x4(0) initial plane withx1(0) =x2(0) = 0. (b) Thex1(0) –x2(0) initial plane withx3(0) =x4(0) = 0. (c) Thex2(0) –x4(0) initial plane withx2(0) =x3(0) = 0. (d) Thex1(0) –x4(0) initial plane withx2(0) =x3(0) = 0.
(a) (b) (c) (d)In addition, lots of unbounded divergent behaviors can be observed in Figure2, which is rarely reported in a general memristive chaotic system [14], indicating that the proposed 4-D memristor-based Colpitts system (3) is less robust to the initials.


### 2.2. Dimensionality Reduction Modeling

To explore dynamical mechanism of the initials-dependent 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 toτ, one gets(4)x1τ-δ1=-aαX2+aX3+aβ∫0τx42ξx2ξdξx2τ-δ2=aX3x3τ-δ3=-0.5X1+X2a-bX3x4τ-δ4=X2where(5)Xiτ=∫0τxiξdξ,δi=xi0i=1,⋯,4Recalling the forth equation of (3), there exists dx4(ξ)=x2(ξ)dξ. Thus the integral term in (4) is signified as(6)∫0τx42ξx2ξdξ=∫0τx42ξdx4ξ=x43τ3-δ433=X2+δ433-δ433=X233+δ4X22+δ42X2Then system (4) can be rewritten as(7)X˙1=-aαX2+aX3+aβX233+δ4X22+δ42X2+δ1X˙2=aX3+δ2X˙3=-0.5X1+X2a-bX3+δ3X˙4=X2+δ4From (7), it is not difficult to find that the right-hand sides of the first three equations do not depend onX4, i.e., the forth equation of (7) is independent of the other three equations. Therefore, an equivalent 3-D dimensionality reduction model can be described as(8)X˙1=aβδ42-aαX2+aX3+aβX233+aβδ4X22+δ1X˙2=aX3+δ2X˙3=-0.5X1+X2a-bX3+δ3Similar to [11], there are correspondences between the state variables of systems (3) and (8) such that(9)x1=X˙1,x2=aX3+δ2,x3=X˙3,x4=X2+δ4Based on the relations in (9), the dynamical behaviors in (8) can be transformed back into those in (3).Noteworthily, the implicit initialsxi(0) of the 4-D memristor-based Colpitts system are mapped as explicitly initials-related system parametersδi appearing in the 3-D dimensionality reduction model. What needs illustration is that, under the situationX1(0) =X2(0) =X3(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 calla,b, α, β as the intrinsic system parameters andδ1,δ2,δ3,δ4 as the extrinsic initials-related system parameters. It follows that the aforementioned 3-D dimensionality reduction model can be utilized for quantitatively investigating the initials-dependent dynamics of the 4-D memristor-based Colpitts system by changing the initials-related system parametersδi.System (8) is a 3-D nonlinear system, whose initials can also influence the dynamical behaviors. Similar to [11, 42], under the fixed initials-related system parameters, system (8) only exhibits two kinds of oscillating states. Takingδ1 = 10−9 andδ2 =δ3 =δ4=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 theX1 −X3 plane, as shown in Figure 3(a). Furthermore, the local basin of attraction in theX1(0) −X2(0) initial plane withX3(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 3-D dimensionality reduction model is less sensitive than the 4-D memristor-based Colpitts system to the initials.Figure 3
Illustrations for the bounded chaotic behavior (red) and unbounded divergent behavior (yellow). (a) Phase portraits under the initials (0, 5, 0) and (−9, 0, 0). (b) Local basin of attraction in theX1(0) −X2(0) initial plane withX3(0) = 0.
(a) (b)


