In this paper it is considered a complex order van der Pol oscillator.
A complex derivative D®§|¯, with ®; ¯ 2 R+ is a generalization of the concept of
integer derivative, where ® = 1; ¯ = 0: By applying the concept of complex derivative,
we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the complex order van der Pol oscillator, are studied for
variation of these parameters.N/

Pinto, Carla

Tenreiro Machado, J. A.

Scientific Repository of the Polytechnic Institute of Porto

English

Complex order van der Pol oscillator

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Complex order van der Pol oscillatorCarla M.A.Pinto1, and J.A. Tenreiro Machado21Centro de Matema´tica da Universidade do Porto andDepartament of Matemathics, Institute ofEngineering of Porto, Rua Dr. Anto´nio Bernardino deAlmeida, 431, 4200-072 Porto, Portugalcap@isep.ipp.pt2Department of Electrical Engineering, Institute ofEngineering of Porto, Rua Dr. Anto´nio Bernardino deAlmeida, 431, 4200-072 Porto, Portugaljtm@isep.ipp.ptAbstract — In this paper it is considered a complex order van der Pol oscillator.A complex derivative Dα±β , with α, β ∈ R+ is a generalization of the concept ofinteger derivative, where α = 1, β = 0. By applying the concept of complex deriva-tive, we obtain a high-dimensional parameter space. Amplitude and period valuesof the periodic solutions of the complex order van der Pol oscillator, are studied forvariation of these parameters.Keywords — van der Pol oscillator, complex order derivative, dynamical behavior1 IntroductionThe van der Pol (VDP) oscillator is an ordinary differential equation that has arisen as a modelof electrical circuits containing vacuum tubes [40] (re-edited [8]). It produces self-sustainingoscillations in which energy is fed into small oscillations while is removed from large oscillations.This is the first relaxation oscillator appearing in the literature [39, 41]. It is given by the followingsecond order differential equationx¨+ µ(x2 − 1) x˙+ x = 0 (1)Parameter µ controls the way the voltage flows through the system. For µ = 0 this is just asimple linear oscillator. For large values of µ, that is for µ À 1, the system exhibits a relaxationoscillation. This means that the oscillator has two distinct phases: a slow recovery phase and afast release phase (vaccuum tubes quickly release or relax their voltage after slowly building uptension).This equation has been used in the design of various systems, from biology, with the modeling ofthe heartbeat [22, 27, 17], the generation of action potentials [21, 20], up to acoustic systems [2]and electrical circuits [3, 12]. The VDP oscillator has also been used in the context of chaostheory [9, 14, 15, 16, 12].PINTO, TENREIRO MACHADOFractional calculus (FC) has been an important research issue in the last few decades. FCis a generalization of the ordinary integer differentiation and integration to an arbitrary, real orcomplex, order [30, 38, 25]. Application of FC have been emerging in different and importantareas of physics and engineering [31, 24, 23, 28, 34, 36, 10, 5, 37]. Fractional order behavior hasbeen found in areas such as fluid mechanics [26], mechanical systems [13], electrochemistry [29],and biology [11, 1], namely in the modeling of the central pattern generators for animal locomotorrhythms [32, 33].There are several definitions of fractional derivatives of order α ∈ R being three of the mostimportant the Riemann - Liouville, the Gru¨nwald - Letnikov, and the Caputo given by:a Dαt f(t) =1Γ(n− α)dndtn∫ taf(τ)(t− τ)α−n+1dτ, n− 1 < α < n (2)a Dαt f(t) = limh→01hα[ t−ah ]∑h=0(−1)k(αk)f(t− kh) (3)a Dαt f(t) =1Γ(α− n)∫ taf (n)(τ)(t− τ)α−n+1dτ, n− 1 < α < n (4)where Γ() is the Euler’s gamma function, [ ] means the integer part of x, and h represents the steptime increment. It is also possible to generalize results based on transforms, yielding:L{ 0 Dαt f(t)} = sαL{f(t)} −n−1∑k=0sk 0 Dα−k−1t f(0+) (5)where s and L represent the Laplace variable and operator, respectively.The definitions demonstrate that fractional derivatives capture the history of the variable, or,by other words, that have memory, contrary to integer derivatives, that are local operators. TheGru¨nwald - Letnikov formulation inspires the numerical calculation of the fractional derivativebased on the approximation of the time increment h through the sampling period T and the seriestruncation at the rth term. This method is often denoted as Power Series Expansion (PSE) yieldingthe equation in the z - domain:Z{Dαx(t)} ≈[1Tαr∑k=0(−1)kΓ(α+ 1)k!Γ(α− k + 1)z−k]X(z) (6)where X(z) = Z{x(t)} and z and Z represent the z-transform variable and operator, respectively.In fact, expression (3) represents the Euler (or first backward difference) approximation in thes → z discretization scheme, being the Tustin approximation another possibility. The most oftenadopted generalization of the generalized derivative operator consists in α ∈ R. The case ofhaving fractional derivative of complex-order α± β ∈ C leads to complex output valued resultsand imposes some restrictions before a practical application. To overcome this problem, it wasproposed recently [18, 19, 6] the association of two complex-order derivatives. In fact, there areseveral arrangements that produce real valued results. For example, with the real part of twocomplex conjugate derivatives Dα±β we get:Z{12[Dα−jβx(t) +Dα+jβx(t)] } ≈ 1Tα { sin [β ln( 1T )] [βz−1 + 12β(1− 2α)z−2 + · · · ]++cos[β ln( 1T )] [−1 + αz−1 − 12β(α2 − α− β2 + · · · )] }X(z)(7)COMPLEX VDPOther combinations and the adoption of a Pade´ fraction, instead of the series for the approxima-tion, are also possible. Nevertheless, in the sequel it is explored the case of expression (7).Having these ideas in mind, this paper is organized as follows. In Section 2, we introduce thetwo approximations of the complex order van der Pol oscillator (CVDP) and we present resultsfrom numerical simulations. In Section 3 we outline the main conclusions of this study.2 Complex order van der Pol systemFractional VDP systems have been studied by many authors [9, 7, 35, 14, 4, 15, 16]. Their workdiffer in the approaches considered to express the fractional derivative. Chen et al [9] considereda forced van der Pol equation with fractional damping of the form:x¨+ µ(x2 − 1)Dαx(t) + x(t) = a sin(ωt) (8)where µ is an endogenous damping parameter, a denotes the amplitude of a periodic forcing, andω is the forcing frequency.Barbosa et al [7] considered the following modified version of the van der Pol equation:x(1+λ) + α(x2 − 1)x(λ) + x = 0 (9)with 0 < λ < 1.Tavazoei et al [35] determined the parametric range for which the fractional VDP system studiedby Barbosa et al [7] can perform as an undamped oscillator. They also showed that, contrary to theinteger order VDP, trajectories in a fractional VDP oscillator do not converge to a unique cycle.Ge et al [14] studied the autonomous and non-autonomous fractional van der Pol oscillator. Thenon-autonomous system in state-space oscillator model is given by:dαx1dtα = x2dβx2dtβ= −x1 − µ(1− x21)(c− ax21)x2 + b sin t(10)where α and β are fractional numbers.To the best knowledge of the authors, little attention has been given to CVDP oscillators. In thispaper, we consider the following complex order state-space model of the VDP