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Existence and Control of Hidden Oscillations in a Memristive Autonomous Duffing Oscillator

  • Vaibhav Varshney
  • S. SabarathinamEmail author
  • K. Thamilmaran
  • M. D. Shrimali
  • Awadhesh Prasad
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 133)

Abstract

Studying the memristor based chaotic circuit and their dynamical analysis has been an increasing interest in recent years because of its nonvolatile memory. It is very important in dynamic memory elements and neural synapses. In this chapter, the recent and emerging phenomenon such as hidden oscillation is studied by the new implemented memristor based autonomous Duffing oscillator. The stability of the proposed system is studied thoroughly using basin plots and eigenvalues. We have observed a different type of hidden attractors in a wide range of the system parameters. We have shown that hidden oscillations can exist not only in piecewise linear but also in smooth nonlinear circuits and systems. In addition, to control the hidden oscillation, the linear augmentation technique is used by stabilizing a steady state of augmented system.

Keywords

Memristor system Hidden oscillations Linear augmentation Memristor stability 

1 Introduction

The new emerging fourth passive element named, memristor whose resistance depends on its internal state variables of the system was proposed by Chua in 1971 (Chua 1971). The concept of memristor is explained by state-dependent Ohm’s law. The dependence is entirely on its past signals (applied voltage/current) across the memristor. The memristors have great paradigmatic usefulness for a circuit functionality because, it is not established with resistors, capacitors and inductors. The nonvolatile memory effect of the memristor is very important for potential applications in dynamics memory, neural synapses, spintronic devices, ultra-dense information storage, neuromorphic circuits, and programmable electronics (Xu et al. 2011; Mouttet 2008; Rajendran et al. 2010; Kim et al. 2011; Thomas 2013; Strukov 2008). This applications leads to a new method of high performance computing. Researchers have been motivated to investigate such memristor based oscillators from the dynamical system theory point of view. In recent years, there has been increasing interest to study the memristor based nonlinear circuits and their dynamics (Strukov 2008; Pershin et al. 2009; Pershin and Ventra 2008, 2009). The dynamics of the memristor based oscillator circuit systems is extraordinarily complex (Tour and He 2008). The basic idea of a memristor is that it is a two terminal resistive device, in which when current passes through it in one direction, the resistance increases, while when current flows in the opposite direction the resistance decreases. This gives the concept that a memristor maintains memory of its resistance, hence its name.

Numerous studies are available for understanding the conceptual background of memristor. The memristor relates the functional relationship of charge (q) and flux (\(\upphi \)) (Chua 1971). Memristor was considered to be the missing fourth circuit element, before it was postulated (the other known three being resistors, capacitors and inductors). Memristor was realized by Stan Williams group of HP Laboratory in 2008, is a passive two-terminal electronic device, described by nonlinear constitutive relation of charge and flux (Wang et al. 2009). The \(v-i\) characteristic of the memristor is inherently nonlinear (pinched hysteresis) and is unique in the sense that no combination of nonlinear resistive, capacitive and inductive components can duplicate their circuit properties. Memristors have generated considerable excitement among circuit theorists. Memristor based chaotic circuit can be constructed using memristor as a nonlinear element and can easily generate chaotic dynamics and some novel features can be observed. Recently, the fabrication of single memristor element, memristive system, memristor circuits, designs and analysis of memristor based application circuit systems, etc. have attracted attention in engineering and biological sciences. Since memristors are commercially unavailable, it would be very useful to have a specific circuit that emulates a memristor. For the simulation of memristor devices there are different nonlinearities available in literature to mimick the feature of the memristor, such as HP memristor model (Radwan et al. 2010; Prodromakis et al. 2011), non-smooth piecewise linearity (Ahamed and Lakshmanan 2013; Chen et al. 2014), smooth cubic nonlinearity (Cheng et al. 2011; Talukdar 2011; Muthuswamy and Kokate 2009), smooth piecewise-quadratic nonlinearity (Bao et al. 2010), and so on. For the first time Itoh and Chua introduced the memristor instead of Chua diode as a nonlinear element based canonical Chua’s oscillator (Itoh and Chua 2008). After that many studies on memristor based nonlinear circuit systems have been done.

