Complex-order forced van der Pol oscillator

In this paper we consider a complex-order forced van der Pol oscillator. The complex derivative D-alpha +/- J beta, with alpha, beta is an element of R+, is a generalization of the concept of an integer derivative, where alpha = 1, beta = 0. The Fourier transforms of the periodic solutions of the complex-order forced van der Pol oscillator are computed for various values of parameters such as frequency omega and amplitude b of the external forcing, the damping mu, and parameters alpha and beta. Moreover, we consider two cases: (i) b = 1, mu = {1.0, 5.0, 10.0}, and omega = {0.5, 2.46, 5.0, 20.0}; (ii) omega = 20.0, mu = {1.0, 5.0, 10.0}, and b = {1.0, 5.0, 10.0}. We verified that most of the signal energy is concentrated in the fundamental harmonic omega(0). We also observed that the fundamental frequency of the oscillations omega(0) varies with alpha and mu. For the range of tested values, the numerical fitting led to logarithmic approximations for system (7) in the two cases (i) and (ii). In conclusion, we verify that by varying the parameter values alpha and beta of the complex-order derivative in expression (7), we accomplished a very effective way of perturbing the dynamical behavior of the forced van der Pol oscillator, which is no longer limited to parameters b and omega.