Dynamics of nonlinear & chaotic systems Lecture 9: Periodic orbits:

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Dynamics of nonlinear & chaotic systems Lecture 9: Periodic orbits:

a detailed consideration

S. Denisov

Theo I, Institut f¨ ur Physik, Universit¨ at Augsburg

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: a definition

Continuous system: X ˙ = F (X , t), F (X , t + T ) = F (X , t) X = {x ₁ , x ₂ , ..., x _N }

Transformation to map: X _k+1 = f (X _k ), X _k = X (kT )

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: a definition

Periodic orbit of period n: X _n = X ₀ , i. e., X (nT) = X (0)

Fixed point of the map: f _n ( ) = f (f (...f ( )...))

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: an example

Duffing oscillator

Equation: x ¨ = −0.1 ˙ x + x − x ³ + sin(ωt)

Vector: X = {x, p = ˙ x}

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: example

Duffing oscillator: bifurcation diagram

0.75 0.76 0.77 0.78 0.79 0.8 0.81

ω

-2 -1 0 1 2

x

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: example

Duffing oscillator: result

-2 -1 0 1

x

-2 -1 0 1 2 3

p

period-one orbit period-one orbit period-three orbit period-four orbit period-seven orbit

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: example

Duffing oscillator: result

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Periodic orbit: controlling chaos

Ott-Grebogi-Yorke (OGY) method (1990)

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Controlling chaos: an example

Belousov-Zhabotinsky reaction

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Belousov-Zhabotinsky reaction

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Belousov-Zhabotinsky reaction: an experiment

Regular dynamics

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Belousov-Zhabotinsky reaction: an experiment

Chaotic dynamics

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Belousov-Zhabotinsky reaction: another experiment

Control

V. Petrov,V. Gaspar, J. Masere, J. Showalter, Nature 381, 240 (1993).

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

Belousov-Zhabotinsky reaction: another experiment

Control

V. Petrov,V. Gaspar, J. Masere, J. Showalter, Nature 381, 240 (1993).

Dynamics of nonlinear &

chaotic systems Lecture 9:

Periodic orbits:

a detailed consideration

S. Denisov

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