Now a days numerous studies are available in the literature about the hidden attractors with no equilibrium point. Leonov et al. studied the hidden attractors in a Chua’s system and suggested special procedure for localization of hidden attractors (Leonov et al. 2011). After that they considered the example of a Lorenz-like system derived from Glukhovsky-Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors, and their characteristics. They also demonstrated the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets (Li et al. 2014; Leonov and Kuznetsov 2013; Leonov et al. 2015). Recently, they investigated the hidden oscillations in dynamical systems, based on the development of numerical methods, computers, and applied bifurcation theory (Leonov et al. 2011). The simple four dimensional equilibrium free autonomous ODE system showed all the attractors are hidden reported by Li et al. (Li and Sprott 2014). Zhouchao Wei et al. found a four-dimensional (4D) non-Sil’nikov autonomous system with three quadratic nonlinearities, which exhibits some behavior previously unobserved: hidden hyperchaotic attractors with only one stable equilibrium (Wei and Zhang 2014). Jafari et al. reviewed several type of new rare chaotic flows with hidden attractors in many dynamical systems. They also explained the flows with no equilibrium, with a line of equilibrium points, and with a stable equilibrium (Jafari et al. 2015). Dawid Dudkowski et al. reviewed the most representative examples of hidden attractors and discussed their theoretical properties and experimental observations. They described numerical methods which allowed the identification of the hidden attractors (Dudkowski et al. 2016). They also discussed the use of perpetual points for tracing the hidden and the rare attractors of dynamical systems (Dudkowski 2015).

Kuznetsov et al. gave some rigorous nonlinear analysis and special numerical methods which should be used for the investigation of nonlinear control systems (Kuznetsov and Leonov 2014). He also described the formation of several different coexisting sets of hidden attractors, including the simultaneous presence of a pair of coinciding quasiperiodic attractors and of two mutually symmetric chaotic attractors (Kuznetsov et al. 2015). Chaudhuri et al. proposed a riddled-like complicated basins of coexisting hidden attractors both in coupled and uncoupled systems and a new route to amplitude death is observed in time-delay coupled hidden attractors (Chaudhuri and Prasad 2014). Brezetskyi et al. presented different types of dynamics for both the single and coupled multistable Vander Pol-Duffing oscillators that have very small basins of attraction considered as hidden or rare (Brezetskyi et al. 2015; Zhao 2014). The control of multistability in the hidden attractor through the scheme of linear augmentation, that can drive the multistable system to a monostable state was proposed by Sharma et al. (2015a, b). Kapitaniak tries to focus on different questions of present day interest in theory and applications of systems with multiple attractors. The particular attention is paid to uncovering and characterizing hidden attractors (Kapitaniak and Leonov 2015). For the real time application point of view, Kiseleva et al. studied the hidden oscillations appearing in electromechanical systems with and without equilibria (Kiseleva et al. 2016).

The above literature is devoted to the brief understanding of hidden attractors. But only limited studies are available for finding the hidden attractor in a memristor based dynamical systems. Chen et al. reported the hidden attractor in memristive Chua’s circuit and also presented the coexisting hidden attractors (Chen et al. 2015). Saha et al. studied memristor based Lorenz system with no equilibrium (Saha et al. 2015). Mo Chen et al. proposed improved memristive Chua’s circuit, from which some hidden attractors are found (Chen et al. 2015). Pham et al. proposed memristor based networks to elucidate the hidden attractors (Pham et al. 2015, 2014; Zhang et al. 2015). Duffing oscillator is well known and is widely used by the researchers for its enigmatic and simplicity. Recently the dynamics of the memristor based Duffing oscillator was studied by Sabarathinam et al. (2017). With the best of our knowledge none of above works studied the dynamics of memristor based autonomous form of the Duffing oscillator. The existence of hidden attractor in the proposed system has been found for wide range of parameters and with different initial condition.

