Dynamics of nonlinear & chaotic systems Lecture 11: Controlling chaos: Ott-Grebogi- Yorke (OGY) method

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Dynamics of nonlinear &

chaotic systems Lecture 11:

Controlling chaos:

Ott-Grebogi- Yorke (OGY) method S. Denisov

Dynamics of nonlinear & chaotic systems Lecture 11: Controlling chaos:

Ott-Grebogi-Yorke (OGY) method

S. Denisov

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

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Controlling chaos:

Poincare section

Continuous system:

X ˙ = f (X , t; p), f (X , t + T ; p ) = f (X , t; p ); X = {x ₁ , x ₂ , x ₃ } p is a control parameter

Map: Z _n+1 = F (Z n ; p)

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Controlling chaos:

Periodic orbit: a fixed pint of the map

Periodic orbit of period one: Z ^∗ = F (Z ^∗ , p) Lianerization at the vicinity of Z ^∗ :

(Z _n+1 − Z ^∗ ) = A · (Z n − Z ^∗ ) + B · (p − p ₀ ) where A = ∂F /∂Z | Z

^∗

,p

₀

and B = ∂F /∂p | Z

^∗

,p

₀

.

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Controlling chaos:

Periodic orbit: the idea of control

p _n − p ₀ = −K ^T · (Z _n − Z ^∗ ) where K is a constant control vector

By substituting it into the linearized deviation equation, we get Z _n+1 − Z ^∗ = (A − BK ^T ) · (Z n − Z ^∗ ) = Σ · (Z n − Z ^∗ )

But what about vector K ? How to choose it?

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Controlling chaos:

Control vector

Matrix A has n eigenvalues, {λ i } _i=1,...,N , among them n _u unstable and n s stable ones, |λ ^l _u | > 1, l = 1, ..., n u , and

|λ ^k _s | < 1, k = 1, ..., n _s , n _s + n _u = N.

Let Σ has n _s eigenvalues identical to n _s stable eigenvalies of A.

The rest of eigenvalues, n r , are set to zero.

From the expression Σ = A − BK ^T one could determine control vector K

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Controlling chaos:

Practical realization

i) How to catch UPO? → reccurence method ii) How to calculate A and B? → least-squares method

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Controlling chaos:

An example: Duffing oscillator

¨

x = −α x ˙ + x − x ³ + sin(ωt)

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Controlling chaos:

An example: Duffing oscillator

¨

x = −α x ˙ + x − x ³ + sin(ωt)

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Controlling chaos:

An example: Duffing oscillator

¨

x = −α x ˙ + x − x ³ + sin(ωt)

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Controlling chaos:

An example: Duffing oscillator

¨

x = −α x ˙ + x − x ³ + sin(ωt)

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Controlling chaos:

Another example: magnetoelastic ribbon

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Controlling chaos:

Another example: magnetoelastic ribbon

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Controlling chaos:

Another example: magnetoelastic ribbon

W. L. Ditto, S. N. Rauseo, and M. L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65, 3211 (1990).

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Controlling chaos:

Another example: magnetoelastic ribbon

W. L. Ditto, S. N. Rauseo, and M. L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65, 3211 (1990).

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Controlling chaos:

Another example: thermal convection loop

J. Singer, Y.-Z. Wang, and Haim H. Bau, Controlling a chaotic system, Phys. Rev. Lett. 66, 1123 (1991).

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Controlling chaos:

Another example: thermal convection loop

J. Singer, Y.-Z. Wang, and Haim H. Bau, Controlling a chaotic system, Phys. Rev. Lett. 66, 1123 (1991).

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Controlling chaos:

Another example: solid-state laser

R. Roy, T. W. Murphy, T. D. Maier, and Z. Gillis, Dynamical control of a chaotic laser, Phys. Rev. Lett. 68, 1259 (1992).

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Controlling chaos:

Another example: solid-state laser

R. Roy, T. W. Murphy, T. D. Maier, and Z. Gillis, Dynamical control of a chaotic laser, Phys. Rev. Lett. 68, 1259 (1992).

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Controlling chaos:

Another example: solid-state laser

R. Roy, T. W. Murphy, T. D. Maier, and Z. Gillis, Dynamical control of a chaotic laser, Phys. Rev. Lett. 68, 1259 (1992).

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Controlling chaos:

Another example: diode resonator

E. R. Hunt, Stabilizing high-period orbits in a chaotic system, Phys. Rev.

Lett. 67, 1953 (1991).

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Controlling chaos:

Another example: diode resonator

E. R. Hunt, Stabilizing high-period orbits in a chaotic system, Phys. Rev.

Lett. 67, 1953 (1991).

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Controlling chaos:

Another example: diode resonator

E. R. Hunt, Stabilizing high-period orbits in a chaotic system, Phys. Rev.

Lett. 67, 1953 (1991).

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Controlling chaos:

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

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Controlling chaos:

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

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Controlling chaos:

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

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Controlling chaos:

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

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Controlling chaos:

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Nonlinear Ddynamics of a flow-focusing bubble generator, Phys. Rev. Lett. 94, 234502 (2005).

