Хаостық Рёсслер жүйесінің аттракторы

Attractor for chaotic Rössler system

The Rössler attractor '/r//ɒ//s//l//ər/ is the attractor for the Rössler system, a system of three non linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy pulling machine. 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 states the Rössler attractor was intended to behave similarly to the Lorenz attractor, but also be easier to analyze qualitatively. How Rössler discovered this set of equations was investigated by Letellier and Messager.

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 value every time it passes through the plane where is changing from negative to positive, commonly done when studying the Lorenz attractor. In the case of the Rössler attractor, the plane is uninteresting, as the map always crosses the plane at due to the nature of the Rössler equations. In the plane for , , , the Poincaré map shows the upswing in values as 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 value is used, the number of points can vary. For example, with a 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 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 be the n th maximum of its x coordinate. Then scatterplot is almost a curve, meaning that knowing one can almost exactly predict .

In the original paper on the Lorenz Attractor, Edward Lorenz analyzed the local maxima of 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 local maximum is plotted against the next local maximum, , the resulting plot (shown here for , , ) 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 , , and 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, is fixed at 0.2, is fixed at 5.7 and changes. As shown in the accompanying diagram, as approaches 0 the attractor approaches infinity (note the upswing for very small values of ). Comparative to the other parameters, varying generates a greater range when period 3 and period 6 orbits will occur. In contrast to and , higher values of converge to period 1, not to a chaotic state.

Here, and changes. The bifurcation diagram reveals that low values of are periodic, but quickly become chaotic as increases. This pattern repeats itself as increases – there are sections of periodicity interspersed with periods of chaos, and the trend is towards higher period orbits as increases. For example, the period one orbit only appears for values of around 4 and is never found again in the bifurcation diagram. The same phenomenon is seen with period three; until , period three orbits can be found, but thereafter, they do not appear. A graphical illustration of the changing attractor over a range of values illustrates the general behavior seen for all of these parameter analyses – the frequent transitions between periodicity and aperiodicity. The above set of images illustrates the variations in the post transient Rössler system as is varied over a range of values. These images were generated with , period 1 orbit. , period 2 orbit. , period 4 orbit. , period 8 orbit. , sparse chaotic attractor. , period 3 orbit. , period 6 orbit. , sparse chaotic attractor. , period 5 orbit. , filled in chaotic attractor.

The attractor is filled densely with periodic orbits: solutions for which there exists a nonzero value of such that These interesting solutions can be numerically derived using Newton's method. Periodic orbits are the roots of the function , where is the evolution by time and is the identity. 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. This lack of degeneracy likely stems from the problem's lack of symmetry. The attractor can be dissected into easier to digest invariant manifolds: 1D periodic orbits and the 2D stable and unstable manifolds of periodic orbits. These invariant manifolds are a natural skeleton of the attractor, just as rational numbers are to the real numbers. For the purposes of dynamical systems theory, one might be interested in topological invariants of these manifolds. Periodic orbits are copies of embedded in , so their topological properties can be understood with knot theory. The periodic orbits with winding numbers 1 and 2 form a Hopf link, showing that no diffeomorphism can separate these orbits.

The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the half twist that occurs in the Rössler attractor only affects a part of the attractor. Rössler showed that his attractor was in fact the combination of a "normal band" and a Möbius strip.