Many physical, biological, and chemical phenomena are well modeled by coupled nonlinear equations (Pikovsky et al. 2001; Fujisaka and Yamada 1983). In most cases these systems are capable of displaying several types of dynamical behaviors such as limit cycle, bistability, birhythmicity, and chaos. In many real-world situations, it is often the case that stable output is required in spite of the nonlinear effects present in the system. Thus, the scope of control or self-regulation in systems with complex dynamics is of considerable interest. Generally, the stabilization of unstable fixed points of an oscillatory system is considered to be an important problem for many practical applications. Over the last two decades, chaos control in dynamical systems and stabilization of unstable dynamical states of the systems have been a topic of intense research from both the theoretical and experimental point of view (Rosa et al. 1998; Ira 1997; Triandaf and Schwartz 2000; Ott et al. 1990; Sinha 1990). Control of chaotic dynamics or stabilization of fixed points is important in many experimental studies; for example, removal of power fluctuation is highly desirable in coupled laser systems (Kim et al. 2005; Kumar et al. 2008, 2009; Prasad 2003). In all previous existing methods (Rosa et al. 1998; Ira 1997; Triandaf and Schwartz 2000; Ott et al. 1990; Sinha 1990) the stabilization of the fixed points can be obtained by changing the accessible internal parameters of the system. However, in many real situations, where the internal parameters of the systems are not accessible, the stabilization of fixed points can be done by using the phenomenon of amplitude death (AD) (Prasad 2003) using interactions between the coupled oscillators.

Recently, linear augmentation has been suggested as an another practical alternative method leading to oscillator suppression, which is achieved by coupling nonlinear systems to a linear system which simply consists of an exponentially decaying function (Bar-Eli 1985) in an uncoupled state. Interestingly, the coupling structure of linear augmentation is quite reminiscent of indirect or environmental coupling procedures (Sharma et al. 2011; Resmi et al. 2010; Sharma et al. 2012) which are motivated by the observations of collective behaviors in several real world systems, namely, behavior of chemical relaxation oscillators globally coupled through the concentration of chemicals in a common solution (Resmi et al. 2012), dynamics of multicell systems where the cells interact through common complex proteins (Toth et al. 2006), and collective behavior of cold atoms in the presence of a coherent electromagnetic field and atomic recoil (Zhang and Zou 2012; Kruse et al. 2003) for instance. These instances therefore also serve as good examples of systems where linear augmentation can exist naturally. Lately, studies have also effectively used linear augmentation in controlling bistability (Javaloyes et al. 2008), the dynamics of drive response systems (Sharma et al. 2013) and in controlling hidden attractors (Sharma et al. 2014). Motivated by the above studies in this chapter, we also applied the linear augmentation techniques in our memristor based autonomous Duffing oscillator to control the hidden oscillations.

The chapter is structured as follows. Section 2 explains the mathematical modeling of the memristor based autonomous Duffing oscillator. Section 3, investigates the systems stability with eigenvalue analysis. Section 4, explores the hidden dynamics with numerical results. Section 5 explains the linear augmentation technique used to stabilize the system. Finally, the chapter concludes with the summary in Sect. 6.

