Rössler attractor

[4] Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams.

[5] How Rössler discovered this set of equations was investigated by Letellier and Messager.

[6] Some of the Rössler attractor's elegance is due to two of its equations being linear; setting

, the eigenvalues are complex and both have a positive real component, making the origin unstable with an outwards spiral on the

The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and eigenvectors.

For the centrally located fixed point, Rössler's original parameter values of a=0.2, b=0.2, and c=5.7 yield eigenvalues of: The magnitude of a negative eigenvalue characterizes the level of attraction along the corresponding eigenvector.

Similarly the magnitude of a positive eigenvalue characterizes the level of repulsion along the corresponding eigenvector.

) are responsible for the steady outward slide that occurs in the main disk of the attractor.

The last eigenvalue/eigenvector pair is attracting along an axis that runs through the center of the manifold and accounts for the z motion that occurs within the attractor.

The blue line corresponds to the standard Rössler attractor generated with

The red line intersecting that fixed point is an illustration of the repulsing plane generated by

The magenta line is generated by stepping backwards through time from a point on the attracting eigenvector which is slightly above

Note that the magenta line nearly touches the plane of the attractor before being pulled upwards into the fixed point; this suggests that the general appearance and behavior of the Rössler attractor is largely a product of the interaction between the attracting

Specifically it implies that a sequence generated from the Rössler equations will begin to loop around

, this influence effectively involves pushing the resulting system towards the general Rössler attractor.

The Poincaré map is constructed by plotting the value of the function every time it passes through a set plane in a specific direction.

increases, as is to be expected due to the upswing and twist section of the Rössler plot.

Knowing that the Rössler attractor can be used to create a pseudo 1-d map, it then follows to use similar analysis methods.

Rössler attractor's behavior is largely a factor of the values of its constant parameters

In general, varying each parameter has a comparable effect by causing the system to converge toward a periodic orbit, fixed point, or escape towards infinity, however the specific ranges and behaviors induced vary substantially for each parameter.

Bifurcation diagrams are a common tool for analyzing the behavior of dynamical systems, of which the Rössler attractor is one.

They are created by running the equations of the system, holding all but one of the variables constant and varying the last one.

Then, a graph is plotted of the points that a particular value for the changed variable visits after transient factors have been neutralised.

values illustrates the general behavior seen for all of these parameter analyses – the frequent transitions between periodicity and aperiodicity.

The attractor is filled densely with periodic orbits: solutions for which there exists a nonzero value of

As the majority of the dynamics occurs in the x-y plane, the periodic orbits can then be classified by their winding number around the central equilibrium after projection.

It seems from numerical experimentation that there is a unique periodic orbit for all positive winding numbers.

For the purposes of dynamical systems theory, one might be interested in topological invariants of these manifolds.

The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint.

Rössler showed that his attractor was in fact the combination of a "normal band" and a Möbius strip.

Rössler attractor as a stereogram with $a=0.2$ , $b=0.2$ , $c=14$

$x,y$ plane of Rössler attractor with $a=0.2$ , $b=0.2$ , $c=5.7$

Examination of central fixed point eigenvectors: The blue line corresponds to the standard Rössler attractor generated with $a=0.2$ , $b=0.2$ , and $c=5.7$ .

Rössler attractor with $a=0.2$ , $b=0.2$ , $c=5.7$

Poincaré map for Rössler attractor with $a=0.1$ , $b=0.1$ , $c=14$

{\displaystyle b} — Bifurcation diagram for the Rössler attractor for varying $b$