Small and Large Forcing Amplitudes

In the limiting case of very large-amplitude forcing, the system is completely dominated by the forcing and the only response possible is a stable limit cycle of period 1 (same period as the forcing). The amplitude at which this becomes the case varies greatly and seems to increase with the forcing frequency. 

An example is shown in figure 3 for section AA near the bottom of the 2/1 resonance horn of figure 2. As the frequency is increased from left to right, the torus becomes phase locked as a pair of period 2 saddle nodes develop on it. The saddle nodes then separate with the saddles alternating with the node and the invariant circle is now composed of the unstable manifolds of the saddles whereas the stable manifolds of the saddles come from the unstable period 1 focus in the middle of the circle and from infinity. As the frequency is increased further, the saddles rotate around the circle and recombine with their neighbouring nodes in another saddle-node bifurcation. 

In general, there should be a resonance horn at every rational value of ailto0 = plq, but as q increases, the apices become more narrow (Hall 1984) and only the larger ones can be located numerically without great difficulty. The resonance horns shown in figure 2 are for resonances of plq = 1/2, 1/1, 3/2, 2/1, 3/1, 4/1, and 5/1. This entrainment behaviour has been recently observed experimentally in several forced oscillatory chemical systems (Dolnik 

(b) The stroboscopic phase portrait in the 2/1 resonance horn for At At, — 0.5, toltoo = 2.0. The period 2 saddles and nodes alternate and the unstable manifolds of the saddles (squares) make up the structure of the phase-locked torus. 

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