The Complementary Case: Nonresonance

While in the present chapter for the most part the resonance cases of the kicked harmonic oscillator have been discussed, in this section I briefly turn to the complementary case of nonresonance, where condition (1.22) is not satisfied. This case is of particular importance for the discussion in chapter 5.

It has been shown in section 1.2 that in order to dynamically construct stochastic webs by means of the POINCARÉ map (1.21), the resonance condition (1.23) needs to be satisfied, thereby restricting the kick period $T$ to a discrete set of values. What is more, the periodic stochastic webs leading to unrestricted diffusion in phase space are obtained for the few values of $T$ specified by equation (1.30) only.

The following typical example demonstrates that for nonresonant $T$, typically the energy grows diffusively as well.

Similar to figure 1.17, figure 1.19

shows the time development of the ensemble averaged energy, but in this case for the nonresonant value of $T=1.0$. Gaussian ensembles of initial conditions centered around some of the initial conditions $(0,p_0)^t$ used for the phase portrait of figure 1.4 are used to compute the averaged energy in this case. For the initial condition with $p_0=5.5$ (close enough to the origin to be encircled by the outer invariant line shown in figure 1.4) there is no energy growth beyond a certain saturation value, as should be expected. But for initial conditions outside of the boundary given by the outermost invariant line, for example for $p_0=12.5$, the energy grows unboundedly as in the case of a periodic stochastic web. The same effect, but even more clearly, can be observed for initial conditions placed further outward, for example with $p_0=16.0$ or $p_0=20.0$.

Figure 1.19a also indicates that, depending on the values of the parameters, it may take a large number of kicks (here: $n{ {\protect\begin{array}{c} >\protect\\ [-0.3cm]\sim \protect\end{array}} }60000$) before it becomes clear that asymptotically the energy growth is linear with time. For larger values of $V_0$, typically this regime of diffusive energy growth is reached after a smaller number of kicks.

The existence of diffusive energy growth is confirmed by considering the rate of growth of $\left< E \right>_n^{\rm cl}$ analytically. As in subsection 1.2.4, the ``random $x_n, p_n$ approximation'' can be applied when unbounded motion is granted, such that the result of equation (1.78) is obtained again. Therefore, in the case of nonresonance for large enough values of $V_0$, or for $(x_0,p_0)^t$ far enough from the origin, the rate of growth of the energy should be expected to increase quadratically with $V_0$, as in the case of resonance.