Rössler attractor

In order to find the fixed points, the three Rössler equations are set to zero and the (${\displaystyle x}$,${\displaystyle y}$,${\displaystyle z}$) coordinates of each fixed point were determined by solving the resulting equations. This yields the general equations of each of the fixed point coordinates:[7]

${\displaystyle {\begin{cases}x={\frac {c\pm {\sqrt {c^{2}-4ab}}}{2}}\\y=-\left({\frac {c\pm {\sqrt {c^{2}-4ab}}}{2a}}\right)\\z={\frac {c\pm {\sqrt {c^{2}-4ab}}}{2a}}\end{cases}}}$

Which in turn can be used to show the actual fixed points for a given set of parameter values:

${\displaystyle \left({\frac {c+{\sqrt {c^{2}-4ab}}}{2}},{\frac {-c-{\sqrt {c^{2}-4ab}}}{2a}},{\frac {c+{\sqrt {c^{2}-4ab}}}{2a}}\right)}$

${\displaystyle \left({\frac {c-{\sqrt {c^{2}-4ab}}}{2}},{\frac {-c+{\sqrt {c^{2}-4ab}}}{2a}},{\frac {c-{\sqrt {c^{2}-4ab}}}{2a}}\right)}$

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 relatively far from the attractor.

The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and eigenvectors. Beginning with the Jacobian:

${\displaystyle {\begin{pmatrix}0&-1&-1\\1&a&0\\z&0&x-c\\\end{pmatrix}}}$

the eigenvalues can be determined by solving the following cubic:

${\displaystyle -\lambda ^{3}+\lambda ^{2}(a+x-c)+\lambda (ac-ax-1-z)+x-c+az=0\,}$

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:

${\displaystyle \lambda _{1}=0.0971028+0.995786i\,}$

${\displaystyle \lambda _{2}=0.0971028-0.995786i\,}$

${\displaystyle \lambda _{3}=-5.68718\,}$

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:

${\displaystyle v_{1}={\begin{pmatrix}0.7073\\-0.07278-0.7032i\\0.0042-0.0007i\\\end{pmatrix}}}$

${\displaystyle v_{2}={\begin{pmatrix}0.7073\\0.07278+0.7032i\\0.0042+0.0007i\\\end{pmatrix}}}$

${\displaystyle v_{3}={\begin{pmatrix}0.1682\\-0.0286\\0.9853\\\end{pmatrix}}}$

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

Rössler attractor with ${\displaystyle a=0.2}$, ${\displaystyle b=0.2}$, ${\displaystyle c=5.7}$

These eigenvectors have several interesting implications. First, the two eigenvalue/eigenvector pairs (${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$) 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 ${\displaystyle a=0.2}$, ${\displaystyle b=0.2}$, and ${\displaystyle c=5.7}$. The red dot in the center of this attractor is ${\displaystyle FP_{1}}$. The red line intersecting that fixed point is an illustration of the repulsing plane generated by ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$. The green line is an illustration of the attracting ${\displaystyle v_{3}}$. The magenta line is generated by stepping backwards through time from a point on the attracting eigenvector which is slightly above ${\displaystyle FP_{1}}$ – 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 ${\displaystyle v_{3}}$ and the repelling ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$ plane. Specifically it implies that a sequence generated from the Rössler equations will begin to loop around ${\displaystyle FP_{1}}$, start being pulled upwards into the ${\displaystyle v_{3}}$ 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 ${\displaystyle a=0.2}$, ${\displaystyle b=0.2}$, and ${\displaystyle c=5.7}$ yield eigenvalues of:

${\displaystyle \lambda _{1}=-0.0000046+5.4280259i}$

${\displaystyle \lambda _{2}=-0.0000046-5.4280259i}$

${\displaystyle \lambda _{3}=0.1929830}$

The eigenvectors corresponding to these eigenvalues are:

${\displaystyle v_{1}={\begin{pmatrix}0.0002422+0.1872055i\\0.0344403-0.0013136i\\0.9817159\\\end{pmatrix}}}$

${\displaystyle v_{2}={\begin{pmatrix}0.0002422-0.1872055i\\0.0344403+0.0013136i\\0.9817159\\\end{pmatrix}}}$

${\displaystyle v_{3}={\begin{pmatrix}0.0049651\\-0.7075770\\0.7066188\\\end{pmatrix}}}$

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 ${\displaystyle \lambda _{1}}$ and ${\displaystyle \lambda _{2}}$, 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 for Rössler attractor with ${\displaystyle a=0.1}$, ${\displaystyle b=0.1}$, ${\displaystyle c=14}$

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 ${\displaystyle y,z}$ value every time it passes through the ${\displaystyle x=0}$ plane where ${\displaystyle x}$ is changing from negative to positive, commonly done when studying the Lorenz attractor. In the case of the Rössler attractor, the ${\displaystyle x=0}$ plane is uninteresting, as the map always crosses the ${\displaystyle x=0}$ plane at ${\displaystyle z=0}$ due to the nature of the Rössler equations. In the ${\displaystyle x=0.1}$ plane for ${\displaystyle a=0.1}$, ${\displaystyle b=0.1}$, ${\displaystyle c=14}$, the Poincaré map shows the upswing in ${\displaystyle z}$ values as ${\displaystyle x}$ 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 ${\displaystyle c}$ value is used, the number of points can vary. For example, with a ${\displaystyle c}$ 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 ${\displaystyle c}$ value is set to 12.8, there would be six points corresponding to a period six orbit.

