Active and Passive Electronic Components
Volume 2016 (2016), Article ID 3426713, 7 pages
http://dx.doi.org/10.1155/2016/3426713

Research Article

Computer and Hardware Modeling of Periodically Forced -Van der Pol Oscillator

A. O. Adelakun,1,2 A. N. Njah,³ O. I. Olusola,³ and S. T. Wara⁴

¹Department of Physics, Faculty of Physical Science, Federal University of Agriculture, Abeokuta, Ogun State, Nigeria
²Department of Physics, School of Science, Federal University of Technology, Akure, Ondo State, Nigeria
³Department of Physics, Faculty of Natural Science, University of Lagos, Akoka, Lagos State, Nigeria
⁴Department of Electrical and Information Engineering, Covenant University, Ota, Ogun State, Nigeria

Received 6 June 2016; Revised 17 October 2016; Accepted 24 November 2016

Academic Editor: Jiun-Wei Horng

Copyright © 2016 A. O. Adelakun 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.

Abstract.

Numerical simulation results for the dynamics of -systems abound in the literature but their experimental results are yet to be known. This paper presents the chaotic dynamics of -Van der Pol oscillator via electronic design, simulation, and hardware implementation. The results obtained are found to be in good agreement with numerical simulation results. The condition for stability of the fixed points is also computed and the method of multiple time scale is used to investigate the dynamical behaviour of the system. Therefore, the -circuits which have rich dynamics and may have important applications in secure communications, random number generations, cryptography, and so forth have been practically implemented.

1. Introduction

In recent years the study of chaotic phenomena in self-excited oscillators with parametric and external-excitation has attracted much attention [1–6]. Among the 2-dimensional periodically forced oscillators, the most extensively studied examples are the Van der Pol oscillators [7, 8], Duffing oscillators [9], and Rayleigh oscillators [10]. Many works on self-excited, parametrically excited, and externally excited oscillators are also investigated in the literature. Recently, the rich dynamics of Van der Pol oscillator have been explored by theoretical analysis and numerical simulations [3, 11].

The -Van der Pol oscillator is a very significant classical model circuit that has been studied and even modified in some studies as reported by King et al. [12, 13], Fotsin et al. [14], and Fodjouong et al. [15]. However, with the advent of the study of chaotic motion by means of strange attractors, Poincare’ map, fractal dimension, it is necessary to seek for a better understanding of nonlinear systems with higher order nonlinear terms such as the -Van der Pol oscillator.

A -Van der Pol oscillator or extended Van der Pol oscillator is the nonlinear oscillator with fifth-order nonlinear term. Its numerical simulation has been studied extensively by Siewe Siewe et al. [16, 17], Tchoukuegno et al. [18–20], and Yu and Li [21] and its synchronization behaviours for possible applications in information processing and secure communication by Njah et al. [22–24], and its application in the modeling of chemical reactions and brain activity by Chen and Dong [25] and Ott et al. [26]. In the past three decades, implementation of chaotic systems in power electronics has gone through intense development in many aspects of technology [27–29], including power devices, control methods, circuit design, computer-aided analysis, passive components, and packaging techniques. Like many areas of engineering, power electronics is mainly motivated by practical applications, and it often turns out that a particular circuit topology or system implementation has found widespread applications long before it is thoroughly analyzed and most of its subtleties are uncovered [30, 31].

The aim of this paper is to design, simulate, and implement periodically forced -Van der Pol oscillator by transforming the state equations into electronic circuit via analog components modeling. We also generate phase portraits for the triple-well and double-well attractors of the system. To the best of our knowledge, computer simulation with electronic software and hardware implementation of -Van der Pol oscillators have not been studied. The rest of the paper is organized as follows: Brief description of the conditions for stability of fixed points for unperturbed system is discussed in the next section. Section 3 explains the numerical simulation of the oscillator for at least six physically relevant situations with corresponding attractors while Section 4 explains the analog simulations with electronic software (multisim) and hardware implementation of the system. Finally, Section 5 concludes the paper.

