Basins of attraction in strongly damped coupled mechanical oscillators: A global example

We consider a finite array of N oscillators with nearest neighbor coupling along a line, and with two types of damping. Friction terms can affect each individual oscillator, separately: local damping. Neighboring damping, in contrast, affects nearest neighbor distances. Although stability of equilibria does not depend on the particular type of damping, global basins of attraction do. We show that basins of attraction can in fact jump discontinuously under continuous variations of local versus neighbor damping. This effect is caused by heteroclinic saddle-saddle connections of equilibria. It occurs even in the limit of strong damping and for only two oscillators, N = 2. The results are based on geometric singular perturbation methods, Sturm type oscillation theory (zero numbers), and the related theory of Jacobi systems. Going beyond the motivating mechanical application, they emphasize the dependence of basins of attraction and heteroclinic orbit connections in gradient systems on the underlying metric.