Effect of intrinsic noise on chimera states in populations of - - PowerPoint PPT Presentation

Effect of intrinsic noise on chimera states in populations of hierarchically coupled oscillators: beyond Ott–Antonsen theory

• D. S. Goldobin, I. V. Tyulkina, L. S. Klimenko, A. Pikovsky

Institute of Continuous Media Mechanics UB RAS (Perm, Russia), Perm State University (Russia), University of Potsdam (Germany)

Funding: Joint RSF–DFG project, RSF grant #19-42-04120

School and Workshop “Patterns of Synchrony: Chimera States and Beyond” ICTP, Trieste — May 10, 2019

2

Systems of the type:

( ) Im(2 ( 1,... ) ), , .

k

i k

t h t e k N

ϕ

ϕ

−

= Ω + = ɺ

Kuramoto-Sakaguchi ensemble:

1sin(

), ( ) .

j

N i i k j k j j

h t e e N

ϕ α

µ ϕ ϕ ϕ α µ

− =

= Ω + − − =

∑

ɺ

Chain of superconducting (Josephson) junctions in parallel with a resistive load:

inp 1

1 ( ) sin sin ,

N k k k j j

I t I N µ ϕ ϕ ϕ ϕ γ

=

+ = + −

∑

ɺɺ ɺ

inp

( ) Im , ( ) /2, 1.

j

i j

I t e h t I

ϕ

µ γ Ω = + = ≫

• S. A. Marvel, S. H. Strogatz, Chaos 19, 013132 (2009)

Active rotators (and theta-neurons):

1

1 sin( ) sin , ( ) ( ). 2

j

N i k j j k k j j j

K B h t B K e N

ϕ

µ ϕ ϕ ϕ ϕ

=

=Ω+ − − = +

∑

ɺ

• Sh. Shinomoto, Y. Kuramoto, Prog. Theor. Phys. 75(5), 1105 (1986)

3

Advance: Watanabe‒Strogatz theory

( ) Im(2 ( 1,... ) ), , .

k

i k

t h t e k N

ϕ

ϕ

−

= Ω + = ɺ

In terms of

k

ψ and z :

1

, : 0, 1

k k k k

N i i i i k

z e e z e z e

ψ ϕ ψ ψ ∗ =

+ = = +

∑

k

ψ evolve with identical rate;

2,

( ) Im(2 ( ) ).

k

z i z h h z t h t z ψ

∗ ∗

= Ω + − = Ω + ɺ ɺ

Dynamics is partially integrable: 3 variables + (N‒3) integrals of motion

• S. Watanabe, S. H. Strogatz, Phys. D 74, 197 (1994)
• A. Pikovsky, M. Rosenblum, Phys. Rev. Lett. 101, 2264103 (2008)
• S. A. Marvel, R. E. Mirollo, S. H. Strogatz, Chaos 19, 043104 (2009)

Interpretation in terms of order parameters?

4

Ott‒Antonsen theory (has simple interpretation)

( ) Im(2 ( 1,... ) ), , .

k

i k

t h t e k N

ϕ

ϕ

−

= Ω + = ɺ

• E. Ott, T. M. Antonsen, Chaos 18, 037113 (2008)

The Master-equation for the probability density

( , ) w t ϕ

:

(( ) ) 0.

i i

w ihe ih e w t

ϕ ϕ

ϕ

− ∗

∂ ∂ + Ω − + = ∂ ∂

Particular solution

1

1 ( , ) 1 ( . .) , 2

m im m

w t a e c c

ϕ

ϕ π

∞ − =

  = + +    

∑

where complex amplitude a is the order parameter

i

a e ϕ =

:

2

a i a h h a

∗

= Ω + − ɺ

• According to W–S theory, OA manifold is not attracting in ideal situations
• In real systems, small detuning of parameters makes the OA manifold attracting

5

Powerful Tool & Expectation of Troubles

( ) Im(2 ( ) ), 1,...,

k

i k

t h t e k N

ϕ

ϕ

−

= Ω + = ɺ

• We can describe and understand the collective dynamics reliably and

in great detail.

• Our main sources of intuition for the theory of collective phenomena

are very specific systems. Here many of collective phenomena (e.g., clustering) are forbidden by conservation laws.

• In reality, the OA conditions are (slightly?) violated.
• The perturbation theory cannot be constructed in a regular way.

A perturbation theory is needed. Preferably, for OA ansatz (not for WS theory).

