# Roessler's model

The equations:

dx/dt = -y -z

dy/dt = x + ay

dz/dt = b + z(x - c)

where    a=b=1/5,  c=5.7

The numeric integration of the system was carried out using an adaptive stepsize controlled runge-kutta solver.

The   Attractor  for t=500.

Poincare sections taken at different angles around the  Z-axis, and the corresponding return maps.

(here t=20000.)

 ps 0 ps 45 ps 90 ps 135 ps 180 ps 225 ps 270 ps 315 rm  0 rm  45 rm  90 rm  135 rm  180 rm  225 rm  270 rm  315

First four prime periodic orbits and their stabilities

The Poincare section is chosen to be in the positive part of the plane (Y,Z).
Prime periodic orbits of length N correspond to N different fixpoints in the N-return maps. These fixed points are defined by the intersection of the curves with the diagonal (Y(t+N)=Y(t)). On the figures the fixed points that correspond to the same periodic orbit are labeled with the same capitol letter. Also indicated in the N-return maps  are the regions with an appropriate symbolic dynamics. The precise location of the periodic orbits and the stability eigenvalues were obtained by using a Newton-routine.

N=1;
return map  and the corresponding  1-cycle periodic orbit  (A) with symbolic dynamics: 1.

Period of the orbit:T=5.88108845586 ;

Poincare section point: X=0;  Y=6.09176831742;  Z=1.29973195919;

eigenvalues of the Jacobian:  1.000000003, -2.403953527, -0.1836169064e-10
liapunov exponent:  0.149141556;

N=2;
return map  and the corresponding 2-cycle  periodic orbit  (B) with symbolic dynamics: 1,0.

Period of the orbit:T=11.7586260717;

One of the Poincare section points (in the 1,0 itinerary):
X=0;   Y=6.91498284608;  Z=0.0757168639342;

eigenvalues of the Jacobian:   1.000000000, -3.512006980, 0.8290005176e-12
liapunov exponent:  0.10683116;

N=3;
return map  and the corresponding 3-cycles:

periodic orbit  (C) with symbolic dynamics: 1,0,0.

Period of the orbit:T=17.5157912663;

One of the Poincare section points (in the 1,0,0 itinerary):
X= 0;   Y=7.54996784577;   Z=0.140432840371;

eigenvalues of the Jacobian:    1.000000019,  -2.341918959,  0.8112502046e-11

liapunov exponent:  0.048583055;

periodic orbit  (D) with symbolic dynamics 1,0,1.

Period of the orbit:T=17.5958658156;

One of the Poincare section points (in the 1,0,1 itinerary):
X=0;  Y=7.29442965395;   Z=0.556169637263;

eigenvalues of the Jacobian:    1.000000001, 5.344908108, 0.1349406670e-10

liapunov exponent:   0.09525785;

N=4;
return map  and the corresponding 4-cycle  periodic orbit  (E) with symbolic dynamics 1,0,1,1.

Period of the orbit:T= 23.508557584;

One of the Poincare section points (in the 1,0,1,1 segment):
X=0;  Y=7.08997393429;  Z= 0 .675113965985;

eigenvalues of the Jacobian:  1.000000001, -16.69674069, -.7258300597e-10

liapunov exponent:  0.119752712;

created by Gabor Simon, Ph.D. student (fall 1999); Feb 7 2000, edits by P. Cvitanovic Sep 2 2000.
E-mail:  simon@nbi.dk