## 2.1. 4-D Memristor-Based Colpitts System and Initials-Dependent Extreme Multistability

The constructing scheme is adopted through imitating the method narrated in [10]. For the inputx and outputy, an incoming ideal memristor with an inner state variableφ can be modeled as(1a)y=Wφxφ˙=xInspired by [24], the memductanceW(φ) chosen here is quadratic inφ, which is characterized by(1b)Wφ=α-βφ2where the parameters α and β are two positive constants. Note that the circuit module ofW(φ) can be synthesized by referring to [24].A classic 3-D Colpitts oscillator model with an exponential nonlinear term was reported in [2, 41], which was described as(2a)x˙1=gQ1-kx3-nx2x˙2=gQkx3x˙3=-Qk1-kgx1+x2-1Qx3whereQ andg are positive real constants,k =0.5, and the exponential nonlinear term(2b)nx2=e-x2-1is used to characterize the voltage-current relation of the bipolar junction transistor in the Colpitts oscillator. When the parameters appearing in (2a) are set asQ = 1.415 andg= 3.1623 [2], the 3-D Colpitts oscillator model (2a) and (2b) is chaotic and displays a spiral attractor.Based on the 3-D Colpitts oscillator model presented in (2a), a novel 4-D memristor-based 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 as(3)x˙1=ax3-aWx4x2x˙2=ax3x˙3=-0.5x1+x2a-bx3x˙4=x2where W(x4)=α-βx42 and two positive parametersa =2g/Q,b =1/Qare introduced for simplicity [41]. To focus on the revelation and reconstitution of extreme multistability, the parameters are determined asa = 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 initials-dependent 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 five-pointed star. Apparently, a variety of disconnected attractors with different topologies, periodicities, and locations are coined in system (3) under different initials. Particularly, asymmetric chaotic double-scroll attractors (Figure 1(a)), symmetric chaotic double-scroll 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 3-D Colpitts oscillator model (2a) and (2b) [2]. It is demonstrated that system (3) has more complex attractor structure.Table 1
Attractor types with different initials of system (3).
Initials Attractor types Phase portraits (10−9, 0, 0, ±3.6) Asymmetric chaotic double-scroll attractors Figure1(a) (10−9, 0, 0, ±3.3) Period-2 limit cycles Figure1(b) (−1, 2, 0, 2.55) Spiral chaotic attractor Figure1(c) (red) (10−9, 0, 0, 0) Symmetric chaotic double-scroll attractor Figure1(c) (blue) (10−9, −1.5, 0, −2) Period-2 limit cycle Figure1(d) (red) (10−9, 0, 0, −0.8) Period-3 limit cycle Figure1(d) (blue) (−1, 0, 0, −3.4) Period-1 limit cycle Figure1(e) (red) (−1, 0, 0, −3.2) Point attractor Figure1(e) (blue) (−1, 2, 0, 2.2) Unbounded orbit Figure1(f)Figure 1
Phase portraits of coexisting infinitely many attractors in thex3 −x4 plane for different initials (x1(0),x2(0),x3(0),x4(0)). (a) Asymmetric chaotic double-scroll attractors for (10−9, 0, 0, ±3.6). (b) Period-2 limit cycles for (10−9, 0, 0, ±3.3). (c) Chaotic attractors with different topologies for (10−9, 0, 0, 0) and (−1, 2, 0, 2.55). (d) Period-2 and period-3 limit cycles for (10−9, −1.5, 0, −2) and (10−9, 0, 0, −0.8). (e) Period-1 limit cycle and point attractor for (−1, 0, 0, −3.2) and (−1, 0, 0, −3.4). (f) Unbounded orbit for (−1, 2, 0, 2.2).
(a) (b) (c) (d) (e) (f)The phase portraits of coexisting infinitely many attractors in Figure1 demonstrate that dynamical behaviors of system (3) are extremely depended on their initials. To inspect the dynamical behaviors distributed in the initial planes, 2-D local basins of attraction in different initial planes are drawn, as shown in Figure 2, where only the periodicities of the state variablex1 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 4-D memristor-based Colpitts system.Figure 2
Two-dimensional local basins of attraction of system (3) in different initial planes for four sets of the initials (x1(0),x2(0),x3(0),x4(0)). (a) Thex3(0) –x4(0) initial plane withx1(0) =x2(0) = 0. (b) Thex1(0) –x2(0) initial plane withx3(0) =x4(0) = 0. (c) Thex2(0) –x4(0) initial plane withx2(0) =x3(0) = 0. (d) Thex1(0) –x4(0) initial plane withx2(0) =x3(0) = 0.
(a) (b) (c) (d)In addition, lots of unbounded divergent behaviors can be observed in Figure2, which is rarely reported in a general memristive chaotic system [14], indicating that the proposed 4-D memristor-based Colpitts system (3) is less robust to the initials.


