Abstract
We have studied the chaotic and strange nonchaotic phenomena of a simple quasiperiodically forced Wien bridge oscillator circuit with diode as the only nonlinearity in this electronic oscillator system responsible for various nonlinear behaviors. Both the experimental results and the numerical simulation results for their confirmation are provided to show the bifurcation process. Various measures used for the numerical confirmation of SNA are power spectrum, maximal Lyapunov exponent, path of translational variables, mean square displacement, projection of poincaré section, log-log plot, and autocorrelation function. Based upon the numerical results, the birth of SNAs has been identified in the band merging route, intermittency route, and blowout bifurcation route. In addition, the birth of SNAs has been analyzed with peculiar mechanism, namely, “0-1 Test” employing the one state dynamical variable.
1. Introduction
Following the theoretical study of Grebogi et al. [1] on SNAs, in the recent years enormous attention has been paid to study the dynamics of SNAs which makes a fractal walk in its complex plane, possessing both discrete and continuous components in its spectral plane [2–4]. As evidence, SNAs are geometric structures possessing characteristic properties existing in the transitional state between periodicity and aperiodic chaotic dynamics. They are often expected to be generic in nonlinear systems excited by at least two periodic forces possessing incommensurate (or) irrational frequencies. Hence, the nature of quasiperiodic force with irrational frequencies plays an essential part in the generation of SNAs. In particular, the SNAs have been realized in a varietal quasiperiodically perturbed physical situations such as in pendulums [5], quantum particles [6], biologically operated oscillators [7], Duffing-type oscillators [8], velocity based oscillators [9], and electronic system [3, 10] and in specific maps [11]. Also the practical observation of these quaint attractors is made possible in the following experiments, quasiperiodically forced system of buckled magneto elastic ribbon [12], in analog simulating model of multistable potential [13] and neon glow discharge experiment [14]. Furthermore, SNAs take part in the Anderson localization in the Schrodinger equation influenced with quasiperiodic potential [15]. Consequently, several routes and their mechanisms leading to the birth of SNAs have been described [9, 10]. The common routes are (i) torus doubling sequence to SNA to chaos [16, 17], (ii) gradual fractalization of torus to SNA to chaos [18], (iii) blowout bifurcation route to SNA [19], (iv) type I intermittency route to SNA [20], (v) type III intermittency route to SNA [8], (vi) birth of SNAs due to reemerging torus doubling bifurcation [9]. The synchronization between SNAs paves an essential application in the field of secure communication, cryptography, and so forth [21–24].
In the present work, we consider standard Wien bridge oscillator which is a familiar electronic linear oscillator. It is driven by a quasiperiodic force and brought into the hierarchy of nonlinear state employing an ordinary component, diode as a nonlinear device. The circuit design and its implementation are described in Section 2. The experimental results obtained from the electronic system are verified with the numerical simulation of the state equations of the modified Wien bridge oscillator system (Section 3). The various dynamical states emerging in the system behavior are discussed in Section 3.1. The “0-1 test” is found useful to analyze the creation of SNA in the system (Section 3.2). The other numerical studies characterizing the property of SNA have been carried out (Section 3.3). The results are concluded in Section 4.
2. Circuit Realization
2.1. System Design
In this paper, the modified nonautonomous Wien bridge oscillator [25] is considered, with additional external sinusoidal force. The quasiperiodically forced system contains a standard Wien bridge oscillator with a diode, used as nonlinear element.
The characteristic function of a diode is a nonlinear piecewise segment (Figure 1). The voltage across the diode is related nonlinearly to its current through the relation, . Hence, it is a nonlinear element that performs switching operation which gets shorted on forward bias with minimal resistance and opened on reverse bias with infinite resistance. Based on these basic properties, diodes can be implicated in oscillatory systems for the investigation of nonlinear phenomena.
The purpose of this design (Figure 2) is to explore the possible complex dynamics obtained from a nonlinear form of simple linear oscillator and to investigate the ensuing chaotic and strange nonchaotic signals, thereby verifying their states with the appropriate characterizing methods.
2.2. System Equations
The electronic system implemented consists of an operational amplifier (OP AMP μA741), four resistors, two capacitors, and a diode (IN4148) and external sinusoidal periodic signals from two function generator (AGILENT 33220A). The chosen parameter fixed values are , , , . The state equations (1) of the designed system (Figure 2) so obtained by applying the Kirchoff’s laws at the appropriate nodes of the circuit are required for dynamical analyses through numerical simulation. The governing normalized equations of the dynamical system are in the form of first order differential equation: where with and .
For our numerical analyses, the following rescaling factors such as , are used where and are the voltages across the capacitors and , respectively. is the nonlinear function of the diode. and are the amplitudes of the external periodic forces with the frequencies and , respectively. The gain of the Wien bridge oscillator is controlled by and such that , which should be maintained above 3 so as to maintain the oscillations. Practically this can be written as where is a small parameter to initiate the oscillations. Hence, we have chosen for our system analysis.
3. Results and Discussion of Experimental and Numerical Investigation
3.1. Analyses Using Phase Diagram, Power Spectrum, and Maximal Lyapunov Exponent
The numerical analyses for the system associated equations are performed with the help of ode45 solver—MATLAB. The experimental and numerical phase trajectories (Figure 3) are accrued for the voltage across capacitor ( axis) and sinusoidal force ( axis) to scrutinize the emanating dynamics.
