Numerical integration of coupled stochastic oscillators driven by Random Forces

Hugo de la Cruz

Escola de Matemática Aplicada (FGV-EMAp) Rio de Janeiro

joint work with: J. C. Jimenez, P.Zubelli

x

(

t

) +

f

(

t

,

x

(

t

)

,

x

0

₍

_t

_{)) =}

_∑

j=1

g

(t

)

η

dx

(

t

) =

y

(

t

)

dt

dy

(

t

) =

f

(

t

,

x

(

t

)

,

y

(

t

))

dt

+

∑

g

(t)dw

x2Rd,y=x0

dx

=

y

dy

= (

α

y

(

1

x

)

x

+

β

x

)

dt

+

γ

(t

)dw

dy

=

(

α

y

(

1

x

)

x

+

β

x1

)

dt

+

γ

(t

)dw

IC:(x0

1,x₂0,y₁0,y₂0)

d

x

(

t

)

y

(

t

)

=

0

1

ω

β

x

(

t

)

y

(

t

)

dt

+

0

ε

u

(

x

(

t

))

dt

+

d

x

(

t

)

y

(

t

)

=

0

1

ω

β

x

(

t

)

y

(

t

)

dt

+

0

ε

u

(

x

(

t

))

dt

+

0

σ

dw

,

K. Burrage, G. Lythe, Accuratestationary densitieswith partitioned numerical methods for stochastic di¤erential Equations, SIAM J. Num. Anal. 47 (3) (2009) 1601–1618.

D. Cohen, M. Sigg, Convergence analysis of trigonometric methodsforsti¤ second-order stochastic di¤erential equations, Num. Math., 121 (2012) 1-29.

A. H. Strømmen, D. J. Higham, Numerical simulation of alinear oscillatorwith additive noise, Appl. Numer. Math. 51 (2004) 89-99.

A. Tocino, On preserving long-time features of alinear stochastic oscillator, BIT Num. Math. 47 (2007) 189-196.

J. Hong, R. Scherer, L. Wang,Predictor-Corrector Methodsfor aLinear Stochastic Oscillatorwith Additive Noise, Mathematical and Computer Modelling. 46 (2007) 738-764.

D. Cohen, On thenumerical discretisationofstochastic oscillators, Math. and Comput in Simulation, 82 (2012) 1478-1495.

d

y(t) = Ω2 ₀ x(t)dt+ _Σ dwt,

x(t) = x(t)

d

y(t) = Ω2 ₀ x(t)dt+ _Σ dwt,

x(t) = x(t)

y(t) 2R2d, Ω2Rd d, Σ2Rd m

-This is a Hamiltonian system,

dx(t) = ∂H

∂ydt

dy(t) = ∂H ∂xdt+

m

∑

∂Hj

∂x dw

j t

with

H(x,y) = 1 2 kΩxk

2₊

kyk2

P1)growth rate of the energy: E x2_{(t) +ω}2_y2_{(t) =E(x}2_(t

0) +ω2y2(t0)) +σ2(t t0),

P2)oscillatory behavior around 0 : x(t) has in…nitely many zeros on [t0,∞),

P3)symplectic structure:dx(t)^dy(t) =dx(t0)^dy(t0), for allt t0,

-L. Markus. Stochastic oscillators, J. Di¤. Eq. 71 (1988)

P1)growth rate of the energy: E x2_{(t) +ω}2_y2_{(t) =E(x}2_(t

0) +ω2y2(t0)) +σ2(t t0),

P2)oscillatory behavior around 0 : x(t) has in…nitely many zeros on [t0,∞),

P3)symplectic structure:dx(t)^dy(t) =dx(t0)^dy(t0), for allt t0,

-L. Markus. Stochastic oscillators, J. Di¤. Eq. 71 (1988)

To study certain properties of the nonlinear oscillator "is enough" to study the linear one (with respect to a new brownian motion)

Previous works:

- The analysis focused on the simple (d=1) harmonic oscillator - No method fordamping coupled oscillator mimics these 3 properties

du(t) = 0d Id

fx(tn,x_n,y_n) f_y(t_n,x_n,y_n) (u(t) xn) +

0

ft (t tn) +

yn

du(t) = 0d Id

fx(tn,x_n,y_n) f_y(t_n,x_n,y_n) (u(t) xn) +

0

ft (t tn) +

yn

f(tn,x_n,y_n) dt

du(t) = 0d Id

fx(tn,x_n,y_n) f_y(t_n,x_n,y_n) (u(t) xn) +

0

ft (t tn) +

yn

f(tn,x_n,y_n) dt

u(t) =xn+LeCn(t tn)r

with

L = [I2d 02d 2],

r| ₌ h₀

(2d+1) 11

i

and

Cn=

2

6 6 4

0d Id 0d yn

fx(tn,x_n,y_n) f_y(t_n,x_n,y_n) f_t(t_n,x_n,y_n) f(t_n,x_n,y_n)

01 d 01 d 0 1

0 0 0 0

3

7 7

x(t) =xn+LeCn(t tn)r+z(t)

dz(t) =q(t,z(t))dt+

m

∑

0

gj(t) dw j t,

x(t) =xn+LeCn(t tn)r+z(t)

dz(t) =q(t,z(t))dt+

m

∑

0

gj(t) dw j t,

z(tn) =0,

q(t,z) = a(t,x_n+LeCn(t tn)r+z) a_x(t_n,x_n)LeCn(t tn)r a_t(t_n,x_n) (t t_n) a(t_n,x_n)

a(t,z) = [0d Id] z|

xn+1=xn+LeCnhr+zn+1

LL-Euler:

zn=

2

4

0

m

∑

j=1

gj(tn)∆wnj

3

5

LL-BE:

zn+1=q(tn+1,zn+1)h+

2 4 0 m ∑ j=1

gj(tn)∆wnj

3

5

LL-MR:

zn+1=q tn+1

+tn

2 ,

zn+1

2 h+

2 4 0 m ∑ j=1

gj(tn)∆wnj

3

xn+1=xn+Le r+zn+1

LL-PC(E,BE):

zn+1=q tn+1, "

0,

m

∑

gj(tn)∆wnj

#! h+ 2 4 0 m ∑ j=1

gj(tn)∆wnj

3

5

LL-PC(E,MR):

zn+1=q tn+tn+1

2 ,

"

0,1 2

m

∑

gj(tn)∆wnj

#! h+ 2 4 0 m ∑

j=1gj

(tn)∆wnj

3 5 LL-PE: ₂ 4 z1 n+1 3

5=q tn, z1_n+1,0 h+ 2

4

0

m

∑g (t )∆wj 3

Theorem

For coupled harmonic oscillators

d x(t) y(t) =

0 I

Ω2 ₀ x(t)dt+ _Σ0 dwt, with initial conditionx(t0) = (x0,y₀), the Locally Linearized integrators satisfy

E kyn+1k2+kΩxn+1k2 =E ky0k2+kΩx0k2 + (kQ2k2+kΩQ1k2) (tn+1 t0),

Proof: It follows that

vn+1=Mvn+Q∆wn, where

vn+1= Ω_yxn+1

n+1 , M=

cos(Ωh) sin(Ωh)

sin(Ωh) cos(Ωh) , and Q=

ΩQ1

Proof: It follows that

vn+1=Mvn+Q∆wn, where

vn+1= Ω_yxn+1

n+1 , M=

cos(Ωh) sin(Ωh)

sin(Ωh) cos(Ωh) , and Q=

ΩQ1

Q2 .