## 2.2. Dimensionality Reduction Modeling

To explore dynamical mechanism of the initials-dependent 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 toτ, one gets(4)x1τ-δ1=-aαX2+aX3+aβ∫0τx42ξx2ξdξx2τ-δ2=aX3x3τ-δ3=-0.5X1+X2a-bX3x4τ-δ4=X2where(5)Xiτ=∫0τxiξdξ,δi=xi0i=1,⋯,4Recalling the forth equation of (3), there exists dx4(ξ)=x2(ξ)dξ. Thus the integral term in (4) is signified as(6)∫0τx42ξx2ξdξ=∫0τx42ξdx4ξ=x43τ3-δ433=X2+δ433-δ433=X233+δ4X22+δ42X2Then system (4) can be rewritten as(7)X˙1=-aαX2+aX3+aβX233+δ4X22+δ42X2+δ1X˙2=aX3+δ2X˙3=-0.5X1+X2a-bX3+δ3X˙4=X2+δ4From (7), it is not difficult to find that the right-hand sides of the first three equations do not depend onX4, i.e., the forth equation of (7) is independent of the other three equations. Therefore, an equivalent 3-D dimensionality reduction model can be described as(8)X˙1=aβδ42-aαX2+aX3+aβX233+aβδ4X22+δ1X˙2=aX3+δ2X˙3=-0.5X1+X2a-bX3+δ3Similar to [11], there are correspondences between the state variables of systems (3) and (8) such that(9)x1=X˙1,x2=aX3+δ2,x3=X˙3,x4=X2+δ4Based on the relations in (9), the dynamical behaviors in (8) can be transformed back into those in (3).Noteworthily, the implicit initialsxi(0) of the 4-D memristor-based Colpitts system are mapped as explicitly initials-related system parametersδi appearing in the 3-D dimensionality reduction model. What needs illustration is that, under the situationX1(0) =X2(0) =X3(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 calla,b, α, β as the intrinsic system parameters andδ1,δ2,δ3,δ4 as the extrinsic initials-related system parameters. It follows that the aforementioned 3-D dimensionality reduction model can be utilized for quantitatively investigating the initials-dependent dynamics of the 4-D memristor-based Colpitts system by changing the initials-related system parametersδi.System (8) is a 3-D nonlinear system, whose initials can also influence the dynamical behaviors. Similar to [11, 42], under the fixed initials-related system parameters, system (8) only exhibits two kinds of oscillating states. Takingδ1 = 10−9 andδ2 =δ3 =δ4=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 theX1 −X3 plane, as shown in Figure 3(a). Furthermore, the local basin of attraction in theX1(0) −X2(0) initial plane withX3(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 3-D dimensionality reduction model is less sensitive than the 4-D memristor-based Colpitts system to the initials.Figure 3
Illustrations for the bounded chaotic behavior (red) and unbounded divergent behavior (yellow). (a) Phase portraits under the initials (0, 5, 0) and (−9, 0, 0). (b) Local basin of attraction in theX1(0) −X2(0) initial plane withX3(0) = 0.
(a) (b)