oscillator:[12(Dα+β +Dα−β)x1x˙2]=[0 1−1 −µ(x21 − 1)] [x1x2](11)where Dα±β, α, β ∈ R+, is a generalization of the concept of the integer derivative, that corre-sponds to α = 1 and β = 0.We adopt the PSE method for the approximation of the complex-order derivative in the discretetime numerical integration. Several experiments demonstrated that it is required a slight adaptionto the standard approach based on a simple truncation of the series. In fact, since our objective isto generate limit cycles, the truncation corresponds to a diminishing of the gain [37] and, conse-quently, leads to difficulties in the promotion of periodic orbits. Therefore, in order to overcomethis limitation, we decided to include a gain adjustment factor corresponding to the sum of themissing truncated series coefficients.The dynamics governing the complex order van der Pol oscillator (11) is given by, respectively:x1(k + 1) = 1ψ(β,∆t)(H(x1(k)) + (∆t)αx2(k))x2(k + 1) = x2(k) + ∆t( −x1(k)− µ(x21(k)− 1)x2(k) ) (12)PINTO, TENREIRO MACHADOwhere ∆t = 0.0005 is the time increment and ψ(β,∆t) = cos[b log(1∆t)]. Function H(xi),i = 1, 2, results from the Taylor series expansion truncation.In Figure 1 we depict periodic solutions of system (12) for α ∈ {0.4, 0.8}, β = 0.8 and µ = 0.5.One can observe the appearance of the relaxation oscillation phenomena as α increases.In Figure 2, we show the phase portraits of solutions of systems (12) for β = 0.8, µ ∈ {0.5, 2}and different values of α. As expected, the larger the value of µ, the more nonlinear the oscillationbecomes. We verify also that we can control the period of the oscillation by varying α.0 0.5 1 1.5 2x 104−2−1012x 10 0.5 1 1.5 2x 104−0.4−0.200.20.4x 2samples*dt0 0.5 1 1.5 2x 104−4−2024x 1  0 0.5 1 1.5 2x 104−2−1012samples*dtx 2  Figure 1: Periodic solutions of the CVDP system (12) for β = 0.8, µ = 0.5 and α ∈{0.4, 0.8}.Figure 2: Phase-space solutions (x1(t), x2(t)) of the CVDP system (12) forα ∈ {0.4, 0.6, 0.8}, β = 0.8 and µ = 0.5 (left) and µ = 2.0 (right).We now simulate the ordinary differential system given by expression (12) for β = 0.8, α ∈{0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, µ = {0.5, 2}, and we measure the amplitude and theperiod of the solutions. The initial conditions are x1(1) = 0.0, x1(2) = 0.005, x1(3) =0.010, x1(4) = 0.015, x1(5) = 0.02, x2(1) = 1.0, x2(2) = 1.005, x2(3) = 1.010, x2(4) =1.015, x2(5) = 1.02.Each simulation is executed until a stable periodic solution is found. The amplitude and theperiod of the solutions are then measured. The results are depicted in Figures 3-4. We find thatthe period increases as α goes from 0.2 to 1.0, for system (12). On the other hand, the amplitudeis almost constant. To be precise, the amplitude shows a very tiny increase as α increases to one.3 ConclusionsIn this paper, a complex-order approximation to the well-known van der Pol oscillator is proposed.The amplitude and the period of solutions produced by this approximation were then measured.COMPLEX VDP0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12.533.544.5Amplitudeα0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.93.63.73.83.944.14.24.34.44.54.6AmplitudeαFigure 3: Amplitude of the periodic solutions x1(t) produced by the CVDP oscillator (12)for β = 0.8, α ∈ {0.2, 0.3, 0.4, 0.5, 06, 0.7, 0.8, 0.9, 1.0} and µ = 0.5 (left) and µ = 2.0(right).0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.500.51α0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.500.51αFigure 4: Period of the solutions x1(t) produced by the CVDP oscillator (12) for β = 0.8,α ∈ {0.2, 0.3, 0.4, 0.5, 06, 0.7, 0.8, 0.9, 1.0} and µ = 0.5 (left) and µ = 2.0 (right).The imaginary part was fixed while the real component was varied, for two distinct values ofparameter µ. It was observed that the waveform period increases as α varies between 0.2 and one.On the other hand, the amplitude values are almost constant as α varies. Moreover, it seems thereis a tiny increase in the amplitude of solutions for system (12), as α approaches one.AcknowledgmentsCP was supported by Fundac¸a˜o para a Cieˆncia e a Tecnologia (FCT) through the Centro deMatema´tica da Universidade do Porto www.fc.up.pt/cmup.References[1] Ahmed, E., Elgazzar, A.S. On fractional order differential equations model for nonlocal epidemicsPhysica A: Statistical Mechanics and its Applications, 379, Issue 2, 607–614, (2007).[2] Altmann, J. Acoustic and seismic signals of heavy military vehicles for co-operative verification.Journal of Sound and Vibration, 273, Issues 4-5, 713–740, (2004).[3] Appelbe, B., Rachinskii, D., Zhezherun, A. Hopf bifurcation in a van der Pol type oscillator withmagnetic hysteresis. Physica B: Condensed Matter, 403, Issues 2-3, 301–304, (2008).[4] Attari, M., Haeri, M., Tavazoei, M.S. Analysis of a fractional order Van der Pol-like oscilattor viadescribing function method. Nonlinear Dynamics. DOI 10.1007/s11071-009-9647-0. (2010).PINTO, TENREIRO MACHADO[5] Baleanu, D. About fractional quantization and fractional variational principles. Communications inNonlinear Science and Numerical Simulations, 14 (6), 2520–2523, (2009).[6] Barbosa, R.S., Tenreiro Machado, J.A. , Silva, M.F. Discretization of Complex-order Algorithms forControl Applications. Journal of Vibration and Control, 14 (9-10), 1349–1361 (2008).[7] Barbosa, R.S., Tenreiro Machado, J.A. , Vinagre, B.M., Caldero´n, A.J. Analysis of the Van der PolOscillator Containing Derivatives of Fractional Order. Journal of Vibration and Control, 13, Issues9–10, 1291–1301, (2007).[8] Bellman, R., Kalaba, R. Selected Papers on Mathematical Trends in Control Theory, Dover eds: NewYork. (1964).[9] Chen, J.-H., Chen, W.-C. Chaotic dynamics of the fractionally damped van der Pol equation. Chaos,Solitons and Fractals, 35 188–198, (2008).[10] Chen, Y.Q., Moore, K.L. Discretization schemes for fractional-order differentiators and integrators.IEEE Trans. Circuits and Systems-I: Fundam. Theory Appl. 49 (3), 363–367, (2002).[11] Darwish, M.A. On quadratic integral equation of fractional orders. Journal of Mathematical Analysisand Applications, 311, Issue 1, 112–119, (2005).[12] Elabbasy, E.M., El-Dessoky, M.M. Synchronization of van der Pol oscillator and Chen chaotic dy-namical system. Chaos, Solitons and Fractals, 36, Issue 5, 1425–1435, (2008).[13] Ge, Z.-M., Jhuang, W.-R. Chaos, control and synchronization of a fractional order rotational mechan-ical system with a centrifugal governor. Chaos, Solitons and Fractals, 33, Issue 1, 270–289, (2007).[14] Ge, Z.-M., Hsu, M.-Y. Chaos in a generalized van der Pol system and in its fractional order system.Chaos, Solitons and Fractals, 33 1711–1745, (2007).[15] Ge, Z.-M., Hsu, M.-Y. Chaos excited chaos synchronizations of integral and fractional order general-ized van der Pol systems. Chaos, Solitons and Fractals, 36 592–604, (2008).[16] Ge, Z.-M., Zhang, A.-R. Chaos in a modified van der Pol system and in its fractional order systems.Chaos, Solitons and Fractals, 32 1791–1822, (2007).[17] Glass, L. Theory of Heart, Springer, New York-Heidelberg-Berlin, (1990).[18] Hartley, T. T., Adams, J. L. and Lorenzo, C. F. Complex-order distributions. Proceedings of theASME International Design Engineering Technical Conferences and Computers and Information inEngineering Conference, Long Beach, CA, DETC2005-84952, on CD-ROM, (2005).