Dynamics of nonlinear & chaotic systems Lecture 8: Bifurcations in continuous systems

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Dynamics of nonlinear &

chaotic systems Lecture 8:

Bifurcations in continuous

systems S. Denisov

Dynamics of nonlinear & chaotic systems Lecture 8: Bifurcations in continuous

systems

S. Denisov

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

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systems S. Denisov

The Selkov system (Glycolysis)

½ x ˙ = −x + ayx ² y

˙

y = b − ay − x ² y

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systems S. Denisov

The Selkov system: vector field

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systems S. Denisov

The Selkov system: eigenvalue dynamics

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systems S. Denisov

The Selkov system: eigenvalue dynamics

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systems S. Denisov

Andronov-Hopf Bifurcation Theorem

Let ˙ v = f _a (v) be a family of systems of differential equations in R ⁿ with a fixed point v = 0 for all a. Let denote c (a) ± id (a) a complex conjugate pair of eigenvalues of the matrix Df _a (0) that crosses the imaginary axis at a = 0; that is

c(0) = 0; d (0) 6= 0. Further assume that no other eigenvalue

of Df _a (0) is an integer multiple of id (0). Then a periodic

orbits bifurcates from v = 0. The period of this orbit

approaches 2π/d as orbit approaches 0.

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systems S. Denisov

Hopf bifurcation: supercritical case

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systems S. Denisov

Hopf bifurcation: supercritical case

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systems S. Denisov

Hopf bifurcation: subrcritical case

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systems S. Denisov

Hopf bifurcations: summary

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systems S. Denisov

Saddle-node bifurcation

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systems S. Denisov

Belousov-Zhabotinsky reaction

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systems S. Denisov

Belousov-Zhabotinsky reaction

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systems S. Denisov

Belousov-Zhabotinsky reaction: model

( x ˙ = a − x − _1+x ^4xy

2

˙

y = bx [1 − _1+x ^y

2

]

x is the concentration of I ⁻ , y is the concentration of ClO ₂ ⁻ ;

a, b > 0.

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systems S. Denisov

Belousov-Zhabotinsky reaction:model

Fixed point: x _c = a/5; y _c = 1 + a ² /25 = 1 − x _c ² .

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Belousov-Zhabotinsky reaction:model

Subcritical Hopf bifurcation.

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systems S. Denisov

Belousov-Zhabotinsky reaction:model

Stability diagram

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systems S. Denisov

Dynamics of nonlinear & chaotic systems Lecture 8: Bifurcations in continuous systems

Dynamics of nonlinear & chaotic systems Lecture 8: Bifurcations in continuous

systems

S. Denisov

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

The Selkov system (Glycolysis)

½ x ˙ = −x + ayx 2 y

˙

y = b − ay − x 2 y

The Selkov system: vector field

The Selkov system: eigenvalue dynamics

The Selkov system: eigenvalue dynamics

Andronov-Hopf Bifurcation Theorem

Let ˙ v = f a (v) be a family of systems of differential equations in R n with a fixed point v = 0 for all a. Let denote c (a) ± id (a) a complex conjugate pair of eigenvalues of the matrix Df a (0) that crosses the imaginary axis at a = 0; that is

c(0) = 0; d (0) 6= 0. Further assume that no other eigenvalue

of Df a (0) is an integer multiple of id (0). Then a periodic

orbits bifurcates from v = 0. The period of this orbit

approaches 2π/d as orbit approaches 0.

Hopf bifurcation: supercritical case

Hopf bifurcation: supercritical case

Hopf bifurcation: subrcritical case

Hopf bifurcations: summary

Saddle-node bifurcation

Belousov-Zhabotinsky reaction

Belousov-Zhabotinsky reaction

Belousov-Zhabotinsky reaction: model

( x ˙ = a − x − 1+x 4xy

˙

y = bx [1 − 1+x y

]

x is the concentration of I − , y is the concentration of ClO 2 − ;

a, b > 0.

Belousov-Zhabotinsky reaction:model

Fixed point: x c = a/5; y c = 1 + a 2 /25 = 1 − x c 2 .

Belousov-Zhabotinsky reaction:model

Subcritical Hopf bifurcation.

Belousov-Zhabotinsky reaction:model

Stability diagram

0

½ x ˙ = −x + ayx ² y

y = b − ay − x ² y

Let ˙ v = f _a (v) be a family of systems of differential equations in R ⁿ with a fixed point v = 0 for all a. Let denote c (a) ± id (a) a complex conjugate pair of eigenvalues of the matrix Df _a (0) that crosses the imaginary axis at a = 0; that is

of Df _a (0) is an integer multiple of id (0). Then a periodic

( x ˙ = a − x − _1+x ^4xy

y = bx [1 − _1+x ^y

x is the concentration of I ⁻ , y is the concentration of ClO ₂ ⁻ ;

Fixed point: x _c = a/5; y _c = 1 + a ² /25 = 1 − x _c ² .