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During the study of dynamics of the system, it is observed that when the external sinusoidal forcing signal boosts the system, the aperiodic chaotic dynamics is prognosticated for chosen fixed parameters in a good frequency range, suitable for high frequency applications. When the dynamics of the system is set at the nonlinear regime by fixing appropriate experimental values discussed in [25] with the frequency , nonlinear phenomena like quasiperiodic torus are observed at , then chaotic attractor at , followed by periodic windows and again chaotic attractor for the slight changes in the variable parameter.
Consequently, the study of the system is extended for further investigation with the additional external periodic force . While the dynamics of the system oscillate in the chaotic regime at and , the amplitude of the second force is slowly increased with the frequency fixed at golden mean ratio . During the variation of the amplitude , the system continues to remain in the chaotic state (Figure 3(a)) for a range, to 1.54, after which it slowly leaves the chaotic state and dynamics starts to exhibit SNAs from to (Figure 3(b)) whereas further variation in the amplitude () makes the system yield the dynamics of quasiperiodic torus attractors. The corresponding power spectra are shown in Figure 4.
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Qualifying this, the maximal Lyapunov exponent (LE) is calculated to characterize the dynamics as the function of as [26] where is the measure of the distance between two trajectories at time and is that of the two initial trajectories.
The plot of the largest LE is shown in Figure 5. The calculation yields positive LE for aperiodic dynamics for the value of below 1.55 due to the exponential divergence of chaotic trajectories. For the range of from 1.55, the system exhibits negative LE indicating quasiperiodic state such that SNA rises as an intermediate dynamics at followed by the torus at .
3.2. Analysis of 0-1 Test and Projection of Poincaré Section in () Plane
Characterizing the regular and irregular dynamics using the binary test, namely, “0-1 test” introduced by Gottwald and Melbourne [27] is a unique and useful method as it depicts the regular and complex dynamics and analyses the presence of SNAs during the transition between regular and chaotic dynamics [11, 27]. The 0-1 test uses only a time series data irrespective of the knowledge of dimension of underlying system. Consequently, this test results in and around unit value for chaotic dynamics, approaches zero for regular dynamics and lies between 0 and 1 for the regime of SNAs. Gottwald and Melbourne’s 0-1 test implemented with modified mean square displacement [27] has proved to be one of the useful tools for our system analyses to distinguish SNA among the chaotic and quasiperiodic regime.
The analysis of 0-1 test which is actually the method of computing the correlation coefficient is calculated as follows.
Let be the total length of time series data and let be an observable with . The definition of correlation coefficient is given as where denote the vectors such as and , the series of modified mean displacement (i.e.) with .
Mean square displacement is determined using the translation variables and (function of ) by applying directly its definition [27] with . As in [27], the modification to the mean square displacement is necessary as it regularizes the linear behavior of its old version. Evidently, its modification permits a good prediction of the asymptotic growth rate .
For our system studied, this test is carried out by considering the series of time dependent variables, . In order to enhance the performance of this test, it is recommended [11] to take the total data of length (5 × 105) divided into 10 segments, each of segments having equal number of data. Then, the correlation coefficient is computed for each segment. Finally it is found by taking average over them and this method is repeated for each of the varying amplitude for a selected range.
The plot of translation variables which makes a fractal walk and modified mean square displacement which oscillates initially, later which increases linearly as the time passes on are shown in Figure 6 for the case of SNA and “0-1 test” in Figure 7.
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It follows that in Figure 7 chaotic regime is indicated by the value oscillating near 1. For a range of amplitude lying between , the transition to SNA takes birth in the route of merging of bands followed by the intermittency route and then blow out bifurcation route for which descends down that lies between 0 and 1 indicated by the arrow. The region of torus emerges for indicated by second arrow. The range of SNA regime can be well defined from that of the chaotic and quasiperiodic dynamics.
The projection of Poincaré section in the ( plane helps to understand the birth route of SNAs which is clearly shown in (Figure 8) for the band merging route, intermittency route, and blowout bifurcation route.
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3.3. Spectral Analysis of Log-Log Plot and Autocorrelation Plot
A qualitative characterizing measure using the relation of partial Fourier transformation reveals singular continuous spectrum for SNA [28], obeying the power law relation, with and being scaling component or the slope of the fitted linear of the spectrum. Indeed the slope of the linear curve in the plot of versus (Figure 9) lies at for SNAs that correspond to the nonchaotic motion with strange behavior, whereas in the case of chaos, the spectrum impart continuous broad band observance with more than 2 indicating irregular dynamics. The autocorrelation plot (Figure 10) of SNA diminishes after a finite interval of time proving the property of SNA.
4. Conclusion
In this paper, a modified nonautonomous model of standard Wien bridge oscillator is investigated to reveal the simplicity of the low dimensional system acting under the force of quasiperiodic signals using a simple diode as a nonlinear device. The system exhibits complex behavior belonging to chaotic attractors and SNAs. In the study, birth of SNAs takes place in three routes—merging of bands followed by intermittency route and blowout bifurcation route. Numerical approach, namely, “0-1 test” which has been newly introduced for characterizing SNA dynamics that occur during the transition between chaotic and regular dynamics is reported in the present work. Additional numerical analyses such as maximal Lyapunov exponent, power spectrum, Poincaré projection, spectral analysis of log-log plot, autocorrelation are carried out supporting the experimental results. We also add that SNAs can meet the applications replacing chaotic attractors. Chaotic attractors are observed for a wide range of variable parameter whereas SNAs reveal their presence only for a short range of system variant [21]. Such signals can be used for the secure transmission of digital signals through the synchronization between two quasiperiodically forced nonlinear systems, provided the frequencies ratio remain irrational and hence the phases of the quasiperiodic driving can be matched. The synchronization property of SNA along with their short range existence may find essential application [21–24] in secure communication processes.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.