Rossler in Close View
Dedicated Professor:
Hashemi Golpaigani
By:
Javad Razjouyan
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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.
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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.
• 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.
• 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 zdimension.
• 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:
• 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.
• 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 :
• 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.
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:
• 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.
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:
a = 0.1 & b = 0.1
a = 0.1 & b = 0.1 & c = 4.0