2 Mathematical Model of Memristor Based Duffing Oscillator

We all know about the six mathematical relations connecting the pairs of four fundamental entities namely, charge (q), current (i), flux (\(\upphi \)) and voltage (v) (Mohanty 2013). Three relations can be understood by the axiomatic definitions of the three classical two terminal circuit elements, namely, resistor (relationship between v and i), inductor (relationship between \(\upphi \) and i) and capacitor (relationship between q and v). One more relationship between \(\upphi \) and q remained undefined. From logical and axiomatic point of view, as well as for the sake of completeness, the necessity for the existence of fourth basic two terminal circuit element was postulated. Chua in 1971, suggested the fourth passive element namely ‘memristor’ (Chua 1971). The memristor is a new passive two terminal element in which there is a functional relationship between the magnetic flux (\(\upphi \)) and electric charge (q). The memristor is governed by the relations \(i=W(\upphi ) v\), and \(v=M(q)i\), where, \(W(\upphi )\), M(q) are called memductance and memristance respectively. The memristor used in this work is a charge controlled memristor that is characterized by its incremental memristance function M(q) describing the charge-dependent rate of flux. We assume that the memristance M(q) is characterized by a monotonically increasing and smooth cubic nonlinearity which is defined by, \(\upphi (q) = \upomega ^2_0 q + {\upbeta }q^3 \), where M(q) is defined as, \(M(q)=\frac{d{\upphi }(q)}{dq}=\upomega ^2_0 + 3{\upbeta }q^2 \) is the memristance function. Figure 1, shows the nonlinearity curve M(q) as a function of charge q of the memristor emulator. With increasing q, the slope changes clockwise (slope value changes drastically) which indicates that the proposed system is very sensitive to q. The effect of the memristor emulator (co-efficient’s of memristor \(\upomega \), \(\upbeta \)) in a single memristive system studied by Sabarathinam and Thamilmaran (2017). They found that the dynamics of the system controlled by the memductance profile of the memristor emulator. Figure 2a, shows that the schematic of the memristor emulator and its characteristic curve in Fig. 2b which is obtained from PSpice simulation. In our case, the memristor based Duffing oscillator is taken with the cubic nonlinearity simply replaced by the charge-controlled memristor characteristic nonlinear system (Fig. 2). Here, we have considered the autonomous form of the Duffing oscillator, so the external force is considered as \(f sin({\upomega }t)=\) 0 from the original Duffing equation (Sabarathinam et al. 2017). Based on the memristance concept (Chua 1971), the state equation for the memristor based autonomous Duffing oscillator is,
$$\begin{aligned} \ddot{x}+\upalpha \dot{x} +M(q)x=0, \end{aligned}$$
(1)
Fig. 1

The variation of memristor nonlinearity M(q) versus x of Eq. (1) for \(q \in \) (0.1,1.0) shown different slopes. The parameters of the memristor emulator consider as \(\upomega ^2_0=\) 0.35 and \(\upbeta =\) 0.85

Fig. 2

PSpice analysis: a Schematic of the memristor nonlinearity with AC sweep analysis (1 V, 500 Hz) and b (\(v-i\)) characteristic curve of memristor. The input and output feed in to the oscilloscope

For our convenience, we considered, \(\upalpha ,\upomega ^2_0,\upbeta =a,b,c\). For the stability analysis as well as in the numerical study, the above equation is splitted into the following system of three first order coupled equations as:
$$\begin{aligned} \dot{x_1}= & {} x_2 \nonumber \\ \dot{x_2}= & {} x_3 \nonumber \\ \dot{x_3}= & {} -a x_3 - b x_2-3cx_1^2x_2. \end{aligned}$$
(2)
Thus by replacing the cubic nonlinearity in the classical Duffing oscillator, a new memristor based autonomous form of Duffing oscillator is designed and its dynamical behavior is investigated in detail.

3 Stability Analysis

Fig. 3

Basin of (i) \(\pm b\) versus (\(x_{10}\)) based on the real part of the eigenvalues (\(\uplambda _{1-3}\)) with fixed parameters \(a,c=0.0001,0.85\) of system (2)

