Adaptive delayed feedback control to stabilize in-phase - - PowerPoint PPT Presentation

Adaptive delayed feedback control to stabilize in-phase synchronization in complex oscillator networks

Viktor Novičenko

September 2019, Rostock

Synchronous behavior can be desirable or harmful. Power grids Parkinson’s disease, essential tremor Pedestrians on a bridge Cardiac pacemaker cells Internal circadian clock The ability to control synchrony in oscillatory network covers a wide range of real-world applications.

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function

G (x, x) = 0

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with

˙ x = f (x) |f (x) − fi (x)| ∼ ε G (x, x) = 0

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t) G (x, x) = 0

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators under delayed feedback control:

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)]

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators under delayed feedback control:

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)]

with time delays |τi − Ti| ∼ |τi − T| ∼ ε

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators under delayed feedback control:

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)]

with time delays |τi − Ti| ∼ |τi − T| ∼ ε phase reduction for system with time delay

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators under delayed feedback control:

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)]

with time delays |τi − Ti| ∼ |τi − T| ∼ ε phase reduction for system with time delay

˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi)

• V. Novičenko: Delayed feedback control of synchronization in weakly coupled
• scillator networks, Phys. Rev. E 92, 022919 (2015)

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators under delayed feedback control:

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)]

with time delays |τi − Ti| ∼ |τi − T| ∼ ε phase reduction for system with time delay

˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi)

effective coupling strength effective frequency

εeff = εα (KC) ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1]

• V. Novičenko: Delayed feedback control of synchronization in weakly coupled
• scillator networks, Phys. Rev. E 92, 022919 (2015)

Weakly coupled near-identical limit cycle oscillators without control:

˙ xi = fi (xi) + ε

N

• j=1

aijG (xj, xi)

coupling function “central” oscillator with has a stable periodic solution and a phase response curve

˙ x = f (x) |f (x) − fi (x)| ∼ ε ξ (t + T) = ξ (t) z (t + T) = z (t)

phase reduction

G (x, x) = 0 ˙ ψi = ωi + ε

N

• j=1

aijh (ψj − ψi)

with frequency and coupling function

ωi = Ωi − Ω h (χ) = 1 T

T

ˆ

• zT s

Ω

• · G
• ξ

s + χ Ω

• , ξ

s Ω

• ds

Weakly coupled near-identical limit cycle oscillators under delayed feedback control:

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)]

with time delays |τi − Ti| ∼ |τi − T| ∼ ε phase reduction for system with time delay

˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi)

effective coupling strength effective frequency

εeff = εα (KC) ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] C =

T

ˆ

• zT (s) · D2f (ξ (s) , 0)

[∇g (ξ (s))]T · ˙ ξ (s)

• ds

the function and the constant

α (KC) = (1 + KC)−1

• V. Novičenko: Delayed feedback control of synchronization in weakly coupled
• scillator networks, Phys. Rev. E 92, 022919 (2015)

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] ˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi) εeff = εα (KC)

where

ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] α (KC) = (1 + KC)−1

(1)

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] ˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi) εeff = εα (KC)

where

ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] α (KC) = (1 + KC)−1

1 2

• 1

α (KC) KC

(1)

(i) All time delays are the same synchronization can not be controled

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] ˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi) εeff = εα (KC)

where

ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] α (KC) = (1 + KC)−1

1 2

• 1

α (KC) KC

τi = τ = T ωeff

i

= α (KC) ωi ⇒

(1)

(i) All time delays are the same synchronization can not be controled (ii) The time delays are equal to the natural periods

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] ˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi) εeff = εα (KC)

where

ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] α (KC) = (1 + KC)−1

1 2

• 1

α (KC) KC

τi = τ = T ωeff

i

= α (KC) ωi ⇒ τi = Ti ⇒ ωeff

i

= ωi

(1)

(i) All time delays are the same synchronization can not be controled (ii) The time delays are equal to the natural periods (iii) The time delays are in-phase synchronization is a stable solution of Eq. (1)

