Research Article  Open Access
Ning Wang, Bocheng Bao, Tao Jiang, Mo Chen, Quan Xu, "ParameterIndependent Dynamical Behaviors in MemristorBased WienBridge Oscillator", Mathematical Problems in Engineering, vol. 2017, Article ID 5897286, 13 pages, 2017. https://doi.org/10.1155/2017/5897286
ParameterIndependent Dynamical Behaviors in MemristorBased WienBridge Oscillator
Abstract
This paper presents a novel memristorbased Wienbridge oscillator and investigates its parameterindependent dynamical behaviors. The newly proposed memristive chaotic oscillator is constructed by linearly coupling a nonlinear active filter composed of memristor and capacitor to a Wienbridge oscillator. For a set of circuit parameters, phase portraits of a doublescroll chaotic attractor are obtained by numerical simulations and then validated by hardware experiments. With a dimensionless system model and the determined system parameters, the initial conditiondependent dynamical behaviors are explored through bifurcation diagrams, Lyapunov exponents, and phase portraits, upon which the coexisting infinitely many attractors and transient chaos related to initial conditions are perfectly offered. These results are well verified by PSIM circuit simulations.
1. Introduction
As the fourth basic circuit element, memristor, describing the relationship between the electric charge and magnetic flux, had been predicted by Chua in 1971 [1]. The remarkable characteristic is that memristor is a new twoterminal element with variable resistance depending on the amount of electric charge passing through it in a determinate direction. Thus, memristors are conveniently led into some nonlinear electronic circuits to replace nonlinear elements to design novel chaotic circuits. In recent years, novel nonlinear dynamical behaviors, like chaos and hyperchaos [2–7], transient chaos [8], selfexcited and hidden attractor [9, 10], coexisting multiple attractors [11, 12], and so on, have been reported in several memristorbased circuits.
Particularly, multistability, that is, coexisting multiple attractors, is an interesting phenomenon appearing in various physical fields, for example, power system [11], nonlinear system/circuit [12–22], and neural networks [23–25], which means that there are more than two different types of coexisting attractors. Sequentially, the phenomenon called extreme multistability occurs when the number of coexisting attractors tends to be infinite for the same set of circuit/system parameters [26–34]. Due to the important and potential values of engineering applications [35–37], study on revealing the phenomenon of coexisting attractors and investigating the internal mechanism of extreme multistability has attracted great attention and become a research focus [26–37]. Furthermore, transient chaos is a common phenomenon in many nonlinear dynamical systems [8, 30, 38–43]. It is necessary to study the transient dynamics to avoid some unwanted thing happening in a finite period of time before settling into final state, for example, the situations of voltage collapse [43]. At the same time, it will be useful to generate more flexible random signals for secret communications and information encryptions.
Chua’s circuit, containing a nonlinear element called Chua’s diode, is the first electronic circuit for generating doublescroll chaotic attractor [44]. By replacing the Chua’s diode with a memristor, some memristorbased chaotic circuits have been reported [8–10, 20, 23, 32–34]. The novel characteristic of memristorbased chaotic circuit is that its dynamical behavior is closely related to the initial state of memristor [19, 20, 29–34]. That is to say, with the variation of memristor initial condition, various kinds of attractors may coexist under the determined circuit/system parameters. Based on the above considerations, a novel memristive chaotic circuit is constructed in this paper. Naturally, complex parameterindependent dynamics, for example, coexisting infinitely many attractors’ behavior and transient transition behavior, can be found.
The layout of the paper is as follows. Section 2 proposes a memristorbased chaotic circuit with line equilibrium and gives the circuit state equations. Numerical simulations and hardware experiments are utilized to observe its strange attractors. In Section 3, a dimensionless model is carried out via introducing four new variables and rescaling the circuit parameters to perform stability analysis of the line equilibrium. In Section 4, coexisting infinitely many attractors’ behavior and transient transition behavior are exhibited through parameterindependent bifurcation diagrams, phase portraits, and timedomain waveform. PSIM circuit simulations are utilized to verify the phenomenon of extreme multistability. Section 5 gives some concluded remarks and outlooks.
2. MemristorBased WienBridge Oscillator
2.1. Circuit Descriptions and State Equations
An active voltagecontrolled memristor reported in [2] is utilized in this paper. The equivalent circuit of the memristor is shown in Figure 1, where the circuit in the dotted box is used to implement the absolute value function and its circuit parameters are selected as = 10 kΩ, = 13.5 kΩ and the gain of the analogue multiplier is assigned as = 0.1. Note that the operational amplifier saturation voltage is = ±13.5 V for ±15 V DC power supplies.
Let and be the voltage and the current at the input port, respectively, and stand for the voltage across the capacitor . The voltagecurrent relationship is thereby characterized aswhere is the gain of the analogue multiplier .
By replacing simplified Chua’s diode reported in [45] with the memristor depicted in Figure 1, a memristorbased Wienbridge oscillator is derived, as shown in Figure 2. The right part of the port 11′ is the active voltagecontrolled memristor, and the left part of the port 22′ is the classical Wienbridge oscillator.