2. Analysis of the Unperturbed System

The general form of second-order differential equation of Van der Pol systems is given as

where is the damping parameter and and are amplitude and angular frequency, respectively, and its potential can be expressed in Taylor series as

where , , and are constant parameters. The condition for stability of the fixed points of oscillator (1) for an unperturbed system can be assumed when . Thus, the system can therefore be expressed as a set of first-order differential equations of the form

which corresponds to an integrable Hamiltonian system with the potential function in (2) and whose associated Hamiltonian function is

From (3) and (4), we can compute for the fixed points and analyze their stabilities after some algebraic manipulations using the following conditions.

(i)For , the unperturbed system has only one fixed point , the centre, for which the system is single-well if and and single-hump potential if and as shown in Figures 1(a) and 1(b), respectively.(ii)For with , , and , the unperturbed systems have three fixed (equilibrium) points, that is, one centre and two saddles connected by two heteroclinic orbits (), (5).(iii)For with , , and , the unperturbed systems have five fixed points: one centre , two saddles connected by two heteroclinic orbits (), and two saddle points connected to each other by one homoclinic orbit (), (5) and (6).

In previous works, Siewe Siewe et al. [16, 17] have shown that, for , system (4) is a Hamiltonian system with a pair of heteroclinic orbits defined as

and a pair of symmetric homoclinic trajectories connecting each unstable point to itself given by

where

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: The smooth potential function, , of the -oscillators for (a) single-well: and , (b) single-hump: and , (c) double-well: , , and , (d) double-hump: , , and , (e) triple-well: , , and , and (f) triple-hump: , , and .

3. Numerical Simulation

The general form of the periodically forced Van der Pol oscillator model is given by the second-order nonautonomous differential equation (1), where is the state variable, dot(s) over denotes the derivative with respect to time , is the amplitude of periodic forcing, is the angular frequency, is the damping parameter, and is the potential. The potential for six physically relevant situations (single, double, and triple-well and hump) is shown in Figure 1. They correspond to different choices of the set of values for the parameter set , , and . For instance, the potential is double-well if (α< 0, β > 0, and λ > 0), double-hump if (α > 0, β < 0, and λ < 0), triple-well if (α > 0, β < 0, and λ > 0), and triple-hump if (α < 0, β > 0, and λ < 0).

Van der Pol oscillator models describe various physical, electrical, mechanical, and engineering devices, for different potentials . The potentials are approximated by finite Taylor series for the -chaotic oscillators as shown in (3), where , , and are constant parameters of the potential. The introduction of a new parameter in (2) yields -Van der Pol equation (9) that can be expressed as

Numerical simulation of (8) using fourth-order Runge-Kutta method gives the phase portraits presented in Figure 2 for double and triple scroll attractors. From the phase portraits it is obvious that the systems are chaotic for the parameter values used.

(a)

(b)

Figure 2: Phase portrait of the chaotic attractor for the -Van der Pol oscillator with parameters values (a) , , , , , and for the double-well potentials and (b) , , , , , and , for the triple-well.

4. Design of Analog Simulator

Circuit realization of -Van der Pol system is shown in Figure 3 (schematic circuit) and Figure 4 (analog circuit) in which the nonlinear resistor (NR) can be modeled by using nonlinear characteristic given by the following relation:

where , , and are parameters of the nonlinear resistor , is the flux over the resistor, is the charge, is the capacitance, is the inductance, is the resistance, , and is the current through the nonlinear resistor. Applying Kirchoff’s law to the circuit shown in Figure 3, we have

Then (10) becomes

Taking , , , , , and , we can rewrite electronic equation (11) in form of flux as

