Rossler in Close View

1. Rossler in Close View Dedicated Professor: Hashemi Golpaigani By: Javad Razjouyan

2. Otto E. Rossler, born on May 20, 1940, to Austrian father in Berlin. Humanistic high-school education (Greek-Latin style), Majored and state exam in medicine, immunological dissertation, Dr. med. 1966. • 1966 Postdoc, Max Planck Institute for behavioral Physiology, Seewiesen, Bavaria. Cooperation with Konrad Lorenz. • 1969 Visiting Appointment Award, Center for Theoretical Biology, State Universitry of new York at Buffalo, New York State. Cooperation with Robert Rosen. • 1969 Professor for Theoretical Biochemistry at the University of Tübingen. • 1973 Habilitation ("Privat-Docent") for Theoretical Biochemistry, University of Tuebingen, 1976 University Docent (tenured). • 1981 Visiting Professor of Mathematics, Guelph University, Canada. • 1983 Visiting Professor of Nonlinear Studies, Center for Nonlinear Studies of the University of California, Los Alamos, New Mexico (non-military). • 1992 Visiting Professor of Chemical Engineering, University of Virginia, Charlottesville, Virginia. • 1993 Visiting Professor of Theoretical Physics, Lyngby University, Denmark. • 1994 Professor of Chemistry by decree. • 1995 Visiting Professor of Complexity Research, Santa Fe Institute, New Mexico. • 1995 Award of the "Systems Research Foundation", Canada. • Author of about 300 scientific papers on : Biogenesis, deductive biology, origin of language, differentiable automata, bacterial brain, chaotic attractors, dripping faucet, heart chaos (with Reimara Rossler), hyperchaos, nowhere-differentiable attractors (with Ichiro Tsuda), flare attractors, endophysics, micro relativity, Platonic computers, micro constructivism, recursive evolution, limitology, interface theory, artificial universes, the hypertext encyclopedia, Lampsacus hometown of all persons, blind-sight experiments in physics, world-change technology.

3. The Rössler attractor 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.

4. 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.

5. 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.

6. The defining equations are: • 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.

7. 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: • The stability in the x,y plane can then be found by calculating the eigenvalues of the Jacobian : • which are :

8. From this, we can see that when 0 < a < 2, 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.

9. 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:

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

11. Eigen values 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:

16. a = 0.1 & b = 0.1

18. a = 0.1 & b = 0.1 & c = 4.0