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Analysis of the Rossler system Chiara Mocenni

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

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With eigenvalues: It is equivalent to the armonic oscillator: For a>0 the oscillator is undamped Moreover, for 00) The third equation (1/2)

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For xc 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)

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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

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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

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

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

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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)

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

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

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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

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Stretching and Folding (1/2)

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Stretching and Folding (2/2)

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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

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Period doubling and chaotic attractor

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The Rossler attractor: a movie