To study the stability of the above mentioned system (Eq. 2) we have used eigenvalue analysis. The system (2), is symmetric with respect to the origin and hence is invariant under the transformation,
$$\begin{aligned} (x_1,x_2,x_3)\rightarrow (-x_1,-x_2,-x_3). \end{aligned}$$
Therefore, the equilibrium point calculated using system Eq. (2) is, (\(x_1^*,x_2^*,x_3^*\)) = (0, 0, 0). In order, to find the eigenvalues, the stability matrix or the Jacobean matrix J is written as,
$$\begin{aligned} J= \begin{bmatrix} 0~~~~~~~~&1 ~~~~~~~~&0 \\ 0 ~~~~~~~~&0 ~~~~~~~~&1 \\ 6c x_{10} x_{20} ~~~~~~~~&-b-3c x_{10}^2 ~~~~~~~~&-a\\ \end{bmatrix} \end{aligned}$$
(3)
In general the characteristic eigenvalue equation written as,
$$\begin{aligned} Det|J-{\uplambda }I|=0\Rightarrow -{\uplambda }^3-{\uplambda }^2 \upalpha +\uplambda (\upomega + 3{\upbeta }x_1^{*2}) + 6{\upbeta }x_1^*x_2^* \end{aligned}$$
(4)
We get eigenvalues for the above equation as, \(\uplambda _{1}=-0.001 + 0.5916i\), \(\uplambda _{2}= 0\), \(\uplambda _{3}=-0.001 - 0.5916i\) for \(\upalpha =\) 0.0001, \(\upomega =\) 0.35, and \(\upbeta =\) 0.85. From the eigenvalues the system have stable equilibrium state. The potential (M(q)) depends on the parameter b. For positive b, we get stable equilibrium points, \(\uplambda _{1-3}=-0.001\pm 0.5916i,0\), and negative b, we get saddle equilibrium pointS, \(\uplambda _{1-3}=0, 0.5916,-0.5917\). Figure 3, shows the changes in stability of the system with respect of \(\pm b\) and \(\pm x_1\) (detailed stability study were made Vaibhav et al. (2017)). For positive regions of b, we get stable fixed points (SFP) everywhere in the basin and for negative b the system have saddle equilibrium (UFP). In this chapter we have taken, \(a>0\) and \(b>0\) case for finding the hidden attractor. In that case, the system fall on stable fixed point. A vector field in the plane (for instance), can be visualized as a collection of different length of arrows which indicates the different magnitude and direction. Vector fields are often used to model, for example, the speed and direction throughout space, or the strength and direction of some force, as it changes from point to point. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow) of the system. Figure 4 shown the vector field of (\(y-z\)) plane of the proposed system in the range of \(\pm 20\). The different length of arrows which tells that the system stability changes by its basin. In the zero line magnitude of eigenvalues is high enough so, if we start the trajectory very near to zero the system drastically fall into the fixed point in a shorter time.
Fig. 4

Vector fields of system (2) in the (\(x_{20}-x_{30}\)) plane shows the arrows depict the field at discrete points, however, the field exists everywhere in the systems basin

4 Hidden Attractor

From the computational point of view, attractors are classified as self excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in the neighborhood of an equilibrium, reaches a state of oscillation, therefore one can easily identify it (Example: Lorenz, Rösseler, Chua oscillators, etc.) (Lakshmanan and Rajaseekar 2012). In contrast, for a hidden attractor, a basin of attraction does not intersect with any small neighborhoods of equilibria (Dudkowski et al. 2016). Hidden attractor can be chaotic as well as periodic e.g. the case of coexistence of the only stationary point which is stable and a stable limit cycle. It can easily predict the existence of self-excited attractor, while for hidden attractor the main problem is how to predict its existence in the phase space. Thus, for localization of hidden attractors it is important to develop special procedures, since there are no similar transient processes leading to such attractors. If the hidden attractor is present in the system dynamics and if coincidentally reached, then device (airplane, electronic circuit, etc.) starts to show the quasi-cyclic behavior that can, based on kind of device cause real disasters. In particular, to the extent that they have been known to exist, dynamical systems with no equilibrium have mostly been considered as nonphysical or mathematically incomplete. However, as experience shows, a system that presents hidden dynamical behaviour doesn’t need to also display an unstable equilibrium state. In this section, the hidden attractor have been revealed in our proposed system (2). The parameters are fixed as \(a=\) 0.0001, \(b=\) 0.35, and \(c=\) 0.85 with the initial conditions \((x_{10},x_{20},x_{30}) = (0, 0, 0)\). From the stability, we start the system anywhere in the basin the trajectories approach to fixed point because of the nature of its stability. For instance in a particular range of initial conditions the system exhibit stable oscillations which does not intersect the neighbourhood of the equilibrium of the system named hidden oscillations. Figure 5 shows three dimensional plot for the hidden oscillation for particular set of initial conditions, (\(x_{10},x_{20},x_{30}\)) = (0.8, 0.6, 0.0) and the parameter are fixed as \(a=\) 0.0001, \(b=\) 0.35, and \(c=\) 0.85 of system (2). The fixed point of the system are also replotted and its indicated as FP in Fig. 5. The different projection of the phase portraits shown in the Fig. 6. Figure 7 shows the time series plot of (a) (\(x_1(t)\)), (b) (\(x_2(t)\)), (c) (\(x_3(t)\)) variables for the corresponding phase plot in Fig. 5 which indicate the periodicity of the system. We got periodic hidden attractor in the wider range of initial conditions, We also intend to study the basin of the system. Figure 8 shows the various type of coexisting hidden attractor for different set of initial conditions indicates as \(H_1-H_6\). This attractors obtained from the initial conditions as (\(H_1,H_6\)):(\(x_{10},x_{20},x_{30}=\pm 4.0,0.6,0\)), (\(H_2,H_5\)):(\(x_{10},x_{20},x_{30}=\pm 2.0,0.6,0\)), and (\(H_3,H_4\)):(\(x_{10},x_{20},x_{30}=\pm 0.5,0.6,0\)). The FP is the equilibrium state of the system obtained from (\(x_{10},x_{20},x_{30}=0,0,0\)). From this Fig. 8, we conclude that our system have many interesting hidden attractors. The Lyapunov exponents were calculated for the periodic attractor where \(\uplambda _{1-3}=\) (0.00053,−0.0432, −0.00004) for very near to the equilibrium and \(\uplambda _{1-3}=\) (0.00052,−0.0232, −0.00022) far away from the equilibrium. The phase space volume was calculated as \(\nabla .F=-a\) which shows that our system is purely dissipative.
Fig. 5

