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Research Papers

Parametric Excitation and Evolutionary Dynamics

[+] Author and Article Information
Rocio E. Ruelas

Center for Applied Mathematics,
Cornell University,
Ithaca, NY 14853

David G. Rand

Program for Evolutionary Dynamics,
Harvard University,
Cambridge, MA 01451

Richard H. Rand

Department of Mathematics and
Department of Mechanical and
Aerospace Engineering,
Cornell University,
Ithaca, NY 14853

1Corresponding author.

Manuscript received April 29, 2012; final manuscript received August 14, 2012; accepted manuscript posted January 22, 2013; published online July 19, 2013. Assoc. Editor: Martin Ostoja-Starzewski.

J. Appl. Mech 80(5), 050903 (Jul 19, 2013) (6 pages) Paper No: JAM-12-1172; doi: 10.1115/1.4023473 History: Received April 29, 2012; Revised August 14, 2012; Accepted January 22, 2013

Parametric excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. Evolutionary dynamics refers to a mathematical model of natural selection (the “replicator” equation) which involves a combination of game theory and differential equations. In this paper we apply perturbation theory to investigate parametric resonance in a replicator equation having periodic coefficients. In particular, we study evolution in the Rock-Paper-Scissors game, which has biological and social applications. Here periodic coefficients could represent seasonal variation. We show that 2:1 subharmonic resonance can destabilize the usual “Rock-Paper-Scissors” equilibrium for parameters located in a resonant tongue in parameter space. However, we also show that the tongue may be absent or very small if the forcing parameters are chosen appropriately.

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References

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Ruelas, R. E., Rand, D. G., and Rand, R. H., 2012, “Nonlinear Parametric Excitation of an Evolutionary Dynamical System,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.226(8), pp. 1912–1920. [CrossRef]
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Figures

Grahic Jump Location
Fig. 1

Integral curves from Eq. (12). Each of these curves represents a motion which is periodic in time.

Grahic Jump Location
Fig. 2

2:1 subharmonic resonance tongue, Eq. (51). The RPS equilibrium point at x1 = x2 = 1/3 is linearly unstable for parameters inside the tongue. The presence of nonlinearities detunes the resonance and prevents unbounded motions which are predicted by the linear stability analysis.

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