FromE(Q∆wn) =0follows that

E kyn+1k2+kΩxn+1k2 =E Mvn 2+2 Mvn,B∆wn + Q∆wn 2

=E v|

nM

|

Mvn +E ∆w|nQ

|

Q∆wn

Theorem

Proof: We have that

xn+1=Mn+1x0+ n

∑

MkQ∆wn k, where

Proof: We have that

xn+1=Mn+1x0+ n

∑

MkQ∆wn k, where

Mk = cos_Ω (kΩh) khsinc(kΩh) sin(kΩh) cos(kΩh) . LetQ=[q1_{, ...,q}m_{], then}

xn+1=cos((n+1)Ωh)x0+sin((n+1)Ωh)Ω 1y0+ n

∑

m

∑

V_ki !

with

Vki= q1icos(kΩh) + (q2iΩ 1)sin(kΩh) ∆win k, and

Vki sN 0,σ2ik , σ2_ik= qi₁cos(kΩh) + (qi₂Ω 1)sin(kΩh)

2

Proof: We have that

xn+1=Mn+1x0+ n

∑

MkQ∆wn k, where

Mk = cos_Ω (kΩh) khsinc(kΩh) sin(kΩh) cos(kΩh) . LetQ=[q1_{, ...,q}m_{], then}

xn+1=cos((n+1)Ωh)x0+sin((n+1)Ωh)Ω 1y0+ n

∑

m

∑

V_ki !

with

Vki= q1icos(kΩh) + (q2iΩ 1)sin(kΩh) ∆win k, and

Vki sN 0,σ2ik , σ2_ik= qi₁cos(kΩh) + (qi₂Ω 1)sin(kΩh)

2

h

What about

n

∑

m

∑

Vki

Proof (Cont): Let us compute

sn2= n

∑

m

∑

σ2ik

!

Proof (Cont): Let us compute

sn2= n

∑

m

∑

σ2ik

!

.

σ2_ik can be rewritten as σ2ik = h qi

2

cos2 kΩh αi ,

with αi _{= arctan} q2iΩ 1

qi 1

forqi

16=0, andαi=

π 2 forq

i

1=0, qi= q i 1

Proof (Cont): Let us compute

sn2= n

∑

m

∑

σ2ik

!

.

σ2_ik can be rewritten as σ2ik = h qi

2

cos2 kΩh αi ,

with αi _{= arctan} q2iΩ 1

qi 1

forqi

16=0, andαi=

π 2 forq

i

1=0, qi= q i 1

qi 2Ω 1

Then

s2 n = h

m

∑

qi 2

_∑

cos2 _{kΩh α}i

!

= h

m

∑

2

n+sin(nΩh+Ωh)cos nΩh 2α

i !

Proof (Cont): ...Since limn!∞s_n2=∞andσ2_in h qi 2, then lim n!∞ σ2 in s2 n =0.

The Law of the Iterated Logarithm may be applied to the sequence

n

∑

∑

V_ki. Then,

i) P 0

B B B @lim supn!∞

0 B B B @ n

∑

m

∑

V_ki

p

2sn(log logsn) 1 2 1 C C C A=1

1

C C C A=1,

ii) P 0

B B B @lim infn!∞

0 B B B @ n

∑

∑

2sn(log logsn) 1 2 1 C C C

A= 1

1

Proof (Cont): ...Since limn!∞s_n2=∞andσ2_in h qi 2, then lim n!∞ σ2 in s2 n =0.

The Law of the Iterated Logarithm may be applied to the sequence

n

∑

∑

V_ki. Then,

i) P 0

B B B @lim supn!∞

0 B B B @ n

∑

m

∑

V_ki

p

2sn(log logsn) 1 2 1 C C C A=1

1

C C C A=1,

ii) P 0

B B B @lim infn!∞

0 B B B @ n

∑

∑

2sn(log logsn) 1 2 1 C C C

A= 1

1

C C C A=1.

Fromi), for 0<_ε<1

Proof (Cont): Thus,

xn+1=cos((n+1)Ωh)x0+sin((n+1)Ωh)Ω 1y0+ n

∑

m

∑

Vki

!

Proof (Cont): Thus,

xn+1=cos((n+1)Ωh)x0+sin((n+1)Ωh)Ω 1y0+ n

∑

m

∑

Vki

!

