Progenitrix of Cycles:
Subharmonic Resonance in Hamiltonian Systems.
In Weakly Dissipative Predator-Prey Systems and The Rainbow Bridge, King and Schaffer [ksgtk96, ks99] explained how subharmonic periodicities of dissipative dynamical systems can be traced to nonintegrable Hamiltonian limits [h83, t89], which are conservative. Thereafter, Schaffer et al. [spmskb01] reviewed the emergence of such periodicities within Hamiltonian limits as one varies a time scale parameter.
The latter transitions break into two broad categories:
So-called weak resonances (Figure 1), the vast majority, emerge from a small amplitude, period-one orbit which is truly a progenitrix of cycles.
In the case of the period-2, -3 and -4 cycles, the so-called strong resonances, the situation is more complicated (Figure 2).
In all cases, the new cycles are produced in pairs, Of these, one is a saddle (denoted by red crosses in the acompanying animations); the other, a neutrally stable center (yellow circles).
Shown here are animations depicting the origin of representative weak resonances (Figure 1) and the period-3 cycle (Figure 2) in the Hamiltonian limit of a seasonally forced predator-prey model displayed as a stroboscopic map. To view an animation, click the figure.
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| Figure 1. Representative weak resonances. |
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| Figure 2. The period-3 cycle, a strong resonance. |
[h83] Hénon, M. 1983. Numerical explorations of Hamiltonian systems. In, Ioos, G., Helleman, R. H. and R. Stora (eds.) Chaotic Behavior of Deterministic Systems. North-Holland. Amsterdam.
[ksgtk96] King, A., W. M. Schaffer, C. Gordon, J. Treat and M. Kot. 1996. Weakly dissipative predator-prey systems. Bull. Math. Biol. 58:835-860.
[ks99] King, A. A. and W. M. Schaffer. 1999. The rainbow bridge: Hamiltonian limits and resonance in predator-prey dynamics. J. Math. Biol. 39: 439-469.
[spmskb01] Schaffer, W. M., Pederson, B., Moore, B. K., Skarpaas, O., King, A. A. and T. V. Bronnikova. 2001. Subharmonic resonance and multi-annual oscillations in northern mammal cycles: A nonlinear dynamics perspective. Chaos, Solitons and Fractals. 12: 251-264.
[t89] Tabor, M. 1989. Chaos and Integrability in Nonlinear Dynamics: An Introduction. J. Wiley. New York.