Homoclinic tangles, phase locking, and chaos in a two-frequency perturbation of Duffing's equation
The perturbation of the Duffing equation (with a saddle at the origin) with two-frequency forcing with small dissipation is studied. The phase equations describing two-frequency phase dependence do not depend on the system variables, and their dynamics is chosen to be very simple: the flow on the torus is decoupled into a rigid rotation in one angular variable and a one-dimensional flow on the circle that can be either periodic or has two equilibria, stable and unstable, that appear via tangent bifurcation. Without perturbation the two-dimensional system has a saddle with two homoclinic orbits to it, after switching the perturbation instead of equilibrium one has a saddle invariant 2-torus whose 3-dimensional stable and unstable manifolds can intersect forming homoclinic tangle. This intersection is detected by Melnikov's method. In dependence on the type of the flow on the phase torus different conclusions about the behavior of orbits are derived.
scientific article; zbMATH DE number 1247986