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Complex Order Van Der Pol Oscillator Carla M Complex order van der Pol oscillator Carla M. A. Pinto, J. A. Tenreiro machado To cite this version: Carla M. A. Pinto, J. A. Tenreiro machado. Complex order van der Pol oscillator. Nonlinear Dynam- ics, Springer Verlag, 2010, 65 (3), pp.247-254. 10.1007/s11071-010-9886-0. hal-00640454 HAL Id: hal-00640454 https://hal.archives-ouvertes.fr/hal-00640454 Submitted on 12 Nov 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. NODY9886_source.tex; 3/11/2010; 13:48 p. 1 Complex order van der Pol oscillator Carla M.A.Pinto Centro de Matem´atica da Universidade do Porto and Departament of Matemathics, Institute of Engineering of Porto, Rua Dr. Ant´onio Bernardino de Almeida, 431, 4200-072 Porto, Portugal [email protected] J.A. Tenreiro Machado Department of Electrical Engineering, Institute of Engineering of Porto, Rua Dr. Ant´onio Bernardino de Almeida, 431, 4200-072 Porto, Portugal [email protected] November 3, 2010 Abstract In this paper it is considered a complex order van der Pol oscilla- tor. The complex derivative Dα±jβ,withα, β ∈ R+ is a generalization of the concept of integer derivative, where α =1,β=0. By applying the concept of complex derivative, we obtain a high-dimensional pa- rameter space. Amplitude and period values of the periodic solutions of the two versions of the complex order van der Pol oscillator, are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed. Keywords — van der Pol oscillator, complex order deriva- tive, dynamical behavior 1 Introduction The van der Pol (VDP) oscillator is an ordinary differential equation that has arisen as a model of electrical circuits containing vacuum tubes [45] (re-edited [8]). It produces self-sustaining oscillations in which energy is fed into small oscillations while is removed from large 1 NODY9886_source.tex; 3/11/2010; 13:48 p. 2 oscillations. This is the first relaxation oscillator appearing in the literature [44, 46]. It is given by the following second order differential equation x¨ + μ x2 − 1 x˙ + x =0 (1) Parameter μ controls the way the voltage flows through the system. For μ = 0 this is just a simple linear oscillator. For large values of μ, that is for μ 1, the system exhibits a relaxation oscillation. This means that the oscillator has two distinct phases: a slow recovery phase and a fast release phase (vaccuum tubes quickly release or relax their voltage after slowly building up tension). This equation has been used in the design of various systems, from biology, with the modeling of the heartbeat [22, 30, 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 chaos theory [9, 14, 15, 16, 12]. Fractional calculus (FC) has been an important research issue in the last few decades. FC is a generalization of the ordinary integer differentiation and integration to an arbitrary, real or complex, or- der [34, 43, 28]. Application of FC have been emerging in different and important areas of physics and engineering [35, 27, 26, 31, 39, 41, 10, 5, 42, 32]. Fractional order behavior has been found in areas such as fluid mechanics [29], mechanical systems [13], electrochem- istry [33], and biology [11, 1], namely in the modeling of the central pattern generators for animal locomotor rhythms [36, 37]. There are several definitions of fractional derivatives of order α ∈ R being three of the most important the Riemann - Liouville, the Gr¨unwald - Letnikov, and the Caputo given by: n t α 1 d f(τ) − Dt f(t)= n α−n+1 dτ, n 1 <α<n (2) Γ(n − α) dt a (t − τ) t−a [h ] α 1 − k α − Dt f(t) = lim ( 1) f(t kh)(3) h→0 hα k h=0 t (n) α 1 f (τ) − Dt f(t)= α−n+1 dτ, n 1 <α<n (4) Γ(α − n) a (t − τ) where Γ() is the Euler’s gamma function, [ ] means the integer part of x,andh represents the step time increment. It is also possible to generalize results based on transforms, yielding: n−1 { α } α { }− k α−k−1 + L Dt f(t) = s L f(t) s Dt f(0 )(5) k=0 2 NODY9886_source.tex; 3/11/2010; 13:48 p. 3 where s and L represent the Laplace variable and operator, respec- tively. The definitions demonstrate that fractional derivatives capture the history of the variable, or, by other words, that have memory, con- trary to integer derivatives, that are local operators. The Gr¨unwald - Letnikov formulation inspires the numerical calculation of the frac- tional derivative based on the approximation of the time increment h through the sampling period T and the series truncation at the rth term. This method is often denoted as Power Series Expansion (PSE) yielding the equation in the z - domain: 1 r (−1)kΓ(α +1) Z{Dαx(t)}≈ z−k X(z)(6) T α k!Γ(α − k +1) k=0 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 the s → z discretiza- tion scheme, being the Tustin approximation another possibility. The most often adopted generalization of the generalized derivative oper- ator consists in α ∈ R. The case of having fractional derivative of complex-order α ± jβ ∈ C leads to complex output valued results and imposes some restrictions before a practical application. To overcome this problem, it was proposed recently [18, 19, 6] the association of two complex-order derivatives. In fact, there are several arrangements that produce real valued results. For example, with the real part of two complex conjugate derivatives Dα±jβ we get: 1 α−jβ α+jβ ≈ 1 1 −1 1 − −2 ··· Z 2 D x(t)+D x(t) T α sin β ln( T ) βz + 2 β(1 2α)z + + 1 − −1 − 1 2 − − 2 ··· +cos β ln( T ) 1+αz 2 β(α α β + ) X(z) (7) Other combinations and the adoption of a Pad´e fraction, instead of the series for the approximation, are also possible. Nevertheless, in the sequel it is explored the case of expression (7). We must remark that it is adopted a scheme of discretized frac- tional derivative commonly accepted in engineering, but not yet fully investigated from a mathematical view point. Let us recall the basic definitions of Riemann-Liouville (R-L) and Caputo (C) derivatives, in equations (2) and (4), based on integral expressions, and of Gr¨unwald- Letnikov (G-L), in equation (3), based on a series expression suitable for discretization. The C derivative is a regularization of the R-L derivative and their G-L derivative is the series representation of the R- L derivative. In equation (5) the Laplace transform of the R-L deriva- 3 NODY9886_source.tex; 3/11/2010; 13:48 p. 4 tives differs from that of the C derivative, as shown by Gorenflo and Mainardi [23, 24], subsequently pointed out in the well known trea- tises on Fractional Calculus by Podlubny [38] and Kilbas-Srivastava and Trujillo [25]. So, in practice it is generalized to complex orders the classical R-L derivative of real order. In case of adopting the Ca- puto derivative (in view of standard initial conditions), it should be adopted a different Laplace transform (see Gorenflo-Mainardi [23, 24]), and also a modified G-L series representation, because the C deriva- tive is a regularization of the R-L derivative in the time origin. For real order less than one the G-L representation of the C derivative has been adopted in the paper by Gorenflo et al. [24]. As a consequence, the Z representation in equation (6) in the complex domain should be modified accordingly for the Caputo derivative. While tackling these matters is not straightforward, we must note that the simulations car- ried in the sequel correspond to steady state responses and that the initial conditions have a minor impact in the resulting charts. Having these ideas in mind, this paper is organized as follows. In Section 2, we introduce the two approximations of the complex order van der Pol oscillator (CVDP) and we present results from numerical simulations. In Section 3 we outline the main conclusions of this study. 2 Complex order van der Pol system Fractional VDP systems have been studied by many authors [9, 7, 40, 14, 4, 15, 16]. Their work differ in the approaches considered to express the fractional derivative. Chen et al [9] considered a 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, and ω 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.
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