Rössler attractor

The Rössler attractor (pronEng|ˈrɒslɚFact|date=December 2007) is the attractor for the Rössler system, a system of three non-linear ordinary differential equations. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. 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. The original Rössler paper says the Rössler attractor was intended to behave similarly to the Lorenz attractor, but also be easier to analyze qualitatively. An orbit within the attractor follows an outward spiral close to the $x,\; y$ plane around an unstable fixed point. Once the graph spirals out enough, a second fixed point influences the graph, causing a rise and twist in the $z$-dimension. In the time domain, it becomes apparent that although each variable is oscillating within a fixed range of values, the oscillations are chaotic. This attractor has some similarities to the Lorenz attractor, but is simpler and has only one manifold. Otto Rössler designed the Rössler attractor in 1976, but the originally theoretical equations were later found to be useful in modeling equilibrium in chemical reactions. The defining equations are:

: $frac\{dx\}\{dt\}\; =\; -y\; -\; z$: $frac\{dy\}\{dt\}\; =\; x\; +\; ay$: $frac\{dz\}\{dt\}\; =\; b\; +\; z(x-c)$

Rössler studied the chaotic attractor with $a\; =\; 0.2$, $b\; =\; 0.2$, and $c\; =\; 5.7$, though properties of $a\; =\; 0.1$, $b\; =\; 0.1$, and $c\; =\; 14$ have been more commonly used since.

An analysis

Some of the Rössler attractor's elegance is due to two of its equations being linear; setting $z\; =\; 0$, allows examination of the behavior on the $x,\; y$ plane

: $frac\{dx\}\{dt\}\; =\; -y$: $frac\{dy\}\{dt\}\; =\; x\; +\; ay$

The stability in the $x,\; y$ plane can then be found by calculating the eigenvalues of the Jacobian , which are $(a\; pm\; sqrt\{a^2\; -\; 4\})/2$. From this, we can see that when , the eigenvalues are complex and at least one has a real component, making the origin unstable with an outwards spiral on the $x,\; y$ plane. Now consider the $z$ plane behavior within the context of this range for $a$. So long as $x$ is smaller than $c$, the $c$ term will keep the orbit close to the $x,\; y$ plane. As the orbit approaches $x$ greater than $c$, the $z$-values begin to climb. As $z$ climbs, though, the $-z$ in the equation for $dx/dt$ stops the growth in $x$.

Fixed points

In order to find the fixed points, the three Rössler equations are set to zero and the ($x$,$y$,$z$) coordinates of each fixed point were determined by solving the resulting equations. This yields the general equations of each of the fixed point coordinates:: $x\; =\; frac\{cpmsqrt\{c^2-4ab\{2\}$: $y\; =\; -left(frac\{cpmsqrt\{c^2-4ab\{2a\}\; ight)$: $z\; =\; frac\{cpmsqrt\{c^2-4ab\{2a\}$

Which in turn can be used to show the actual fixed points for a given set of parameter values:: $left(frac\{c+sqrt\{c^2-4ab\{2\},\; frac\{-c-sqrt\{c^2-4ab\{2a\},\; frac\{c+sqrt\{c^2-4ab\{2a\}\; ight)$: $left(frac\{c-sqrt\{c^2-4ab\{2\},\; frac\{-c+sqrt\{c^2-4ab\{2a\},\; frac\{c-sqrt\{c^2-4ab\{2a\}\; ight)$

As shown in the general plots of the Rössler Attractor above, one of these fixed points resides in the center of the attractor loop and the other lies comparatively removed from the attractor.

Eigenvalues and eigenvectors

The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and eigenvectors. Beginning with the Jacobian: , the eigenvalues can be determined by solving the following cubic:: $-lambda^3+lambda^2(a+x-c)\; +\; lambda(ac-ax-1-z)+x-c+az\; ,$

For the centrally located fixed point ($FP\_\{1\}$), Rössler’s original parameter values of a=0.2, b=0.2, and c=5.7 yield eigenvalues of:: $lambda\_\{1\}=\; 0.0970\; +\; 0.9952i\; ,$: $lambda\_\{2\}=\; 0.0970\; -\; 0.9952i\; ,$: $lambda\_\{3\}=\; -5.6870\; ,$

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.

The eigenvectors corresponding to these eigenvalues are:: : : These eigenvectors have several interesting implications. First, the two eigenvalue/eigenvector pairs ($v\_\{1\}$ and $v\_\{2\}$) 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. This effect is roughly demonstrated with the figure below.