The state equations for the proposed circuit can be written by four firstorder differential equations aswhere, , , , and are four state variables representing the voltages of capacitors , , , and , respectively, = /, and .
The typical circuit parameters in Figures 1 and 2 are listed in Table 1.

2.2. Numerical Simulations and Hardware Experiments
In this part, RungeKutta ODE23 algorithm is utilized to draw phase portraits. The circuit parameters listed in Table 1 are used for numerical simulations and hardware experiments. The initial conditions of four state variables are set to (10^{−9} V, 0 V, 0 V, 0 V). The four Lyapunov exponents are calculated by Wolf et al.’s method [46] as = 2.0759 × 10^{3}, = 11.7050, = −8.6321, and = −3.2043 × 10^{4}, indicating that chaos emerges in the proposed memristorbased Wienbridge oscillator. A doublescroll chaotic attractor can be numerically simulated and plotted in different phase planes, as shown in Figure 3.
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According to the equivalent circuit of the memristor in Figure 1 and the circuit schematic of the memristorbased Wienbridge oscillator in Figure 2, a common hardware circuit on breadboards is designed to validate the dynamical behaviors. Figure 4 is a photograph of the experimental hardware circuit and a doublescroll chaotic attractor captured by the digital oscilloscope. In this circuit, operational amplifier AD711KN and analogue multiplier AD633JN are selected and supplied with ±15 V DC power supplies. The experimental results are captured by fourchannel digital oscilloscope. The circuit parameters listed in Table 1 are utilized; experimentally measured phase portraits of the typical chaotic attractor are shown in Figure 5, which are well consistent with numerical simulations plotted in Figure 3.
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3. Dimensionless System Model
To preferably investigate parameterindependent dynamical behaviors in memristorbased Wienbridge oscillator, a dimensionless system model is necessary, by introducing four new variables and rescaling the circuit parameters in a dimensionless form asEquation (2) can be rewritten aswhere . Therefore, the normalized parameters for (4) are obtained by (3) as
By solving the equations , we can obtain the equilibrium of system (4). Clearly, the system has a line equilibrium, which is expressed aswhere is an uncertain constant. For the line equilibrium , the Jacobian matrix is given asCorrespondingly, the characteristic equation used to solve four eigenvalues is obtained aswhere
There are one zero root and three nonzero roots in characteristic equation (8). For these nonzero roots, the stability conditions via RouthHurwitz criterion are given asIf the three conditions of (10) are satisfied, the line equilibrium is stable, whereas if any one of the three conditions is not satisfied, is unstable, which may induce the emergence of periodic and chaotic behaviors in system (4). When varies in the region system (4) is unstable, and the corresponding four eigenvalues at equilibrium have a positive real root, two complex conjugate roots with negative real parts, and a zero root.
4. ParameterIndependent Bifurcation Behaviors
In this section, bifurcation behaviors associated with the initial conditions are investigated by numerical simulations. The normalized parameters listed in (5) are fixed, and the state variable is chosen as bifurcation variable. RungeKutta ODE23 algorithm is used for numerical calculations and Wolf et al.’s method [46] is used to calculate Lyapunov exponents.
4.1. Memristor Initial ConditionDependent Dynamics
In order to study the dynamical behavior of the memristorbased Wienbridge oscillator under different memristor initial condition, three initial conditions = 0, = 0, and = 10^{−9} are fixed, and the initial condition is taken to vary in the region [−8, 8]. With increasing, the bifurcation diagram and the four Lyapunov exponents are plotted in Figures 6(a) and 6(b), respectively.
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(b)
From Figure 6, complex bifurcation behaviors including tangent bifurcation, period doubling bifurcation, reverse period doubling bifurcation, and chaotic crisis can be observed, and the coexistence of point attractors with different locations, periodic limit cycles with different periodicities, and chaotic attractors with different topologies can be found. When is increased continuously, system (4) starts from the chaotic state and breaks into period 4 state through tangent bifurcation at = −6.54 and then goes into chaotic state via period doubling bifurcation. Afterwards, at = −4.66, system (4) breaks into period 5 state via the second tangent bifurcation and then goes through period doubling bifurcation and reverse period doubling bifurcation with the emergence of five bubbles in the initial region of [31]. Furthermore, a chaotic crisis occurs at = −2.58, which results in the fact that system (4) jumps from chaotic state into periodic state. It is interesting that, within the initial region of , system (4) goes through period doubling bifurcation to reverse period doubling bifurcation and exhibits symmetrical dynamics about = 0 which leads to the coexistence of symmetrical attractors. Finally, system (4) tends to the stable state with point attractor at = 5.07.
To confirm the coexisting attractors, phase portraits under different memristor initial conditions are plotted, as shown in Figure 7. Figures 7(a) and 7(b) show the route to chaos, for example, period doubling bifurcation route and reverse period doubling bifurcation route. Both the two routes are symmetrical in the regions and . It can be observed from Figure 7(a) that when is set to 2.4, 1.4, and 0.5 and −2.4, −1.4, and −0.5, respectively, the corresponding attractors in the plane are originsymmetric. Figure 7(b) is a doublescroll chaotic attractor under the memristor initial condition = 0.001. Furthermore, Figures 7(c) and 7(d) are plotted to show other coexisting attractors in the regions of and .