• V. Vlasov, M. Rosenblum, A. Pikovsky, J. Phys. A Math. Theor. 49, 31LT02 (2016)

6

Populations of Oscillators with Intrinsic Noise

( ) Im(2 ( ) )

k

i k k

t h t f e

ϕ

ϕ

−

= Ω + + ɺ ( ), ( )

k k k

f t t σξ ξ =

: Gaussian noise signals,

( ) ( ') 2 ( ')

k m km

t t t t ξ ξ δ δ = −

. Fokker–Planck equation:

2 2

[( ) ]

i i

w w ihe ih e w t

ϕ ϕ

σ ϕ ϕ

− ∗

∂ ∂ ∂ + Ω − + = ∂ ∂ ∂

In Fourier space,

1 1

( , ) (2 ) [1 ( . .)]

im m m

w t a e c c

ϕ

ϕ π

∞ − − =

= + +

∑

:

2 2 1 1

.

m m m m m

a im a mha mh a m a σ

∗ − +

= Ω + − − ɺ

With Lorentzian distribution of frequencies

[ ]

2 2

( ) ( ) g γ π γ Ω = Ω − Ω −

:

2 2 1 1

( ) .

m m m m m

a m i a mha mh a m a γ σ

∗ − +

= Ω − + − − ɺ

7

‘Circular’ Cumulants

I.V.T., D.S.G., L.S.K., A.P., Phys. Rev. Lett. 120, 264101 (2018) Perturbation to O-A manifold

1 m m

a a =

?

[ ] [ ] [ ]

1 2 2 3 3

. ( . , ) .2 ... a a a a smthg a a G smthg a smthg  =    = +     = + +     

For

( )

j

i m m

a e ϕ =

, moment-generating function

( ) !

j

m ke m m

k F k e a m

ϕ

∞ =

= = ∑

Cumulant-generating function:

1

( ) ln ( ) .

m m m

k k F k k k κ

∞ =

∂ Φ = = ∂

∑

2 3 1 1 2 2 1 3 3 2 1 1

1 , , ( 3 2 ). 2 a a a a a a a κ κ κ = = − = − +

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Cumulant Expansion

1

ln ( )

m m m

k F k k k κ

∞ =

∂ = ∂

∑

1 2 1 1 1 2 2 2 1 1

( ) ( ) ( )

m m m m m m j j j m m m j j

m i h h m m m m κ γ κ δ κ κ κ σ κ κ κ

− ∗ + − + = − − + =

= Ω − + − + − +

∑ ∑

ɺ

instead of

2 2 1 1

( ) .

m m m m m

a m i a mha mh a m a γ σ

∗ − +

= Ω − + − − ɺ

OA solution = wrapped Cauchy distribution of phases:

1 1 1

,

m

a κ κ

>

= =

Expansion

1 1

1

m m

κ ε κ ε

>

∼ ≪ ∼

1 2( 1) 1

1

m m

κ κ σ

− >

∼ ∼

9

Conventional Cumulants versus ‘Circular’ Cumulants

Gaussian distribution

j

K ∈ ℝ,

1

K ,

2

K ,

2 j

K > =

1

K : centered (mean),

2

K : width

wrapped Cauchy distribution on circle

j

κ ∈ ℂ,

1 i

e ϕ κ =

,

1 j

κ > =

1

arg( ) κ : centered,

1

| | e κ

−

=

[width]

Deviation from Gaussian distribution:

3

K : skewness

4

K : kurtosis

Deviation from wrap. Cauchy distr.:

2

κ :

2

| | κ

,

2 2 1

arg( / ) κ κ = Θ

10

1 2 1 1 1 2 2 2 1 1

( ) ( ) ( )

m m m m m m j j j m m m j j

m i h h m m m m κ γ κ δ κ κ κ σ κ κ κ

− ∗ + − + = − − + =

= Ω − + − + − +

∑ ∑

ɺ

Model Reductions: Series Truncations

Expansion

1 1

1

m m

κ ε κ ε

>

∼ ≪ ∼

1 2( 1) 1

1

m m

κ κ σ

− >

∼ ∼

Two-cumulant truncations:

3 3 2 2 2 1 3 2 1

3 3 2 , 2 , κ κ κ κ κ κ κ

∗

= = =

2 2 1 2 2 2 1 2 1 3 1 2 1 2 1

( 2 2 4 ( ) ) ( ( ) . 4 ) 4 , 2 i h h h i h κ σ κ κ γ κ κ σ κ γ κ κ κ κ κ

∗ ∗ ∗

= Ω − + − + − = Ω − − − + − ɺ ɺ

D.S.G., I.V.T., L.S.K., A.P., Chaos 28, 101101 (2018)

