Resonance Overlap

Isolated resonances in a Dynamical System can cause considerable distortion of preserved Tori in their Neighborhood, but they do not introduce any Chaos into a system. However, when two or more resonances are simultaneously present, they will render a system nonintegrable. Furthermore, if they are sufficiently ``close'' to each other, they will result in the appearance of widespread (large-scale) Chaos.

To investigate this problem, Walker and Ford (1969) took the integrable Hamiltonian

$$H_0(I_1,I_2) = I_1+I_2-I_1^2-3I_1 I_2+{I_2}^2$$

and investigated the effect of adding a 2:2 resonance and a 3:2 resonance

$$H({\bf I},\theta) = H_0({\bf I})+\alpha I_1 I_2\cos(2\theta_1-2\theta_2) +\beta {I_1}^{3/2}I_2\cos(2\theta_1-3\theta_2).$$

At low energies, the resonant zones are well-separated. As the energy increases, the zones overlap and a ``macroscopic zone of instability'' appears. When the overlap starts, many higher-order resonances are also involved so fairly large areas of Phase Space have their Tori destroyed and the ensuing Chaos is ``widespread'' since trajectories are now free to wander between regions that previously were separated by nonresonant Tori.

Walker and Ford (1969) were able to numerically predict the energy at which the overlap of the resonances first occurred. They plotted the $\theta_2$-axis intercepts of the inner 2:2 and the outer 2:3 separatrices as a function of total energy. The energy at which they crossed was found to be identical to that at which 2:2 and 2:3 resonance zones began to overlap.

See also Chaos, Resonance Overlap Method

References

Walker, G. H. and Ford, J. ``Amplitude Instability and Ergodic Behavior for Conservative Nonlinear Oscillator Systems.'' Phys. Rev. 188, 416-432, 1969.