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] ˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi) εeff = εα (KC)

where

ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] α (KC) = (1 + KC)−1

1 2

• 1

α (KC) KC

τi = τ = T ωeff

i

= α (KC) ωi ⇒ τi = Ti ⇒ ωeff

i

= ωi τi − Ti = T − Ti 1 − α (KC) ⇒ ωeff

i

= 0

(1)

ψ1 = ψ2 = . . . = ψN

(i) All time delays are the same synchronization can not be controled (ii) The time delays are equal to the natural periods (iii) The time delays are in-phase synchronization is a stable solution of Eq. (1)

˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] ˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi) εeff = εα (KC)

where

ωeff

i

= ωi + Ωτi − Ti T [α (KC) − 1] α (KC) = (1 + KC)−1

1 2

• 1

α (KC) KC

τi = τ = T ωeff

i

= α (KC) ωi ⇒ τi = Ti ⇒ ωeff

i

= ωi τi − Ti = T − Ti 1 − α (KC) ⇒ ωeff

i

= 0

(1)

ψ1 = ψ2 = . . . = ψN

Our goal is to derive an algorithm for authomatic adjusment

• f the time delays to acheve in-phase synchronization

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011)

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011) Main idea:

• construct potential which is equal to zero at desirable state and positive at other

states

• use gradient descent method for the time delay

V ≥ 0 ˙ τ = −∂V ∂τ

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011) Main idea:

• construct potential which is equal to zero at desirable state and positive at other

states

• use gradient descent method for the time delay

V ≥ 0 ˙ τ = −∂V ∂τ ˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] V = 1 2

N

• i,j=1

aij [sj (t) − si (t)]2

potential for the in-phase synchronization

• V. Novičenko and I. Ratas: In-phase synchronization in complex oscillator networks by

adaptive delayed feedback control, Phys. Rev. E 98, 042302 (2018)

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011) Main idea:

• construct potential which is equal to zero at desirable state and positive at other

states

• use gradient descent method for the time delay

V ≥ 0 ˙ τ = −∂V ∂τ ˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] V = 1 2

N

• i,j=1

aij [sj (t) − si (t)]2

potential for the in-phase synchronization

si (t) ≈ g

• ξ
• t + ψi

Ω

• ⇒ ∂V

∂τk = Ω−1

N

• i,j=1

aij [sj (t) − si (t)]

• ˙

sj (t) ∂ψj ∂τk − ˙ si (t) ∂ψi ∂τk

• V. Novičenko and I. Ratas: In-phase synchronization in complex oscillator networks by

adaptive delayed feedback control, Phys. Rev. E 98, 042302 (2018)

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011) Main idea:

• construct potential which is equal to zero at desirable state and positive at other

states

• use gradient descent method for the time delay

V ≥ 0 ˙ τ = −∂V ∂τ ˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] V = 1 2

N

• i,j=1

aij [sj (t) − si (t)]2

potential for the in-phase synchronization by assuming that is small and do not change in time we get

|ψj − ψi| si (t) ≈ g

• ξ
• t + ψi

Ω

• ⇒ ∂V

∂τk = Ω−1

N

• i,j=1

aij [sj (t) − si (t)]

• ˙

sj (t) ∂ψj ∂τk − ˙ si (t) ∂ψi ∂τk

• Lψ ∼ τ

L = D − A

here

• V. Novičenko and I. Ratas: In-phase synchronization in complex oscillator networks by

adaptive delayed feedback control, Phys. Rev. E 98, 042302 (2018)

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011) Main idea:

• construct potential which is equal to zero at desirable state and positive at other

states

• use gradient descent method for the time delay

V ≥ 0 ˙ τ = −∂V ∂τ ˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] V = 1 2

N

• i,j=1

aij [sj (t) − si (t)]2

potential for the in-phase synchronization by assuming that is small and do not change in time we get

|ψj − ψi| si (t) ≈ g

• ξ
• t + ψi

Ω

• ⇒ ∂V

∂τk = Ω−1

N

• i,j=1

aij [sj (t) − si (t)]

