Basins of attraction in forced systems with time-varying dissipation

We consider periodically forced systems with dissipation depending on time and study how the sizes of the basins of attraction are modified with respect to the case of constant dissipation. In particular, we investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies over a certain time interval before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients growing linearly between two given values, in the time interval where it varies, for a large number of choices of the interval: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins for all values of the time interval, and discuss its range of applicability. Second, we observe that sometimes very slight changes in the time interval may produce abrupt large variations in the relative areas of the basins of attraction of the surviving attractors: we show how comparing the contracted phase space at a time after the final value of dissipation has been reached with the basins of attraction corresponding to that value of constant dissipation can explain the presence of such variations. Both procedures are illustrated by application to a pendulum with periodically oscillating support.