1. Analysis of the Rossler system Chiara Mocenni

2. Considering only the first two equations and ssuming small z The Rossler equations

3. With eigenvalues: It is equivalent to the armonic oscillator: For a>0 the oscillator is undamped Moreover, for 0<a<2 the origin is an unstable spiral The reduced system

4. But it is non sufficient by itself to produce the chaotic behavior because in the linear case this instability is not compensated by any other mechanism and the system is globally unstable In the 3 dimensional phase space this fact induces the stretching mechanism, typical of choatic systems A nonlinear stabilizing term is necessary for maintaining the trajectories confined in a region of the 3D phase space Geometry of the phase space

5. The folding is produced by the term associated to parameter c in the third equation Consider only the equation for z It has the steady state (assume b>0) The third equation (1/2)

6. For x<c the coefficient of z in the equation is negative and the (positive) steady state z * is stable For x>c the coefficient of z in the equation is positive and the system diverges. For positive b the instability is developing along increasing z. The third equation (2/2)

7. Assume fixed b and c and vary parameter a b=2 c=4 Assume fixed b and c and vary parameter a b=2 c=4 The system has two steady states, that exist for a ≠ 0 a < c 2 /4b The system has two steady states, that exist for a ≠ 0 a < c 2 /4b The complete system

8. Assume 0 < a < c 2 /4b Let S 1 be the steady state (x 1,y 1,z 1 ) Let P 2 be the steady state (x 2,y 2, z2 ) Let A the Jacobian matrix Assume 0 < a < c 2 /4b Let S 1 be the steady state (x 1,y 1,z 1 ) Let P 2 be the steady state (x 2,y 2, z2 ) Let A the Jacobian matrix Steady states and linear analysis

9. For a = c 2 /4b S 1 and P 2 coincide and for a > c 2 /4b disappear The steady states

10. Steady state S 1 is a saddle because it has 3 real eigenvalues with opposite signs Linear analysis (1/2) Saddle-node bifurcation

11. Steady state S 2 is a stable spiral for a < 0.125. For a=0.125 a Hopf bifurcation occurs From this value S 2 is an unstable spiral Steady state S 2 is a stable spiral for a < 0.125. For a=0.125 a Hopf bifurcation occurs From this value S 2 is an unstable spiral Linear analysis (2/2)

12. The limit cycle S 2 is an unstable spiral and a stable limit cycle (with period 6.2) is formed through the Hopf bifurcation

13. The limit cycle is then destabilized through a PERIOD DOUBLING bifurcation of cycles Bifurcations of cycles

14. Explaining the mechanisms This fact is due to the mechanism shown by equation 3, that is activated for x = c = 4 The limit cycle grows for increasing a, and when x = 4 the variable z is destabilized, producing a divergence of the trajectories along z At this point a stabilizing mechanism on x is induced by equation 1 and leading again z to enter the stable region The result of this combination of mechanisms induces that a double period limit cycle is formed This fact is due to the mechanism shown by equation 3, that is activated for x = c = 4 The limit cycle grows for increasing a, and when x = 4 the variable z is destabilized, producing a divergence of the trajectories along z At this point a stabilizing mechanism on x is induced by equation 1 and leading again z to enter the stable region The result of this combination of mechanisms induces that a double period limit cycle is formed

15. Stretching and Folding (1/2)

16. Stretching and Folding (2/2)

17. The complete picture Parameter a Stable node/spiral Stable limit cycle Stable limit cycle of period 2 Stable limit cycle of period 4 Stable limit cycle of period 8 CHAOS Unstable system

18. Period doubling and chaotic attractor

19. The Rossler attractor: a movie