Figure 1.

Illustration of different time scales of nonlinear dynamics captured by a nested set of SSMs. (a) Dynamics near a slow two-dimensional SSM $\mathcal{W}(E_S^2)$ with three trajectories in the phase space. (b,c) The three trajectories, shown with consistent colours, and their spectrogram. These trajectories were generated by a mechanical system in which $1,2.7,4.2$ are the first three linearized frequencies. In (b,c), the top trajectory has initial condition $\mathbf{\text{x}}(0)$ on the slow two-dimensional SSM $\mathcal{W}(E_S^2)$ and decays on it, the middle is initialized with a small perturbation off $\mathcal{W}(E_S^2)$ and the bottom one decays on $\mathcal{W}(E_S^4)$. (Online version in colour.)