## 3. Dynamical Mechanism Illustrations for Extreme Multistability

### 3.1. Equilibria and Stabilities Depending on the Initials-Related System Parameters

By settingX.1 = X.2 = X.3 = 0 and solving for the equilibrium of system (8), one gets(10)S=-X-2+2bδ2+2aδ3,X-2,-δ2ain which X-2 is solved by(11)X-23+3δ4X-22+3δ42-αβX-2+3δ1-δ2aβ=0DefineP andQ as(12a)P=-3αβ(12b)Q=3δ1-δ2aβ-3δ4δ42-αβ+2δ43In pursuance of the classical Cardan discriminant Δ = (Q/2)2 + (P/3)3 [14, 43], the roots of (11) are derived as(13a)X-2,1=-0.5Q+Δ3+-0.5Q-Δ3-δ4(13b)X-2,2=0.5-1+j3-0.5Q+Δ3+0.5-1-j3-0.5Q-Δ3-δ4(13c)X-2,3=0.5-1-j3-0.5Q+Δ3+0.5-1+j3-0.5Q-Δ3-δ4The detailed breakdowns of the equilibriumS 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 equilibriumS of system (8), the stability analysis of system (8) can be effectively performed. By the Routh-Hurwitz criterion, if and only if(14)αβ<X-2+δ4<aα+2baβis satisfied, the determined equilibriumS 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 initials-related system parametersδi (i= 1, 2, 3, 4). Thus, the initials-dependent extreme multistability presented in 4-D memristor-based Colpitts system can be deduced from the evolutions of the determined equilibrium in the 3-D dimensionality reduction model [40, 42].Table 2
The equilibrium of the dimensionality reduction model (8).
Δ X - 2 X - 1 Equilibrium Δ > 0 X - 2,1 - X - 2,1 + 2 b δ 2 + 2 a δ 3 S 1 = ( - X - 2,1 + 2 b δ 2 + 2 a δ 3 , X - 2,1 , - δ 2 / a ) Δ = 0 X - 2,1, --0.5Q3 - X - 2,1 + 2 b δ 2 + 2 a δ 3 S 2,1 = ( - X - 2,1 + 2 b δ 2 + 2 a δ 3 , X - 2,1 , - δ 2 / a ) - 0.5 Q 3 + 2 b δ 2 + 2 a δ 3 S 2,2 = ( - 0.5 Q 3 + 2 b δ 2 + 2 a δ 3 , - - 0.5 Q 3 , - δ 2 / a ) Δ < 0 X - 2,1, X-2,2, X-2,3 - X - 2,1 + 2 b δ 2 + 2 a δ 3 S 3,1 = ( - X - 2,1 + 2 b δ 2 + 2 a δ 3 , X - 2,1 , - δ 2 / a ) - X - 2,2 + 2 b δ 2 + 2 a δ 3 S 3,2 = ( - X - 2,2 + 2 b δ 2 + 2 a δ 3 , X - 2,2 , - δ 2 / a ) - X - 2,3 + 2 b δ 2 + 2 a δ 3 S 3,3 = ( - X - 2,3 + 2 b δ 2 + 2 a δ 3 , X - 2,3 , - δ 2 / a )Takeδ1 = 10−9 andδ2 =δ3 = 0 as an example. When the initials-related system parameterδ4 varies within [-4,4], system (8) invariably has three equilibriaS3,1,S3,2, andS3,3; the X-2 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 dash-dot lines denote the unstable saddle-focus (USF), stable node-focus (SNF), and unstable node-focus (UNF), respectively. More specifically, the USF denotes the equilibriumS has one negative real root and a pair of conjugated complex roots with positive real parts; the SNF indicates the equilibriumS is of one negative real root and a pair of conjugated complex roots with negative real parts; the UNF represents the equilibriumS has one positive real root and a pair of conjugated complex roots with negative real parts. The corresponding bifurcation diagram of the state variableX1 is presented in Figure 4(b), in which [X1(0),X2(0),X3(0)] = [0,0,0] are determined; the upper is the bifurcation diagram such thatδ4 varies from −4 to 0, and the lower is bifurcation diagram such thatδ4 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).Figure 4
Dynamics with the variation ofδ4 atδ1 =10−9,δ2 =δ3 = 0. (a) The X-2 coordinates and stabilities of the three equilibria. (b) Bifurcation diagram ofX1, the upper:δ4 ∈ [-4,0]; the lower:δ4 ∈ [0,4].
(a) (b)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 positiveδ4 in the region I, but are asymmetric in the regions II and III. More narrowly, in the region I, whenδ4 varies within [-4,-2.9488], the three equilibriaS3,1,S3,2, andS3,3 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 period-doubling bifurcation route. In the region [-2.9488,-2.2592] of I and region II,S3,2 becomes a stable equilibrium, S3,1 and S3,3 are still unstable equilibria. The dynamical behaviors of system (8) are mainly determined by the stable equilibrium S3,2, 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 [2.2592,4], system (8) displays the almost symmetric dynamical behaviors as those in [-4,-2.2592]. Accordingly, the stability distributions of these three determined equilibria related to the initials-related system parameterδ4 lead to the emergence of complex dynamical behaviors in system (8).