• ˙

sj (t) ∂ψj ∂τk − ˙ si (t) ∂ψi ∂τk

• Lψ ∼ τ

L = D − A

here

• V. Novičenko and I. Ratas: In-phase synchronization in complex oscillator networks by

adaptive delayed feedback control, Phys. Rev. E 98, 042302 (2018)

˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi)

• V. Pyragas and K. Pyragas: Adaptive modification of the delayed feedback control

algorithm with a continuously varying time delay, Phys. Lett. A 375, 3866 (2011) Main idea:

• construct potential which is equal to zero at desirable state and positive at other

states

• use gradient descent method for the time delay

V ≥ 0 ˙ τ = −∂V ∂τ ˙ xi =fi (xi, ui (t)) + ε

N

• j=1

aijG (xj, xi) si (t) =g (xi (t)) ui (t) =K [si (t − τi) − si (t)] V = 1 2

N

• i,j=1

aij [sj (t) − si (t)]2

potential for the in-phase synchronization by assuming that is small and do not change in time we get

|ψj − ψi| si (t) ≈ g

• ξ
• t + ψi

Ω

• ⇒ ∂V

∂τk = Ω−1

N

• i,j=1

aij [sj (t) − si (t)]

• ˙

sj (t) ∂ψj ∂τk − ˙ si (t) ∂ψi ∂τk

• Lψ ∼ τ

∂ψi ∂τk ∼ L†

ik

finally the derivative

L = D − A

here

• V. Novičenko and I. Ratas: In-phase synchronization in complex oscillator networks by

adaptive delayed feedback control, Phys. Rev. E 98, 042302 (2018)

˙ ψi = ωeff

i

+ εeff

N

• j=1

aijh (ψj − ψi)

fi (x, u) =   x(1)

• 1 − x2

(1) − x2 (2)

• − Ωix(2) + u

x(2)

• 1 − x2

(1) − x2 (2)

• + Ωix(1)

  G (y, x) =

• 2 y(1) − x(1)
• ⇒ h (χ) = sin (χ)

6 4 3 5 2 1

r = 1 6

6

• i=1

exp [iψi]

fi (x, u) =   x(1)

• 1 − x2

(1) − x2 (2)

• − Ωix(2) + u

x(2)

• 1 − x2

(1) − x2 (2)

• + Ωix(1)

  G (y, x) =

• 2 y(1) − x(1)
• ⇒ h (χ) = sin (χ)

6 4 3 5 2 1

r = 1 6

6

• i=1

exp [iψi]

2 6.23 6.27 6.31 1 2 3 10 4 0.5 1 (a) (b) (c)

fi (x, u) =

• x(1) − x3

(1)/3 − x(2) + 0.5

ǫi x(1) (1 + u) + 0.7 − 0.8x(2)

• G (y, x) =
• y(1)/ 2 + y(2)
• − x(1)/ 2 + x(2)
• ⇒

2 4 6

• 0.2

0.2 0.4

h( )

fi (x, u) =

• x(1) − x3

(1)/3 − x(2) + 0.5

ǫi x(1) (1 + u) + 0.7 − 0.8x(2)

• G (y, x) =
• y(1)/ 2 + y(2)
• − x(1)/ 2 + x(2)
• ⇒

2 4 6

• 0.2

0.2 0.4

h( )

39 39.2 39.4 39.6 39.4 39.5 39.6 0.0798 0.08 0.0802 1 2 3 4 5 6 105 5000 10000 (a) (b) (c) (d)

fi (x, u) =

• x(1) − x3

(1)/3 − x(2) + 0.5

ǫi x(1) (1 + u) + 0.7 − 0.8x(2)

• G (y, x) =
• y(1)/ 2 + y(2)
• − x(1)/ 2 + x(2)
• ⇒

2 4 6

• 0.2

0.2 0.4

h( )

39 39.2 39.4 39.6 39.4 39.5 39.6 0.0798 0.08 0.0802 1 2 3 4 5 6 105 5000 10000 (a) (b) (c) (d)

• 2

2 10 20 30 40 50

• 2

2 (a) (b)

The end