Three dimensional phase plot for hidden oscillation at (\(x_{10},x_{20},x_{30}\))\( = \)(0.8, 0.6, 0.0) with the parameters fixed at \(a =\) 0.0001, \(b =\) 0.35, and \(c =\) 0.85 of system (2). FP indicates fixed point of the system

Fig. 6

Different projection of the typical phase portraits in the a \(x_1-x_2\), b \(x_1-x_3\), and c \(x_2-x_3\) planes of the same parameter used Fig. 5

Fig. 7

Time series of a (\(x_1(t)\)), b (\(x_2(t)\)), c (\(x_3(t)\)) variables corresponding phase plot of Fig. 5

Fig. 8

Three dimensional phase plot for different coexisting hidden oscillations for different initial conditions as (\(H_1,H_6\)):(\(x_{10},x_{20},x_{30}=\pm 4.0,0.6,0\)), (\(H_2,H_5\)):(\(x_{10},x_{20},x_{30}=\pm 2.0,0.6,0\)), and (\(H_3,H_4\)):(\(x_{10},x_{20},x_{30}=\pm 0.5,0.6,0\))

5 Controlling Hidden Attractor

In the above section the existence of the hidden oscillation is presented. In this section, the control of hidden attractor (the stabilization of fixed points) is examined. For that we take linear augmentation technique (Resmi et al. 2010). The scheme is generalized and can be applied to any other system as well. Here we coupled memristor based duffing oscillator Eq. (2) with the linear system to control the dynamics of hidden attractor in it. Initially we applied the augmentation in \(x_1\). The coupled equation is given by:
$$\begin{aligned} \dot{x_1}= & {} x_2 + {\upvarepsilon }u \nonumber \\ \dot{x_2}= & {} x_3 \nonumber \\ \dot{x_3}= & {} -a x_3 - b x_2-c x_1^2 x_2 \nonumber \\ \dot{u}= & {} -ku - \upvarepsilon (x_1-b) \end{aligned}$$
(5)
Fig. 9

Phase space trajectories of coupled memristor based duffing oscillator \(x_1\) versus \(x_2\) for \(\upvarepsilon =2.0\) a \(k=0.01\) and b \(k=1.0\)