11

1 2 1 1 1 2 2 2 1 1

( ) ( ) ( )

m m m m m m j j j m m m j j

m i h h m m m m κ γ κ δ κ κ κ σ κ κ κ

− ∗ + − + = − − + =

= Ω − + − + − +

∑ ∑

ɺ

Two-cumulant truncations:

3 3 2 2 2 1 3 2 1

3 3 2 , 2 , κ κ κ κ κ κ κ

∗

= = =

:

2 2 1 2 2 2 1 2 1 3 1 2 1 2 1

( 2 2 4 ( ) ) ( ( ) . 4 ) 4 , 2 i h h h i h κ σ κ κ γ κ κ σ κ γ κ κ κ κ κ

∗ ∗ ∗

= Ω − + − + − = Ω − − − + − ɺ ɺ

Wrapped Gaussian distribution:

2 2 2

1 2 1 m m im im m

a R e e e

σ ψ ψ −

= =

Zaks et al., Phys. Rev. E 68, 066206 (2003); Hannay et al., Sci. Adv. 4, e1701047 (2018)

1 1

1

m m

κ ε κ ε

>

∼ ≪ ∼

1 1 1

1

m m

κ κ ε

− >

∼ ∼

12

Fig: Kuramoto model with intrinsic noise OA ansatz: blue squares

• Gauss. ans.: red diamonds

3

κ =

: open squares

2 3 2 1

3 2 κ κ κ =

: filled circles

2 3 2 1

3 2κ κ κ∗ =

: open circles D.S.G., I.V.T., L.S.K., A.P., Chaos 28, 101101 (2018)

13

1 2 1 1 1 2 2 2 1 1

( ) ( ) ( )

m m m m m m j j j m m m j j

m i h h m m m m κ γ κ δ κ κ κ σ κ κ κ

− ∗ + − + = − − + =

= Ω − + − + − +

∑ ∑

ɺ

For

σ =

, extension of Ott–Antonsen solution

1 2 2 1

0, 0,

n n

κ κ κ

+

≠ ≠ =

:

{ }

2 1 1 1 2 2 2 1 2 2

( 2 2 4 ) ( ) ( ) . 1 2 17 ( ( )) , { } 1, , , , ... 3 15 315

n n n n

i h h C i h t C κ κ γ κ κ γ κ κ κ κ κ

∗ ∗

+ = Ω − − = Ω − + − = = − − ɺ ɺ

This is a two-bunch solution with equipartition of elements between bunches. IVT, DSG, LSK, AP, Radiophys. Quantum Electron. 61, no.8–9, 640–649 (2019) Analytical study of some problems without expansion truncation

14

Chimera States in Coupled Kuramoto Ensembles

1 1 1 1

1 1 sin( ) sin( ) ( ), 2 2 1 1 sin( ) sin( ) ( ). 2 2

N N k j k j k k j j N N k j k j k k j j

A A t N N A A t N N ϕ ϕ ϕ α φ φ α σξ φ φ φ α ϕ ϕ α σζ

= = = =

+ − = Ω + − − + − − + + − = Ω + − − + − − +

∑ ∑ ∑ ∑

ɺ ɺ σ =

:

• D. M. Abrams, R. Mirollo, S. H. Strogatz,
• D. A. Wiley, PRL 101, 084103 (2008)

1 2

... , , 1

N i

Z e Z

ϕ

φ φ φ = = = = Φ = < <

i

Ze− Φ: (i) static or (ii) periodic in time

I.V.T. et al., PRL 120, 264101 (2018)

15

• A. Pikovsky, M. Rosenblum, PRL 101, 264103 (2008)

if

j

ϕ are not on the OA manifold, + 1 frequency:

static -> periodic; periodic -> quasiperiodic

0.3, / 2 0.15 A α π = = −

. Left panel: Two-bunch state for

σ =

. Right panel: Two-bunch state for

0.01 σ =

.

16

Conclusion

• ‘Circular’ cumulant approach for the extension of OA theory
• Hierarchies of cumulants

PRL 120, 264101 (2018); Chaos 28, 101101 (2018);

• Radiophys. Quantum Electron. 61, no.8–9, 640–649 (2019).

Open questions:

• Weak intrinsic noise makes the heir of OA manifold attracting. Or?
• Numerical instability of truncation with more than 2 cumulants.
• Relationships between the higher-order cumulants and

( , ) W t ψ

.

D.S. Goldobin, Fluct. Noise Lett. 18, no.2, 1940002 (2019).

• Perturbation theory for other situations (small inertia, ‘deterministic’

perturbations)?

• Constrains on cumulant values; e.g.,

2 2 1

| | 1 | | Z κ ≤ −

.