The Rössler attractor 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. 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. An orbit within the attractor follows an outward spiral close to the 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 -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. Definition The defining equations of the Rössler system are: Rössler studied the chaotic attractor with , , and , though properties of , , and have been more commonly used since. Another line of the parameter space was investigated using the topological analysis. It corresponds to , , and was chosen as the bifurcation parameter. How Rössler discovered this set of equations was investigated by Letellier and Messager. An analysis Some of the Rössler attractor's elegance is due to two of its equations being linear; setting , allows examination of the behavior on the plane The stability in the plane can then be found by calculating the eigenvalues of the Jacobian , which are . From this, we can see that when , the eigenvalues are complex and both have a positive real component, making the origin unstable with an outwards spiral on the plane. Now consider the plane behavior within the context of this range for . So as long as is smaller than , the term will keep the orbit close to the plane. As the orbit approaches greater than , the -values begin to climb. As climbs, though, the in the