A pair of van der Pol oscillators coupled by fractional derivatives

作者：Suchorsky M K^*; Rand R H

来源：Nonlinear Dynamics, 2012, 69(1-2): 313-324.

DOI：10.1007/s11071-011-0266-1

• Fractional calculus
• Fractional derivative
• Coupled oscillators
• Perturbation methods

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators:
x" - epsilon(1 - x(2))x' + x = epsilon gamma D-alpha(y - x) (1)
y" - epsilon(1 - y(2))y' + y = epsilon gamma D-alpha(x - y) (1)
where D (alpha) x is the order alpha derivative of x(t), and 0 <alpha < 1. We use a two-variable perturbation method on the system's corresponding variational equations to derive expressions for the transition curves separating regions of stability from instability in the alpha, gamma parameter plane. The perturbation results are validated with numerics and through direct comparison with known results in the limiting cases of alpha=0 and alpha=1, where the fractional coupling reduces to position coupling and velocity coupling, respectively.

• 出版日期2012-7