The figure examines the central fixed point eigenvectors. The blue line corresponds to the standard Rössler attractor generated with $a=0.2$, $b=0.2$, and $c=5.7$. The red dot in the center of this attractor is $FP\_\{1\}$. The red line intersecting that fixed point is an illustration of the repulsing plane generated by $v\_\{1\}$ and $v\_\{2\}$. The green line is an illustration of the attracting $v\_\{3\}$. The magenta line is generated by stepping backwards through time from a point on the attracting eigenvector which is slightly above $FP\_\{1\}$ – it illustrates the behavior of points that become completely dominated by that vector. 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 $v\_\{3\}$ and the repelling $v\_\{1\}$ and $v\_\{2\}$ plane. Specifically it implies that a sequence generated from the Rössler equations will begin to loop around $FP\_\{1\}$, start being pulled upwards into the $v\_\{3\}$ vector, creating the upward arm of a curve that bends slightly inward toward the vector before being pushed outward again as it is pulled back towards the repelling plane.

For the outlier fixed point, Rössler’s original parameter values of $a=0.2$, $b=0.2$, and $c=5.7$ yield eigenvalues of:: $lambda\_\{1\}=\; -0.0000046\; +\; 5.4280259i$: $lambda\_\{2\}=\; -0.0000046\; -\; 5.4280259i$: $lambda\_\{3\}=\; 0.1929830$

The eigenvectors corresponding to these eigenvalues are:: : :

Although these eigenvalues and eigenvectors exist in the Rössler attractor, their influence is confined to iterations of the Rössler system whose initial conditions are in the general vicinity of this outlier fixed point. Except in those cases where the initial conditions lie on the attracting plane generated by $lambda\_\{1\}$ and $lambda\_\{2\}$, this influence effectively involves pushing the resulting system towards the general Rössler attractor. As the resulting sequence approaches the central fixed point and the attractor itself, the influence of this distant fixed point (and its eigenvectors) will wane.

Poincaré map

The Poincaré map is constructed by plotting the value of the function every time it passes through a set plane in a specific direction. An example would be plotting the $y,\; z$ value every time it passes through the $x\; =\; 0$ plane where $x$ is changing from negative to positive, commonly done when studying the Lorenz attractor. In the case of the Rössler attractor, the $x\; =\; 0$ plane is uninteresting, as the map always crosses the $x\; =\; 0$ plane at $z\; =\; 0$ due to the nature of the Rössler equations. In the $a=0.1$ plane for $a=0.1$, $b=0.1$, $c=14$, the Poincaré map shows the upswing in $z$ values as $x$ increases, as is to be expected due to the upswing and twist section of the Rössler plot. The number of points in this specific Poincaré plot is infinite, but when a different $c$ value is used, the number of points can vary. For example, with a $c$ value of 4, there is only one point on the Poincaré map, because the function yields a periodic orbit of period one, or if the $c$ value is set to 12.8, there would be six points corresponding to a period six orbit.

Mapping local maxima

In the original paper on the Lorenz Attractor, Edward Lorenz analyzed the local maxima of $z$ against the immediately preceding local maxima. When visualized, the plot resembled the tent map, implying that similar analysis can be used between the map and attractor. For the Rössler attractor, when the $z\_n$ local maximum is plotted against the next local $z$ maximum, $z\_\{n+1\}$, the resulting plot (shown here for $a=0.2$, $b=0.2$, $c=5.7$) is unimodal, resembling a skewed Henon map. Knowing that the Rössler attractor can be used to create a pseudo 1-d map, it then follows to use similar analysis methods. The bifurcation diagram is specifically a useful analysis method.

Variation of parameters

Rössler attractor's behavior is largely a factor of the values of its constant parameters ($a$, $b$, and $c$). 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. Periodic orbits, or "unit cycles," of the Rössler system are defined by the number of loops around the central point that occur before the loops series begins to repeat itself.

Bifurcation diagrams are a common tool for analyzing the behavior of chaotic systems. Bifurcation diagrams for the Rössler attractor are created by iterating through the Rössler ODEs holding two of the parameters constant while conducting a parameter sweep over a range of possible values for the third. The local $x$ maxima for each varying parameter value is then plotted against that parameter value. These maxima are determined after the attractor has reached steady state and any initial transient behaviors have disappeared. This is useful in determining the relationship between periodicity and the selected parameter. Increasing numbers of points in a vertical line on a bifurcation diagram indicates the Rössler attractor behaves chaotically that value of the parameter being examined.