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Controlling chaos:

A new candidate: bubble generator

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Controlling chaos:

A new candidate: bubble generator

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Dynamics of nonlinear & chaotic systems Lecture 11: Controlling chaos: Ott-Grebogi- Yorke (OGY) method

Dynamics of nonlinear & chaotic systems Lecture 11: Controlling chaos:

Ott-Grebogi-Yorke (OGY) method

S. Denisov

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

Poincare section

Continuous system:

X ˙ = f (X , t; p), f (X , t + T ; p ) = f (X , t; p ); X = {x 1 , x 2 , x 3 } p is a control parameter

Map: Z n+1 = F (Z n ; p)

Periodic orbit: a fixed pint of the map

Periodic orbit of period one: Z ∗ = F (Z ∗ , p) Lianerization at the vicinity of Z ∗ :

(Z n+1 − Z ∗ ) = A · (Z n − Z ∗ ) + B · (p − p 0 ) where A = ∂F /∂Z | Z

,p

and B = ∂F /∂p | Z

,p

.

Periodic orbit: the idea of control

p n − p 0 = −K T · (Z n − Z ∗ ) where K is a constant control vector

By substituting it into the linearized deviation equation, we get Z n+1 − Z ∗ = (A − BK T ) · (Z n − Z ∗ ) = Σ · (Z n − Z ∗ )

But what about vector K ? How to choose it?

Control vector

Matrix A has n eigenvalues, {λ i } i=1,...,N , among them n u unstable and n s stable ones, |λ l u | > 1, l = 1, ..., n u , and

|λ k s | < 1, k = 1, ..., n s , n s + n u = N.

Let Σ has n s eigenvalues identical to n s stable eigenvalies of A.

The rest of eigenvalues, n r , are set to zero.

From the expression Σ = A − BK T one could determine control vector K

Practical realization

i) How to catch UPO? → reccurence method ii) How to calculate A and B? → least-squares method

An example: Duffing oscillator

¨

x = −α x ˙ + x − x 3 + sin(ωt)

An example: Duffing oscillator

¨

x = −α x ˙ + x − x 3 + sin(ωt)

An example: Duffing oscillator

¨

x = −α x ˙ + x − x 3 + sin(ωt)

An example: Duffing oscillator

¨

x = −α x ˙ + x − x 3 + sin(ωt)

Another example: magnetoelastic ribbon

Another example: magnetoelastic ribbon

Another example: magnetoelastic ribbon

W. L. Ditto, S. N. Rauseo, and M. L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65, 3211 (1990).

Another example: magnetoelastic ribbon

W. L. Ditto, S. N. Rauseo, and M. L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65, 3211 (1990).

Another example: thermal convection loop

J. Singer, Y.-Z. Wang, and Haim H. Bau, Controlling a chaotic system, Phys. Rev. Lett. 66, 1123 (1991).

Another example: thermal convection loop

J. Singer, Y.-Z. Wang, and Haim H. Bau, Controlling a chaotic system, Phys. Rev. Lett. 66, 1123 (1991).

Another example: solid-state laser

R. Roy, T. W. Murphy, T. D. Maier, and Z. Gillis, Dynamical control of a chaotic laser, Phys. Rev. Lett. 68, 1259 (1992).

Another example: solid-state laser

R. Roy, T. W. Murphy, T. D. Maier, and Z. Gillis, Dynamical control of a chaotic laser, Phys. Rev. Lett. 68, 1259 (1992).

Another example: solid-state laser

R. Roy, T. W. Murphy, T. D. Maier, and Z. Gillis, Dynamical control of a chaotic laser, Phys. Rev. Lett. 68, 1259 (1992).

Another example: diode resonator

E. R. Hunt, Stabilizing high-period orbits in a chaotic system, Phys. Rev.

Lett. 67, 1953 (1991).

Another example: diode resonator

E. R. Hunt, Stabilizing high-period orbits in a chaotic system, Phys. Rev.

Lett. 67, 1953 (1991).

Another example: diode resonator

E. R. Hunt, Stabilizing high-period orbits in a chaotic system, Phys. Rev.

Lett. 67, 1953 (1991).

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Oscillations with uniquely long periods in a microfluidic bubble generator, Nature Physics 1, 168 (2005).

A new candidate: bubble generator

P. Garstecki, M. J. Fuerstman, and G.M. Whitesides, Nonlinear Ddynamics of a flow-focusing bubble generator, Phys. Rev. Lett. 94, 234502 (2005).

A new candidate: bubble generator

A new candidate: bubble generator

X ˙ = f (X , t; p), f (X , t + T ; p ) = f (X , t; p ); X = {x ₁ , x ₂ , x ₃ } p is a control parameter

Map: Z _n+1 = F (Z n ; p)

Periodic orbit of period one: Z ^∗ = F (Z ^∗ , p) Lianerization at the vicinity of Z ^∗ :

(Z _n+1 − Z ^∗ ) = A · (Z n − Z ^∗ ) + B · (p − p ₀ ) where A = ∂F /∂Z | Z

p _n − p ₀ = −K ^T · (Z _n − Z ^∗ ) where K is a constant control vector

By substituting it into the linearized deviation equation, we get Z _n+1 − Z ^∗ = (A − BK ^T ) · (Z n − Z ^∗ ) = Σ · (Z n − Z ^∗ )

Matrix A has n eigenvalues, {λ i } _i=1,...,N , among them n _u unstable and n s stable ones, |λ ^l _u | > 1, l = 1, ..., n u , and

|λ ^k _s | < 1, k = 1, ..., n _s , n _s + n _u = N.

Let Σ has n _s eigenvalues identical to n _s stable eigenvalies of A.

From the expression Σ = A − BK ^T one could determine control vector K

x = −α x ˙ + x − x ³ + sin(ωt)

x = −α x ˙ + x − x ³ + sin(ωt)

x = −α x ˙ + x − x ³ + sin(ωt)

x = −α x ˙ + x − x ³ + sin(ωt)