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.)
 
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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