Attractiveness of periodic orbits in parametrically forced systems with time-increasing friction

Abstract

We consider dissipative one-dimensional systems subject to a periodic force and study numerically how a time-varying friction affects the dynamics. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction. We find that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. From a quantitative point of view, the scenario depends both on the final value and the growth rate of the damping coefficient. The relevance of the results for the spin-orbit model are discussed in some detail.

Comparisons between the pendulum with varying length and the pendulum with oscillating support

“…Decreasing γ and ζ further, the periodic attractors attract less of the phase space and the basin of attraction belonging to the downward fixed point increases again. This was not observed for the cubic oscillator in [9], where decreasing the damping coefficient causes the basin of attraction of the fixed point to decrease. This is linked to the bifurcation structure of the two systems and the choice of parameters α, δ and ε.…”

“…The relative areas of the basins of attraction corresponding to the fixed points in both systems follow a similar curve, as do the period-1 rotations and period-2 oscillations; compare Figure 4 with Figure 6. The shape of these curves is different from those the cubic oscillator, see [9], where decreasing the damping coefficient leads to a decrease in the fixed points basin of attraction. The reason behind the increase in the size of the basin of attraction associated with the fixed point is related to the choice of parameters and the bifurcation structure of the two systems.…”

“…Moreover the threshold values may be calculated using perturbation techniques. The method is described, for instance, in [7], where it was used for a forced cubic oscillator, and has since been applied to the spin-orbit system [9] and the pendulum with oscillating support [64]. To implement the method, one writes γ and ζ as power series in δ and ε, respectively, as γ = δB 1 + δ 2 B 2 + δ 3 B 3 + .…”

“…Applying the Liouville transformation (9), with h instead of f , and setting x(τ ) = θ(t), the system may be written in terms of the new time τ as…”

The high-order Euler method and the spin–orbit model.

“…The time-asymptotic behaviour, that is, the solution after any transient has decayed, is determined for each of these initial conditions, and the probability of capture by each of the possible attractors is thereby estimated. The challenges of this approach are (a) that realistic values of the dissipation parameter γ are small, so transient times, which are O(1/γ )-see Bartuccelli et al (2012) for an argument in a similar case-are long; and (b), in order to obtain low-error estimates of capture probabilities, a large number I of initial conditions must be considered: in fact, the width of the 95 % confidence interval (CI) for the probabilities is proportional to I −1/2 -see Eq. (10).…”