The electronic analog simulator for -Van der Pol oscillator can be designed using the electronic multiplier, , which perform the multiplication operations. The analog circuit operates over a dynamic range of The integrators are operational amplifiers, and (UA741CD) with feedback capacitors, and summations of the state variables are obtained using operational amplifiers with the help of feedback resistors and multiple input resistors. Also, the simulator output showing the dynamics when varying the input components can be viewed directly on a software oscilloscope using an appropriate time scaling. The designed circuitry realization for the oscillator consists of two channels A and B which realize the two state variables and , respectively. The attractors were generated from -Van der Pol circuit of Figure 4 by varying

The computer simulation of -Van der Pol circuit on electronic software (multisim) (Figure 5(a)) and its hardware implementation on electronic breadboard (Figure 5(b)) both reproduced the numerical results in Figure 2 on the oscilloscope. Indeed, the practical implementation of the aforementioned system has application in technology such as in secure communication.

Figure 3: Schematic circuit of Van der Pol model.

Figure 4: Analog circuit of -Van der Pol oscillator.

(a)

(b)

Figure 5: Phase portrait of -Van der Pol attractors for (a) computer simulation and (b) experimental implementation for (i) double-well at and and (ii) triple-well at and .

5. Conclusion

In this paper, we have used the analog electronic software (multisim) for the electronic circuit design and simulation of -Van der Pol oscillator and have practically implemented the designed circuit on electronic breadboard using analog components. Our experimental results reproduced those obtained via numerical simulation and multisim simulation. Therefore the rich dynamics of extended Van der Pol oscillator which have wide application in science and technology have been practically implemented, for possible technological application such as in secure communication.

Competing Interests

The authors declare that they have no competing interests.

References

J. Warminski and J. M. Balthazar, “Vibrations of a parametrically and self-excited system with ideal and non-ideal energy sources,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 25, no. 4, 2003.

J. Warminski, “Complexity and chaos of parametrically and self-excited system with non-ideal energy source,” Journal of Vibration and Control, vol. 11, pp. 371–379, 2005.

J. Warminski, “Regular and chaotic vibrations of a parametrically and self-excited system under internal resonance condition,” Meccanica, vol. 40, no. 2, pp. 181–202, 2005.

M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific Publishing, Singapore, 1996.

A. N. Njah and U. E. Vincent, “Chaos synchronization between single and double wells Duffing-Van der Pol oscillators using active control,” Chaos, Solitons and Fractals, vol. 37, no. 5, pp. 1356–1361, 2008.

J. Yu, W. Zhang, and X. Gao, “Dynamical behavior in the perturbed compound KdV-Burgers equation,” Chaos, Solitons & Fractals, vol. 33, no. 4, pp. 1307–1313, 2007.

F. Xie and X. Lin, “Asymptotic solution of the van der Pol oscillator with small fractional damping,” Physica Scripta, vol. 2009, no. 136, Article ID 014033, 2009.

R. S. Barbosa, J. A. T. Machado, B. M. Vinagre, and A. J. Calderón, “Analysis of the Van der Pol oscillator containing derivatives of fractional order,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1291–1301, 2007.

J. Cao, C. Ma, H. Xian, and Z. Jiang, “Nonlinear dynamics of Duffing system with fractional order damping,” in Proceedings of the ASME IDETC/CIE Conference, DETC2009-86401, San Diego, Calif, USA, August 2009.

M. S. Siewe, H. Cao, and M. A. F. Sanjuán, “On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 772–782, 2009.

W. Dongping, Z. Shangbo, and Y. Juebang, “The existence of closed trajectory in the van der Pol oscillator,” in Proceedings of the International Conference on Communications, Circuits and Systems and West Sino Exposition Proceedings, vol. 2, pp. 1629–1632, Institute of Electrical and Electronics Engineers, Chengdu, China, July 2002.

G. P. King and S. T. Gaito, “Bistable chaos. I. unfolding the cusp,” Physical Review. A. Third Series, vol. 46, no. 6, article no. 3092, 1992.

