Diverse Tools and Functionalities
This page gives an overview over the additional functionalities which have been introduced for handling and analysing the dynamics on invariant manifolds. They aid in visualisation of results, manipulation of the manifold parametrisation, among others.
Contents
Dynamical System Analysis
For the general analysis of the behaviour of the dynamical system, and the dynamics on it in relation to the dynamics on the invariant manifolds, the following functions have been written.
- Linear response of the full dynamical system.
- Project to autonomous SSM which projects a point in phase space directly onto the autonomous SSM.
- Transient trajectory on autonomous SSM projects the initial condition of a full system trajectory onto the SSM and performs forward integration on it.
- Distance of a point on the SSM to a point in the full dynamical system. This can be used to analyse the convergence of a full system trajectory onto the SSM.
ROMs
Several functions are especially dedicated to the analysis of the reduced dynamics on the SSM and to mapping the results of the analysis from the parametrisation space back to the physical phase space.
- Autonomous reduced dynamics This function evaluates the autonomous reduced dynamics on a specific point of the parametrisation space.
- Symbolic reduced dynamics for displaying a symbolic representation of the autonomous redcued dynamics.
- Reduced to full, which maps the parametrisation coordinates onto the SSM in phase space.
- Reduced to full (complex), which maps the parametrisation coordinates onto the SSM in phase space using a complex valued leading order representation of the non-autonomous manifold.
- Reduced to full trajectory maps a trajectory from the parametrisation space onto the invariant manifold in full phase space.
Visualisation
The following functions serve for the visualisation of results and of the manifold itself.
- Plot stability lines for including the stability behaviour of branches of response directly in the format of the plot.
- Plot invariant torus on SSM in full phase space.
- Stability plot for plotting the stability behaviour of solutions, for instance in a FRC plot.
Nonlinear Maps
The following functions are designed to handle and convert nonlinear functions, such as the internal forces or the parametrisation coefficients. They are employed to evaluate mult-index expansions, convert them into a tensor - based representation, and for multi-variable calculus.
- NsumK a combinatorics function.
Further tools
- Monitor memory of active workspace variables.
- Read solution of numerical integration.
- Read initial condition for a full system periodic orbit obtained via SSM computation. Can be used to verify the periodic orbit in the full system.
- Read equilibrium point on SSM that have been computed for further analysis.
- Read periodic orbits on SSM that have been computed for further analysis.
- Read invariant tori on SSM that have been computed for further analysis.