Fractional van der Pol Oscillator Equations

Details Fractional van der Pol Oscillator Equations Fractional van der Pol Oscillator Equations

Apr 2002

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2002aps..aprb17026o&link_type=abstract

American Physical Society, April Meeting, Jointly Sponsored with the High Energy Astrophysics Division (HEAD) of the American As

Astronomy and Astrophysics

Astrophysics

Scientific paper

The van der Pol oscillator differential equation has an elastic restoring force which is harmonic, i.e., it is a linear function of the dependent variable. The fractional van der Pol oscillator (FVDPO) is one for which the linear term is replaced by the dependent variable to one-third power: ddot x + x^1\over 3 = k(1-x^2)dot x,<=no (1) where k is a positive parameter. Another type of FVDPO is ddot x + x=k(1-x^2)dot x^1\over 3.<=no(2) (Note that both equations are of odd parity, i.e., they are invariant under the transformation x→-x.) We show in both cases that an essentially unique limit-cycle exists and use the method of harmonic balance to calculate approximations to their periodic solutions. A detailed analysis is made of the properties of their respective trajectories in phase space including the stability nature of the fixed-points.

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