Lyapunov exponent computation

The computation of the largest lyapunov exponent is based on a long time integration of two nearby trajectories, in which the separation in calculated at fixed time intervals via a recursive rescaling of the mutual distance between the two trajectories.

The following formalism applies.

The leading trajectory starts from an initial value \(\mathbf{x}_1\). Fixed a sampling time \(\Delta t\), the \((i+1)-th\) point of the reference trajectory is:

\begin{equation} \mathbf{x}_{i+1} = f\big(\mathbf{x}_{i}\big) \Big \rvert_{t_i}^{t_{i} + \Delta t} \end{equation}

The neighbouring trajectory is initially perturbed with a perturbation \(\delta \mathbf{x}\). At every sampling point this trajectory is riscaled with the module of the initial perturbation at time \(t_0\).

The exact separation of the trajectories is:

\begin{equation} \mathbf{d}_{i+1} = f\big(\mathbf{z}_{i}\big) \Big \rvert_{t_i}^{t_{i} + \Delta t}- \mathbf{x}_{i+1} \end{equation}

the vector \(\mathbf{d}_{i+1}\) is re-scaled with the module of the initial perturbation \(|\delta \mathbf{x}|\).

The next integration point for the perturbed trajectory is therefore:

\begin{equation} \mathbf{z}_{i+1} = \mathbf{x}_{i+1} + \mathbf{d}_{i+1} \frac{|\delta \mathbf{x}|}{|\mathbf{d}_{i+1}|} \end{equation}

By recursively finding the \(\mathbf{z}_{i}\) points via rescaling the distance, and then measuring the deviation of the two trajectories, the estimation of the largest Lyapunov exponent is calculated then from the series:

\begin{equation} \lambda_{t} = \frac{1}{N \Delta t} \sum_{i=1}^{N} \ln \Big(\frac{\delta \mathbf{x}}{|\mathbf{d}_{i}|}\Big) \end{equation}