### 3.2. Extreme Multistability Reconstitution

Observed from Figure4(b), we know that system (8) can display rich dynamical behaviors hinging on the initials-related system parametersδ1,δ2,δ3, andδ4. For intuitively manifesting the coexistence of infinitely many attractors, two-parameter bifurcation plots in different initials-related parameter planes are plotted, as shown in Figure 5. Here the two-parameter bifurcation plots are depicted by examining the periodicities of the state variableX1, which are different from the parameter-space 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 3-D dimensionality reduction model can be utilized for quantitatively investigating the initials-dependent dynamics of the 4-D memristor-based Colpitts system by changing the initials-related 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 3-D dimensionality reduction model is the equivalent representation of the 4-D memristor-based Colpitts system.Figure 5
Two-parameter bifurcation plots depicted by the periodicities of the state variableX1 in different initials-related parameter planes for four sets of the initials-related system parameters (δ1,δ2,δ3,δ4). (a) Bifurcation plot in theδ3 –δ4 plane withδ1 =δ2 = 0. (b) Bifurcation plot in theδ1 –δ2 plane withδ3 =δ4 = 0. (c) Bifurcation plot in theδ2 –δ4 plane withδ1 =δ3 = 0. (d) Bifurcation plot in theδ1 –δ4 plane withδ2 =δ3 = 0.
(a) (b) (c) (d)When the initials-related system parametersδ1 =δ2 = 0, the coexistence of infinitely many attractors in theδ3 −δ4 parameter plane can be observed in Figure 5(a). In the regions [-5,-3] and [3,5] ofδ4, the system can generate asymmetric chaotic double-scroll attractors. In contrast, the intuition of the region [-1,1] ofδ4 is that the system can generate symmetric chaotic double-scroll attractors. Furthermore, whenδ3 =δ4 = 0,δ1 =δ3 = 0, andδ2 =δ3 = 0, Figures 5(b), 5(c), and 5(d) reveal the coexistence of infinitely many attractors in different initials-related 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 Figure5, different types of coexisting attractors are listed in Table 3. Automatically, referring to Figure 1, for eleven sets of different initials-related system parameters (δ1,δ2,δ3,δ4) in different color areas of Figure 5, the phase portraits of coexisting attractors in theX1 −X2 plane are numerically simulated, as displayed in Figure 6, where the point attractor in Figure 6(e) is marked as five-pointed 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 initials-related parameters-dependent dynamics featured by Figure 6 intuitively verify the initials-dependent extreme multistability in the proposed 4-D memristor-based Colpitts system.Table 3
Different color regions and the coexisting attractor types.
Colors in Figure5 Coexisting attractor types Examples in Figure6 Red Upper- and lower-asymmetric chaotic double-scroll attractors Figure6(a) Cyan Upper- and lower-period-2 limit cycles Figure6(b) Red Symmetric chaotic double-scroll attractor and chaotic spiral attractor Figure6(c) Cyan and yellow Period-2 and period-3 limit cycles Figure6(d) Cadet blue and blue Period-1 limit cycle and point attractor Figure6(e) Black Unbounded orbit Figure6(f)Figure 6
MATLAB simulated phase portraits of coexisting infinitely many attractors in theX1 −X2 plane for different initials-related system parameters (δ1,δ2,δ3,δ4). (a) Upper- and lower-asymmetric chaotic double-scroll attractors. (b) Upper- and lower-period-2 limit cycles. (c) Chaotic attractor with different topologies. (d) Period-2 and period-3 limit cycles. (e) Period-1 limit cycle and point attractor. (f) Unbounded orbit.
(a) (b) (c) (d) (e) (f)


