Analysis of the Rossler system Chiara Mocenni The

Analysis of the Rossler system Chiara Mocenni

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

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

Geometry of the phase space In the 3 dimensional phase space this fact induces the stretching mechanism, typical of choatic systems 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 A nonlinear stabilizing term is necessary for maintaining the trajectories confined in a region of the 3 D phase space

The third equation (1/2) 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 (2/2) 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 complete system 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 2/4 b

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

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

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

Linear analysis (2/2) • 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

The limit cycle • S 2 is an unstable spiral and a stable limit cycle (with period 6. 2) is formed through the Hopf bifurcation

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

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

Stretching and Folding (1/2)

Stretching and Folding (2/2)

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

Period doubling and chaotic attractor

The Rossler attractor: a movie