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(b)
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It is obvious that the many infinitely coexisting attractors’ behavior can be found via memristor initial conditionbased bifurcation dynamics in Figure 6 and confirmed by phase portraits in Figure 7. Considering that the attractors plotted in Figure 7 are shaped under the partly given system initial conditions, practically, there are infinitely many coexisting attractors when the initial condition changes continuously. The coexisting phenomenon of infinitely many attractors is also known as extreme multistability [26–31].
Comparing Figure 6(a) with Figure 6(b), we find that when the initial condition = −7.35, bifurcation dynamics in Figure 6(a) shows the chaotic behavior, but the largest Lyapunov exponent in Figure 6(b) drops to zero and indicates the periodic behavior. Different dynamical behaviors imply that there is a transition from transient chaotic to steady periodic behaviors with the evolution of the time.
To reveal this interesting transition behavior, the timedomain waveform and phase portraits are plotted in Figure 8. Figure 8(a) shows the timedomain waveform of the variable in the time interval [0 ks, 2 ks], from which it can be obviously observed that the system trajectory starts from a chaotic orbit and then enters into a periodic orbit at 1.33 ks. Figures 8(b) and 8(c) show the phase portraits in two different time intervals of [0.6 ks, 1 ks] and [1.6 ks, 2 ks], respectively.
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4.2. Other Initial ConditionsRelevant Dynamical Behaviors
Three initial values are chosen as = 0, = 10^{−9}, and = 0. When the initial condition is gradually adjusted from −2 to 2, the bifurcation diagram and the corresponding four Lyapunov exponent spectra are shown in Figures 9(a) and 9(b), respectively.
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(b)
While increasing the initial condition from −2 to 2, such dynamical behaviors as period, chaos, period doubling bifurcation, reverse period doubling bifurcation, and tangential bifurcation can be observed from Figure 9(a). In the region , period doubling bifurcation and reverse period doubling bifurcation can be observed clearly, leading to the occurrence of symmetrical coexisting attractors, as plotted in Figure 10, where Figures 10(a)–10(d) show the dynamics from period to chaos through period doubling bifurcation, and Figures 10(e)–10(h) display limit cycles with different periodicities and chaotic spiral attractors. Note that phase portraits with the red and blue orbits represent their initial conditions as and , respectively. In the regions and , several symmetrical periodic windows with different periodicities can be also seen from Figure 9. Consequently, Figure 9 shows a global symmetrical bifurcation behavior with respect to = 0, and the phase portraits under the initial conditions of and are symmetric about the origin, which confirms coexisting infinitely many attractors’ behavior with the continuous variation of the initial condition .
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When three initial conditions are chosen as = 0, = 0, and = 10^{−9}, and is gradually adjusted from −1.6 to 1.6, the bifurcation diagram is shown in Figure 11(a). Similarly, when three initial conditions are set as = 0, = 0, and = 10^{−9}, and is varied from −1.6 to 1.6, the bifurcation diagram is depicted in Figure 11(b). In Figures 11(a) and 11(b), there exists symmetrical bifurcation behavior with similar dynamics including chaos, period, forward and reverse period doubling bifurcation, tangential bifurcation, chaotic crisis, and periodic windows.
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4.3. Confirmation by PSIM Circuit Simulations
Considering that it is difficult to achieve the specified initial voltages of four capacitors in hardware circuit experiment, PSIM 9.0.3 is utilized to validate the coexisting attractors for some given initial voltages [20, 29, 30].
The circuit simulation model of the memristorbased Wienbridge oscillator is established, as shown in Figure 12. Take the results of numerical simulations in Figure 10 as examples. Corresponding to several specified initial conditions of , PSIM circuit simulations are performed, as plotted in Figure 13. Comparing Figure 13 with Figure 10, it can be seen that PSIM circuit simulations are well consistent with numerical simulations, which conform that the coexisting phenomenon of infinitely many attractors indeed exists in the memristorbased Wienbridge oscillator.
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5. Conclusions
When a nonlinear active filter consisting of an active voltagecontrolled memristor and a capacitor is connected to a classical Wienbridge oscillator in parallel, a memristorbased Wienbridge chaotic oscillator is constructed. The proposed memristive oscillator has a line equilibrium and thereby its dynamical behaviors depend on the circuit parameters and the initial conditions. Focusing on the parameterindependent dynamical behaviors in the memristive oscillator, a set of circuit parameters are determined and only the initial conditiondependent dynamical behaviors are explored by using some conventional methods of bifurcation diagrams, Lyapunov exponents, and phase portraits, from which the extreme multistability phenomenon of coexisting infinitely many attractors is effectively exhibited. Furthermore, hardware experiments and PSIM circuit simulations are performed to well verify the numerical simulations. Such a work may be attributed to some technological applications, for an example, multirandom signal generator, in future days.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundations of China under Grant nos. 51777016, 61601062, and 51607013 and the Natural Science Foundation of Jiangsu Province, China, under Grant no. BK20160282.
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Copyright © 2017 Ning Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.