## 3.1. Equilibria and Stabilities Depending on the Initials-Related System Parameters

By settingX.1 = X.2 = X.3 = 0 and solving for the equilibrium of system (8), one gets(10)S=-X-2+2bδ2+2aδ3,X-2,-δ2ain which X-2 is solved by(11)X-23+3δ4X-22+3δ42-αβX-2+3δ1-δ2aβ=0DefineP andQ as(12a)P=-3αβ(12b)Q=3δ1-δ2aβ-3δ4δ42-αβ+2δ43In pursuance of the classical Cardan discriminant Δ = (Q/2)2 + (P/3)3 [14, 43], the roots of (11) are derived as(13a)X-2,1=-0.5Q+Δ3+-0.5Q-Δ3-δ4(13b)X-2,2=0.5-1+j3-0.5Q+Δ3+0.5-1-j3-0.5Q-Δ3-δ4(13c)X-2,3=0.5-1-j3-0.5Q+Δ3+0.5-1+j3-0.5Q-Δ3-δ4The detailed breakdowns of the equilibriumS 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 equilibriumS of system (8), the stability analysis of system (8) can be effectively performed. By the Routh-Hurwitz criterion, if and only if(14)αβ<X-2+δ4<aα+2baβis satisfied, the determined equilibriumS 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 initials-related system parametersδi (i= 1, 2, 3, 4). Thus, the initials-dependent extreme multistability presented in 4-D memristor-based Colpitts system can be deduced from the evolutions of the determined equilibrium in the 3-D dimensionality reduction model [40, 42].Table 2
The equilibrium of the dimensionality reduction model (8).
Δ X - 2 X - 1 Equilibrium Δ > 0 X - 2,1 - X - 2,1 + 2 b δ 2 + 2 a δ 3 S 1 = ( - X - 2,1 + 2 b δ 2 + 2 a δ 3 , X - 2,1 , - δ 2 / a ) Δ = 0 X - 2,1, --0.5Q3 - X - 2,1 + 2 b δ 2 + 2 a δ 3 S 2,1 = ( - X - 2,1 + 2 b δ 2 + 2 a δ 3 , X - 2,1 , - δ 2 / a ) - 0.5 Q 3 + 2 b δ 2 + 2 a δ 3 S 2,2 = ( - 0.5 Q 3 + 2 b δ 2 + 2 a δ 3 , - - 0.5 Q 3 , - δ 2 / a ) Δ < 0 X - 2,1, X-2,2, X-2,3 - X - 2,1 + 2 b δ 2 + 2 a δ 3 S 3,1 = ( - X - 2,1 + 2 b δ 2 + 2 a δ 3 , X - 2,1 , - δ 2 / a ) - X - 2,2 + 2 b δ 2 + 2 a δ 3 S 3,2 = ( - X - 2,2 + 2 b δ 2 + 2 a δ 3 , X - 2,2 , - δ 2 / a ) - X - 2,3 + 2 b δ 2 + 2 a δ 3 S 3,3 = ( - X - 2,3 + 2 b δ 2 + 2 a δ 3 , X - 2,3 , - δ 2 / a )Takeδ1 = 10−9 andδ2 =δ3 = 0 as an example. When the initials-related system parameterδ4 varies within [-4,4], system (8) invariably has three equilibriaS3,1,S3,2, andS3,3; the X-2 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 dash-dot lines denote the unstable saddle-focus (USF), stable node-focus (SNF), and unstable node-focus (UNF), respectively. More specifically, the USF denotes the equilibriumS has one negative real root and a pair of conjugated complex roots with positive real parts; the SNF indicates the equilibriumS is of one negative real root and a pair of conjugated complex roots with negative real parts; the UNF represents the equilibriumS has one positive real root and a pair of conjugated complex roots with negative real parts. The corresponding bifurcation diagram of the state variableX1 is presented in Figure 4(b), in which [X1(0),X2(0),X3(0)] = [0,0,0] are determined; the upper is the bifurcation diagram such thatδ4 varies from −4 to 0, and the lower is bifurcation diagram such thatδ4 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).Figure 4
Dynamics with the variation ofδ4 atδ1 =10−9,δ2 =δ3 = 0. (a) The X-2 coordinates and stabilities of the three equilibria. (b) Bifurcation diagram ofX1, the upper:δ4 ∈ [-4,0]; the lower:δ4 ∈ [0,4].
(a) (b)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 positiveδ4 in the region I, but are asymmetric in the regions II and III. More narrowly, in the region I, whenδ4 varies within [-4,-2.9488], the three equilibriaS3,1,S3,2, andS3,3 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 period-doubling bifurcation route. In the region [-2.9488,-2.2592] of I and region II,S3,2 becomes a stable equilibrium, S3,1 and S3,3 are still unstable equilibria. The dynamical behaviors of system (8) are mainly determined by the stable equilibrium S3,2, 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 [2.2592,4], system (8) displays the almost symmetric dynamical behaviors as those in [-4,-2.2592]. Accordingly, the stability distributions of these three determined equilibria related to the initials-related system parameterδ4 lead to the emergence of complex dynamical behaviors in system (8).