Here \(\upvarepsilon \) is the coupling strength between the oscillatory and the linear systems, k is the decay parameter of the linear system u, and b is a control parameter of the augmented system. Here, we have taken only one linear system, in some cases we can take two linear systems for stabilizing both co-ordinates. We are not able to control the dynamics by augmenting the \(x_1\) variable. The phase space plot of \(x_1\) versus \(x_2\) for different \(\upvarepsilon \) for this case is shown in Fig. 9a, b. Next, we augmented the \(x_2\) variable and dynamics is controlled by it. The equation is given by:
$$\begin{aligned} \dot{x_1}= & {} x_2 \nonumber \\ \dot{x_2}= & {} x_3 + {\upvarepsilon }u \nonumber \\ \dot{x_3}= & {} -a x_3 - b x_2-c x_1^2 x_2 \nonumber \\ \dot{u}= & {} -ku - \upvarepsilon (x_2-b ). \end{aligned}$$
(6)
Fig. 10

Phase space diagram in the parameter space (\(k-\upvarepsilon \)), where A and B are the uncontrolled and controlled region respectively

Fig. 11

Phase space trajectories of coupled memristor based duffing oscillator \(x_1\) versus \(x_2\) for a \(\upvarepsilon =2.0\) and c \(\upvarepsilon =5.0\). Time series for \(x_1\) b \(\upvarepsilon =2.0\) and d \(\upvarepsilon =5.0\)

Now shown in Fig. 10 is the phase space diagram in parameter space (\(k-\upvarepsilon \)) in which A is the uncontrolled region and B is the region in which dynamics is controlled. The dashed line in it is demarcated when largest Lyapunov exponent changes its sign. The important thing to note here is that we are not getting the fixed point of the system, but the system is going to new fixed points created due to coupling. Now shown in Fig. 11 is the phase space (a) and (b) and time series (c) and (d) showing how the system is approaching the fixed point. Left panel is for \(\upvarepsilon =2\) and right panel is for \(\upvarepsilon =5\). It is clear from the figure that in both cases system is going to fixed point, but transient trajectory is different in both cases. So, in this section by augmentation technique we are able to control the dynamics of hidden attractor in memristor based Duffing oscillator and stabilize it to a fixed point which is not a equilibrium point of the uncoupled system.

6 Conclusion

We have observed a very interesting and emerging phenomenon of hidden oscillations in memristor based autonomous Duffing oscillator. The stability studied with the help of eigenvalues analysis. The hidden oscillations are obtained and presented in phase plot and its time series. We found our proposed system have wide range of coexisting hidden oscillations in its basin. We observed periodic form of hidden oscillation which is confirmed by Lyapunov exponents. In many applications, knowing the property of periodic oscillatory solutions is very interesting and valuable as many biological and cognitive activities require repetition. But in case of periodic hidden oscillations which leads to the new area of research by its repetition. Controlling of hidden attractors using linear augmentation technique is performed. The linear augmentation validate for all variables of our proposed system. We found that the linear augmentation will stabilize the system for only one variable mode (\(x_2\) of Eq. (2)). The memristor emulator does not allows to stabilize the charge component (\(x_1\) of Eq. (2)) of the system. In future, the detailed study of the hidden oscillations will be made by our proposed system with different physical situation which will be reported elsewhere (Vaibhav et al. 2017).

Notes

Acknowledgements

Vaibhav Varshney, Awadhesh Prasad and acknowledge DST, Government of India for the financial support. S. Sabarathinam acknowledges the DST-SERB for the financial assistance through National Post Doctoral Fellow (NPDF) scheme. K. Thamilmaran acknowledge DST PURSE scheme for financial assisitance and Awadhesh Prasad and M. D. Shrimali also acknowledge joint project DST-RFBR for funding.

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Vaibhav Varshney
    • 1
  • S. Sabarathinam
    • 1
    Email author
  • K. Thamilmaran
    • 3
  • M. D. Shrimali
    • 2
  • Awadhesh Prasad
    • 1
  1. 1.Department of Physics and AstrophysicsUniversity of DelhiDelhiIndia
  2. 2.Department of PhysicsCentral University of RajasthanAjmerIndia
  3. 3.School of PhysicsCentre for Nonlinear Dynamics, Bharathidasan UniversityTiruchirappalliIndia

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