The Wada property and visualization of strange attractors

This page presents my master thesis and highlights some results by nice pictures.

Master thesis

Download the original master thesis

Propriété de Wada et visualisation des attracteurs étranges

in French. This discusses two dynamical systems: the Plykin attractor and Hubbard's pendulum. An A0 sized poster

A strange attractor

presenting Hubbard's pendulum is also available for download.

The Plykin attractor

This is a dynamical system on the torus with four attractive fixed points. The following pictures demonstrate the dependence of the attraction basins on the system parameters. Click on them for larger images. Download the document above for further details.

The four attraction basins are colored in white, red, green and blue. The darker a pixel is, the more iterations are necessary to reach a given neighbourhood of the attractive fixed point from the corresponding initial condition.

Decreasing the perturbation radius shrinks the attraction basin.

The perturbation amplitude influences one of the eigenvalues at the fixed points.

Hubbard's original pendulum

This is a dynamical system governed by an ordinary differential equation of order two. Thus the phase space is of dimension two. The equation describes the ideal behaviour of a forced damped pendulum. Click on the pictures for a higher resolution. Download the document above for further details.

$fractal_basin_flag$ Three attraction basins are colored red, white and green. These have discrete translation invariance by the nature of the pendulum.

$fractal_angle_hsv$ Each point in a disc has the same color as its preimages. This presents numerical evidence of that both eigenvalues are negative.

The preimages of a circle are shown to present the behaviour of the mapping and the way the attraction basins intertwine.

One attraction basin is red, while darkness is determined by the number of iterations that have passed since the last time in the past the pendulum has had a high angular speed. This makes the hyperbolic periodic orbits visible on the accessible boundary of the attraction basin, as well as their unstable manifolds.

Variation of the pendulum parameters

For certain values of the parameters, no attractive fixed point (stable equilibrium state) has been found, however, repellors can be visualized by simply iterating the system towards the past. The shape of the repellors depend on the parameters as shown below.

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