## 3.2. Extreme Multistability Reconstitution

Observed from Figure4(b), we know that system (8) can display rich dynamical behaviors hinging on the initials-related system parametersδ1,δ2,δ3, andδ4. For intuitively manifesting the coexistence of infinitely many attractors, two-parameter bifurcation plots in different initials-related parameter planes are plotted, as shown in Figure 5. Here the two-parameter bifurcation plots are depicted by examining the periodicities of the state variableX1, which are different from the parameter-space 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 3-D dimensionality reduction model can be utilized for quantitatively investigating the initials-dependent dynamics of the 4-D memristor-based Colpitts system by changing the initials-related 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 3-D dimensionality reduction model is the equivalent representation of the 4-D memristor-based Colpitts system.Figure 5
Two-parameter bifurcation plots depicted by the periodicities of the state variableX1 in different initials-related parameter planes for four sets of the initials-related system parameters (δ1,δ2,δ3,δ4). (a) Bifurcation plot in theδ3 –δ4 plane withδ1 =δ2 = 0. (b) Bifurcation plot in theδ1 –δ2 plane withδ3 =δ4 = 0. (c) Bifurcation plot in theδ2 –δ4 plane withδ1 =δ3 = 0. (d) Bifurcation plot in theδ1 –δ4 plane withδ2 =δ3 = 0.
(a) (b) (c) (d)When the initials-related system parametersδ1 =δ2 = 0, the coexistence of infinitely many attractors in theδ3 −δ4 parameter plane can be observed in Figure 5(a). In the regions [-5,-3] and [3,5] ofδ4, the system can generate asymmetric chaotic double-scroll attractors. In contrast, the intuition of the region [-1,1] ofδ4 is that the system can generate symmetric chaotic double-scroll attractors. Furthermore, whenδ3 =δ4 = 0,δ1 =δ3 = 0, andδ2 =δ3 = 0, Figures 5(b), 5(c), and 5(d) reveal the coexistence of infinitely many attractors in different initials-related 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 Figure5, different types of coexisting attractors are listed in Table 3. Automatically, referring to Figure 1, for eleven sets of different initials-related system parameters (δ1,δ2,δ3,δ4) in different color areas of Figure 5, the phase portraits of coexisting attractors in theX1 −X2 plane are numerically simulated, as displayed in Figure 6, where the point attractor in Figure 6(e) is marked as five-pointed 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 initials-related parameters-dependent dynamics featured by Figure 6 intuitively verify the initials-dependent extreme multistability in the proposed 4-D memristor-based Colpitts system.Table 3
Different color regions and the coexisting attractor types.
Colors in Figure5 Coexisting attractor types Examples in Figure6 Red Upper- and lower-asymmetric chaotic double-scroll attractors Figure6(a) Cyan Upper- and lower-period-2 limit cycles Figure6(b) Red Symmetric chaotic double-scroll attractor and chaotic spiral attractor Figure6(c) Cyan and yellow Period-2 and period-3 limit cycles Figure6(d) Cadet blue and blue Period-1 limit cycle and point attractor Figure6(e) Black Unbounded orbit Figure6(f)Figure 6
MATLAB simulated phase portraits of coexisting infinitely many attractors in theX1 −X2 plane for different initials-related system parameters (δ1,δ2,δ3,δ4). (a) Upper- and lower-asymmetric chaotic double-scroll attractors. (b) Upper- and lower-period-2 limit cycles. (c) Chaotic attractor with different topologies. (d) Period-2 and period-3 limit cycles. (e) Period-1 limit cycle and point attractor. (f) Unbounded orbit.
(a) (b) (c) (d) (e) (f)


