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A pair of van der Pol oscillators coupled by fractional derivatives

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Abstract

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators:

(1)
(2)

where D α x is the order α derivative of x(t), and 0<α<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 α, γ parameter plane. The perturbation results are validated with numerics and through direct comparison with known results in the limiting cases of α=0 and α=1, where the fractional coupling reduces to position coupling and velocity coupling, respectively.

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Authors and Affiliations

  1. Dept. Mechanical and Aerospace Engineering, Cornell University, 1485, Ithaca, NY, USA

    M. K. Suchorsky & R. H. Rand

  2. Dept. Mathematics, Cornell University, 14850, Ithaca, NY, USA

    R. H. Rand

Authors
  1. M. K. Suchorsky
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  2. R. H. Rand
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Correspondence to M. K. Suchorsky.

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Suchorsky, M.K., Rand, R.H. A pair of van der Pol oscillators coupled by fractional derivatives. Nonlinear Dyn 69, 313–324 (2012). https://doi.org/10.1007/s11071-011-0266-1

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