[19] Hartley, T. T., Lorenzo, C. F., and Adams, J. L. Conjugated-order differintegrals. Proceedings of theASME International Design Engineering Technical Conferences and Computers and Information inEngineering Conference, Long Beach, CA, DETC2005-84951, on CD-ROM, (2005).[20] FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. BiophysicalJ. 1, 445–466, (1961).[21] FitzHugh R. Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Bio-physics, 17, 257–278, (1955).[22] FitzHugh R. Motion picture of nerve impulse propagation using computer animation. Journal of Ap-plied Physiology, 25, 628–630, (1968).[23] Machado, J. T. Analysis and design of fractional-order digital control systems. Journal Systems Anal-ysis, Modelling, Simulation, 27, 107–122, (1997).[24] Mainardi, F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Soli-tons and Fractals, 7, 1461–1477, (1996).[25] Miller, K. S., Ross, B. An introduction to the fractional calculus and fractional differential equations,John Wiley and Sons, (1993).[26] Momani, S., Odibat, Z. Analytical approach to linear fractional partial differential equations arisingin fluid mechanics. Physics Letters A, 355, Issues 4-5, 271–279, (2006).COMPLEX VDP[27] Nagumo J., Arimoto S., and Yoshizawa S. An active pulse transmission line simulating nerve axon.Proc IRE. 50 2061-2070, (1962).[28] Nigmatullin, R. The statistics of the fractional moments: Is there any chance to ”read quantitatively”any randomness?. Signal Processing, 86 (10), 2529-2547, (2006).[29] Oldham, K. B. Fractional differential equations in electrochemistry. Advances in Engineering Soft-ware, 41, Issue 1, 9–12, (2010).[30] Oldham, K. B., Spanier, J. The fractional calculus: theory and application of differentiation andintegration to arbitrary order. Academic Press (1974).[31] Oustaloup, A. La commande CRONE: commande robuste d’ordre non entier. Hermes. (1991).[32] Pinto, C.M.A., Tenreiro Machado, J.A. Complex order biped rhythms. International Journal of Bi-furcation and Chaos. Submitted, (2010).[33] Pinto, C.M.A., Tenreiro Machado, J.A. Fractional Central Pattern Generators for Bipedal Locomo-tion. Nonlinear Dynamics. Accepted for publication, (2010).[34] Podlubny, I. Fractional-order systems and PID-controllers. IEEE Transactions on Automatic Control,44 (1), 208–213, (1999).[35] M.S. Tavazoei, M. Haeri, M. Attari, S. Bolouki, and M. Siami. More Details on Analysis ofFractional-order Van der Pol Oscillator. Journal of Vibration and Control 15 (6), 803–819 (2009).[36] Tenreiro Machado, J. Discrete-time fractional-order controllers. Journal of Fractional Calculus andApplied Analysis, 4, 47–66, (2001).[37] Tenreiro Machado, J. A. Fractional Derivatives: Probability Interpretation and Frequency Responseof Rational Approximations. Communications in Nonlinear Science and Numerical Simulations, El-sevier, 14, Issues 9-10, 3492–3497, September-October, (2009).[38] Samko, S. G., Kilbas, A. A., Marichev, O. I. Fractional integrals and derivatives: theory and appli-cations. Gordon and Breach Science Publishers (1993).[39] B. van der Pol. On relaxation-oscillations, The London, Edinburgh, and Dublin Philosophical Maga-zine and Journal of Science Ser., 7, 2, 978–992, (1926).[40] B. van der Pol. Forced oscillations in a circuit with nonlinear resistance (receptance with reactivetriode), London, Edinburgh and Dublin Phil. Mag., 3, 65–80, (1927).[41] B. van der Pol e J. van der Mark. Frequency Demultiplication, Nature, 120, 363–364 (1927).

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Complex order van der Pol oscillator

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