## 4. PSIM Circuit Simulations

The 3-D dimensionality reduction model described by (8) is equivalently implemented in an analog circuit form, as manifested in Figure 7, where the gains of two multipliersM1 andM2 are set as 1. According to basic circuit theory, the circuit state equations are formulated in a general form as(15)RCv˙1=aβδ42-aαv2+av3+aβv233+aβδ4v22+δ1RCv˙2=av3+δ2RCv˙3=-0.5v1+v2a-bv3+δ3wherev1,v2, andv3 represent the state variables andRC is the integrating time constant. The initials-related system parametersδ1,δ2,δ3, andδ4 are implemented by additional DC voltage sources or directly linking to the ground. Note that theR6 in Figure 7 is only a negative feedback resistor to ensure the dissipativity of the physical circuit without self-excited oscillation, which is used to implement the self-feedback term in Equation (15). More details about the circuit design principle can refer to the operational amplifier stability in [46, 47].Figure 7
Physical circuit implementing the 3-D dimensionality reduction model (8).To better confirm the extreme multistability generated from the equivalent circuit in Figure7, 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 asR= 10 kΩ,R1 =R/(aβδ42− aα),R2 =R/a= 1.9231 kΩ,R3 =R/aβδ4,R4 = 3R/aβ= 57.6923 kΩ,R5 = 2aR= 104 kΩ,R6 =R/b= 11.1111 kΩ, andC = 100 nF. The initials [v1(0),v2(0),v3(0)] are assigned as (0 V, 0 V, 0 V) and the initials-related system parameters (δ1,δ2,δ3,δ4) are assigned as the same values by referring to those in Figure 6. PSIM intercepted phase plane plots in thev1 −v2 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 4-D memristor-based Colpitts system. What needs to be specified is that the initials of the 4-D memristor-based Colpitts system are in the explicit form in the physical circuit of the 3-D dimensionality reduction model described by (8), which can be used to easily achieve the controllable strategy for extreme multistability in the 4-D memristor-based Colpitts system [14, 23].Figure 8
PSIM simulated phase portraits of coexisting infinitely many attractors in thev1 −v2 plane for different initials-related system parameters (δ1,δ2,δ3,δ4). (a) Upper- and lower-asymmetric chaotic double-scroll attractors. (b) Upper- and lower-period-2 limit cycles. (c) Chaotic attractor with different topologies. (d) Period-2 and period-3 limit cycles. (e) Period-1 limit cycle and point attractor. (f) Unbounded orbit.
(a) (b) (c) (d) (e) (f)


## 5. Conclusion

In this paper, a dimensionality reduction reconstitution scheme for extreme multistability in memristor-based Colpitts system was introduced. By employing an ideal memristor to substitute the exponential nonlinear term of original 3-D Colpitts oscillator model, a novel 4-D memristor-based Colpitts system was obtained. The initials-dependent extreme multistability of the proposed system was exhibited via phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3-D dimensionality reduction model was constructed using SVM method. As a consequence, the implicit initials of the 4-D memristor-based Colpitts system were transformed into the explicitly initials-related system parameters of the 3-D dimensionality reduction model. Meanwhile, the dynamical mechanism was quantitatively explored by deriving the initials-related equilibria and discussing the equilibrium stabilities in the 3-D dimensionality reduction model. Furthermore, the initials-dependent extreme multistability was verified by two-parameter 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:(1) the proposed 4-D memristor-based Colpitts system has great practical importance, as it has much smaller size, lower power consumption, and more complex attractor structure; (2) the dimensionality reduction model greatly reduces the computational overhead, as the system goes from the 4-D to 3-D; (3) the traditional quantitative analyses can be used for exploring the extreme multistability phenomenon, because the implicit initials of the 4-D memristor-based Colpitts system are transformed into the explicitly initials-related system parameters of the 3-D dimensionality reduction model; (4) the physical control and mechanism explanation for extreme multistability are realized through dimensionality reduction reconstitution.



---

*Source: 4308549-2019-11-26.xml*