Gallery of Computations
Some graphical examples are presented which were computed by numerical methods I have developed and/or implemented. Each of the computations is an outer covering of the solution. The applied methods are integrated into the AnT software package and accessible for download. Further information about the techniques and the numerical case studies is available in my Ph.D. thesis.
Computation of stable manifolds
The RIM method is used in order to compute stable manifolds for continuous maps. Remarkably, the method is capable to compute higher-dimensional stable manifolds of non-smooth and non-invertible maps.
Nien-Wicklin Map: Approximation of parts of the global stable set belonging to the saddle P0. The underlying system is a non-invertible 2D map. The global stable set is the union of successive pre-images of the local stable manifold of P0.
Nien-Wicklin Map: Parts of the global stable set belonging to the saddle P0.
Nonlinear Leslie Model: Approximation of the global stable manifold of a period-4 saddle which separates a Henon-like attractor and a period-2 attractor. The underlying system is a non-invertible 3D map. The computed manifold is 2D. Remarkably, the applied method is capable to compute several parts of the manifold which are disconnected in the areas of investigation.
Nonlinear Leslie Model: Computed outer covering of the stable manifold belonging to the period-4 saddle orbit in different views and areas of investigation.
Computation of the chain recurrent set
The concepts of symbolic analysis are applied for the computation of the chain recurrent set. Due to the fact that the chain recurrent set contains all kinds of return trajectories, this technique can also be used to compute quasi-periodic trajectories. The method implies the construction and analysis of a symbolic image graph. For the presented computations, it is necessary to apply specific tuning techniques for the graph construction which are not in accordance with the basic theoretical concepts.
Discrete Food Chain Model: The computation of the chain recurrent set reveals that the system, which is a piecewise-smooth 3D map, has one main attractor, which is a quasi-periodic cycle, and an unstable invariant set of saddle type which is probably also a quasi-periodic cycle. Though the computation of the attractor is a simple task of numerical investigation, the computation of an outer covering of the unstable invariant set is only possible by our methods if sophisticated tunings are applied.
Discrete Food Chain Model: Different views of an approximation of the chain recurrent set. The main attractor (red), fixed points (blue) and an unstable invariant set of saddle type (green) are shown.
Lorenz System: The computation of the chain recurrent set for the Lorenz system is an example for the application of the techniques to a system continuous in time. The unstable limit cycles can be revealed.
Lorenz System: An outer covering of the chain recurrent set at two different positions in parameter space.
Localization of periodic points
Application of the RIM method in order to compute periodic points.
Tinkerbell Map: An outer covering of all detected periodic points with period lower equal 14.
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