Varying $a$

In order to examine the behavior of the Rössler attractor for different values of $a$, $b$ was fixed at 0.2, $c$ was fixed at 5.7. Numerical examination of attractor's behavior over changing $a$ suggests it has a disproportional influence over the attractor's behavior. Some examples of this relationship include:
* $a\; leq\; 0$: converges to the centrally located fixed point
* $a\; =\; 0.1$: unit cycle of period 1
* $a\; =\; 0.2$: standard parameter value selected by Rössler, chaotic
* $a\; =\; 0.3$: chaotic attractor, significantly more Möbius strip-like (folding over itself).
* $a\; =\; 0.35$: similar to .3, but increasingly chaotic
* $a\; =\; 0.38$: similar to .35, but increasingly chaotic

If $a$ gets even slightly larger than .38, it causes MATLAB to hang. Note this suggests that the practical range of $a$ is very narrow.

Varying $b$

The effect of $a$ on the Rössler attractor’s behavior is best illustrated through a bifurcation diagram. This bifurcation diagram was created with $a=0.2$, $c=5.7$. As shown in the accompanying diagram, as $b$ approaches 0 the attractor approaches infinity (note the upswing for very small values of $b$. Comparative to the other parameters, varying $b$ seems to generate a greater range when period-3 and period-6 orbits will occur. In contrast to $a$ and $c$, higher values of $b$ systems that converge on a period-1 orbit instead of higher level orbits or chaotic attractors.

Varying $c$

The traditional bifurcation diagram for the Rössler attractor is created by varying $c$ with $a=b=.1$. This bifurcation diagram reveals that low values of $c$ are periodic, but quickly become chaotic as $c$ increases. This pattern repeats itself as $c$ increases – there are sections of periodicity interspersed with periods of chaos, although the trend is towards higher order periodic orbits in the periodic sections as $c$ increases. For example, the period one orbit only appears for values of $c$ around 4 and is never found again in the bifurcation diagram. The same phenomena is seen with period three; until $c=12$, period three orbits can be found, but thereafter, they do not appear.

A graphical illustration of the changing attractor over a range of $c$ values illustrates the general behavior seen for all of these parameter analyses – the frequent transitions from ranges of relative stability and periodicity to completely chaotic and back again.

The above set of images illustrates the variations in the post-transient Rössler system as $c$ is varied over a range of values. These images were generated with $a=b=.1$ (a) $c\; =\; 4$, periodic orbit. (b) $c\; =\; 6$, period-2 orbit. (c) $c\; =\; 8.5$, period-4 orbit. (d) $c\; =\; 8.7$, period-8 orbit. (e) $c\; =\; 9$, sparse chaotic attractor. (f) $c\; =\; 12$, period-3 orbit. (g) $c\; =\; 12.6$, period-6 orbit. (h) $c\; =\; 13$, sparse chaotic attractor. (i) $c\; =\; 18$, filled-in chaotic attractor.

Links to other topics

The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the half-twist in the Rössler attractor makes it similar to a Möbius strip.

ee also

* Lorenz attractor
* List of chaotic maps
* Chaos theory
* Dynamical system
* Fractals
* Otto Rössler

References

* Cite journal
author = E. N. Lorenz
year = 1963
title = Deterministic nonperiodic flow
journal = J. Atmos. Sci.
volume = 20
pages = 130–141
doi = 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
* Cite journal
author = O. E. Rössler
title = An Equation for Continuous Chaos
journal = Physics Letters
volume = 57A
issue = 5
pages = 397&ndash;398
year = 1976
* Cite journal
author = O. E. Rössler
title = An Equation for Hyperchaos
journal = Physics Letters
volume = 71A
issue = 2,3
pages = 155&ndash;157
year = 1979
* Cite book
author = Steven H. Strogatz
title = Nonlinear Dynamics and Chaos
publisher = Perseus publishing
year = 1994

External links

* [http://lagrange.physics.drexel.edu/flash/rossray Flash Animation using PovRay]
* [http://to-campos.planetaclix.pt/fractal/lorenz_eng.html Lorenz and Rössler attractors] - Java animation
* [http://scholarpedia.org/article/Rossler_attractor Rössler attractor in Scholarpedia]