M. G. M. Gomes and G. P. King, “Bistable chaos. II. Bifurcation analysis,” Physical Review A, vol. 46, no. 6, pp. 3100–3110, 1992.

H. Fotsin, S. Bowong, and J. Daafouz, “Adaptive synchronization of two chaotic systems consisting of modified Van der Pol-Duffing and Chua oscillators,” Chaos, Solitons & Fractals, vol. 26, no. 1, pp. 215–229, 2005.

G. J. Fodjouong, H. B. Fotsin, and P. Woafo, “Synchronizing modified van der Pol-Duffing oscillators with offset terms using observer design: application to secure communications,” Physica Scripta, vol. 75, no. 5, pp. 638–644, 2007.

M. S. Siewe, F. M. M. Kakmeni, C. Tchawoua, and P. Woafo, “Bifurcations and chaos in the triple-well -Van der Pol oscillator driven by external and parametric excitations,” Physica A: Statistical Mechanics and Its Applications, vol. 357, no. 3-4, pp. 383–396, 2005.

M. Siewe Siewe, F. M. Moukam Kakmeni, and C. Tchawoua, “Resonant oscillation and homoclinic bifurcation in a Φ⁶-Van der Pol oscillator,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 841–853, 2004.

R. Tchoukuegno and P. Woafo, “Dynamics and active control of motion of a particle in a φ ⁶ potential with a parametric forcing,” Physica D: Nonlinear Phenomena, vol. 167, no. 1-2, pp. 86–100, 2002.

R. Tchoukuegno, B. R. Nbendjo, and P. Woafo, “Resonant oscillations and fractal basin boundaries of a particle in a φ ⁶ potential,” Physica A. Statistical Mechanics and its Applications, vol. 304, no. 3-4, pp. 362–378, 2002.

R. Tchoukuegno, B. R. Nana Nbendjo, and P. Woafo, “Linear feedback and parametric controls of vibrations and chaotic escape in a ϕ ⁶ potential,” International Journal of Non-Linear Mechanics, vol. 38, no. 4, pp. 531–541, 2003.

J. Yu and J. Li, “Investigation on dynamics of the extended Duffing-Van der Pol system,” Zeitschrift fur Naturforschung A, vol. 64, no. 5-6, pp. 341–346, 2009.

A. N. Njah and K. S. Ojo, “Synchronization via backstepping nonlinear control of externally Van der Pol and -Duffing oscillators,” Far East Journal of Dynamical Systems, vol. 11, no. 2, pp. 143–159, 2009.

A. N. Njah, “Synchronization via active control of identical and non-identical ϕ ⁶ chaotic oscillators with external excitation,” Journal of Sound and Vibration, vol. 327, no. 3–5, pp. 322–332, 2009.

A. N. Njah, “Synchronization via active control of parametrically and externally excited φ ⁶ Van der Pol and Duffing oscillators and application to secure communications,” Journal of Vibration and Control, vol. 17, no. 4, pp. 493–504, 2011.

G. Chen and X. Dong, From Chaos to Order, vol. 24 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1998.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990.

B. K. Bose, Modern Power Electronics: Evolution, Technology and Applications, IEEE Press, New York, NY, USA, 1992.

R. D. Middlebrook and S. Cuk, “A general unified approach to modelling switching-converter power stages,” International Journal of Electronics, vol. 42, no. 6, pp. 521–550, 1977.

M. di Bernardo and C. K. Tse, “Chaos in power electronics: an overview,” in Chaos in Circuits and Systems, chapter 16, pp. 317–340, World Scientific, New York, NY, USA, 2002.

C. K. Tse and M. Di Bernardo, “Complex behavior in switching power converters,” Proceedings of the IEEE, vol. 90, no. 5, pp. 768–781, 2002.

J. C. Chedjou, H. B. Fotsin, P. Woafo, and S. Domngang, “Analog simulation of the dynamics of a van der pol oscillator coupled to a duffing oscillator,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 6, pp. 748–757, 2001.