The Lorenz map is the relation between successive maxima of a coordinate in a trajectory. Consider a trajectory on the attractor, and let ${\displaystyle x_{max}(n)}$ be the n-th maximum of its x-coordinate. Then ${\displaystyle x_{max}(n)}$-${\displaystyle x_{max}(n+1)}$ scatterplot is almost a curve, meaning that knowing ${\displaystyle x_{max}(n)}$ one can almost exactly predict ${\displaystyle x_{max}(n+1)}$.[8]

Lorenz map for Rössler attractor with a = 0.2, b = 0.2, c = 5.

${\displaystyle Z_{n}}$ vs. ${\displaystyle Z_{n+1}}$

In the original paper on the Lorenz Attractor,[9] Edward Lorenz analyzed the local maxima of ${\displaystyle z}$ 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 ${\displaystyle z_{n}}$ local maximum is plotted against the next local ${\displaystyle z}$ maximum, ${\displaystyle z_{n+1}}$, the resulting plot (shown here for ${\displaystyle a=0.2}$, ${\displaystyle b=0.2}$, ${\displaystyle c=5.7}$) is unimodal, resembling a skewed Hénon 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 a particularly useful analysis method.

Rössler attractor's behavior is largely a factor of the values of its constant parameters ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$. 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 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. Chaotic regions are indicated by filled-in regions of the plot.

Here, ${\displaystyle b}$ is fixed at 0.2, ${\displaystyle c}$ is fixed at 5.7 and ${\displaystyle a}$ changes. Numerical examination of the attractor's behavior over changing ${\displaystyle a}$ suggests it has a disproportional influence over the attractor's behavior. The results of the analysis are:

• ${\displaystyle a\leq 0}$: Converges to the centrally located fixed point
• ${\displaystyle a=0.1}$: Unit cycle of period 1
• ${\displaystyle a=0.2}$: Standard parameter value selected by Rössler, chaotic
• ${\displaystyle a=0.3}$: Chaotic attractor, significantly more Möbius strip-like (folding over itself).
• ${\displaystyle a=0.35}$: Similar to .3, but increasingly chaotic
• ${\displaystyle a=0.38}$: Similar to .35, but increasingly chaotic.

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

Here, ${\displaystyle a}$ is fixed at 0.2, ${\displaystyle c}$ is fixed at 5.7 and ${\displaystyle b}$ changes. As shown in the accompanying diagram, as ${\displaystyle b}$ approaches 0 the attractor approaches infinity (note the upswing for very small values of ${\displaystyle b}$. Comparative to the other parameters, varying ${\displaystyle b}$ generates a greater range when period-3 and period-6 orbits will occur. In contrast to ${\displaystyle a}$ and ${\displaystyle c}$, higher values of ${\displaystyle b}$ converge to period-1, not to a chaotic state.

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

Here, ${\displaystyle a=b=0.1}$ and ${\displaystyle c}$ changes. The bifurcation diagram reveals that low values of ${\displaystyle c}$ are periodic, but quickly become chaotic as ${\displaystyle c}$ increases. This pattern repeats itself as ${\displaystyle c}$ increases – there are sections of periodicity interspersed with periods of chaos, and the trend is towards higher-period orbits as ${\displaystyle c}$ increases. For example, the period one orbit only appears for values of ${\displaystyle c}$ around 4 and is never found again in the bifurcation diagram. The same phenomenon is seen with period three; until ${\displaystyle c=12}$, period three orbits can be found, but thereafter, they do not appear.

A graphical illustration of the changing attractor over a range of ${\displaystyle c}$ values illustrates the general behavior seen for all of these parameter analyses – the frequent transitions between periodicity and aperiodicity.

Varying c

c = 4, period 1

c = 6, period 2

c = 8.5, period 4

c = 8.7, period 8

c = 9, chaotic

c = 12, period 3

c = 12.6, period 6

c = 13, chaotic

c = 18, chaotic

The above set of images illustrates the variations in the post-transient Rössler system as ${\displaystyle c}$ is varied over a range of values. These images were generated with ${\displaystyle a=b=.1}$.

• ${\displaystyle c=4}$, period-1 orbit.
• ${\displaystyle c=6}$, period-2 orbit.
• ${\displaystyle c=8.5}$, period-4 orbit.
• ${\displaystyle c=8.7}$, period-8 orbit.
• ${\displaystyle c=9}$, sparse chaotic attractor.
• ${\displaystyle c=12}$, period-3 orbit.
• ${\displaystyle c=12.6}$, period-6 orbit.
• ${\displaystyle c=13}$, sparse chaotic attractor.
• ${\displaystyle c=18}$, filled-in chaotic attractor.