>_{0 a.s, in…nitely often.}

From

ii) P 0

B B B @lim infn!∞

0 B B B @ n

∑

∑

2sn(log logsn) 1 2 1 C C C

A= 1

1

C C C A=1,

for 0<_ε<₁

n

∑

m

∑

Vki <( 1+ε)

p

2sn(log logsn) 1

2 _{a.s, for in…nite values ofn,}

which implies that

1

_∑

_∑

Theorem

For the coupled harmonic oscillators with initial conditionx(t0) = (x0,y0)2R2d, the Locally

Proof: Let us consider the di¤erential 2-form

w2:=

d

∑

dxni ^dyni,

where

dxni^dyni(ξ,η) =det ξ i 1 ηi1

ξi2 ηi2

, ξ= ξ1 ξ₂ 2R

2d

,η= η1 η₂ 2R

2d

In matrix notation,

w2_{(ξ,η) =ξ}|_Jη

, withJ= 0 I

Proof: Let us consider the di¤erential 2-form

w2:=

d

∑

dxni ^dyni,

where

dxni^dyni(ξ,η) =det ξ i 1 ηi1

ξi2 ηi2

, ξ= ξ1 ξ₂ 2R

2d

,η= η1 η₂ 2R

2d

In matrix notation,

w2_{(ξ,η) =ξ}|_Jη

, withJ= 0 I

I 0 .

Letψ(t,t₀)the ‡ow determined by the LL integrators, i.e.,[x_ny_n]|=ψ(t_n,t₀) [x₀y₀]|. All we need to prove is thatψ0=∂(xn,yn)

∂(x0,y0) satis…es

ψ0 | Jψ0=J.

Since the noise is additive, we have thatψ0=M.

d

x

(

t

)

y

(

t

)

=

0

1

ω

0

x

(

t

)

y

(

t

)

dt

+

2650 2660 2670 2680 2690 2700 2710 - 15 - 10 - 5 0 5 10 15

390 400 410 420 430 440 450 460 470 - 4 - 3 - 2 - 1 0 1 2 3 4 h2

750 760 770 780 790 800

- 6 - 4 - 2 0 2 4 6

180 190 200 210 220 230

- 5 0 5

-10 -5 0 5 10 15 -8

-6 -4 -2 0 2 4 6

Exact Sol. LL-P(EC)¹ P(EC)¹ T

1= 30

T

2= 70

T

Pendulum without damping perturbed by additive noise:Ex. (Gitterman05)

dX1(t) =X2(t)dt,

dX2_{(t) = sin X}1_{(t) dt+σdW}

t.

- 2 - 1 0 1 2 - 2 - 1 0 1 2 L L -PC (E, B E)

- 1 0 1 2

- 2 - 2 - 1 0 1 2

- 2 - 1 0 1 2

- 2 - 1 0 1 2

- 2 - 1 0 1 2

- 2 - 1 0 1 2 P C( E ,BE ) h₃

- 2 - 1 0 1 2

- 2 - 1 0 1 2 2 h₂

- 2 - 1 0 1 2

Ex. (Schurz09, Cohen12)

dx(t) =y(t)dt

dy(t) = (w2_{x(t) +x}4_(t))dt+σdw

Ex. (Schurz09, Cohen12)

dx(t) =y(t)dt

dy(t) = (w2_{x(t) +x}4_(t))dt+σdw

t

Expected value of the energy along the solution

E(1 2(y

2_{(t) +w}2_x2_{(t)) +}ε 5x

5_{(t)) =}1 2(y

2

0 50 100 150 0

10 20 30 40 50 60 70 80

E

ner

gy

Coupled Van der Pol oscillators driven by external random forces:(Artemiev et.al)

dx1=y1

dx2=y2 dy1= (1

2y1(1 x 2

1) x1+0.1x2)dt+1.5dw1 dy2=(5y2(1 x22) x2+0.1x1)dt+2.5dw2

σ1=1.5andσ2=2.5, initial condition(x0 1,x0

2,y0 1,y0

2) = (1,1,0,0).

- 2 - 1 0 1 2 3 - 10 - 5 0 5 10 T=17.5 L L P C( E ,M R)

- 3 - 2 - 1 0 1 2 3

- 15 - 10 - 5 0 5 10 T=19.375

- 3 - 2 - 1 0 1 2 3

- 15 - 10 - 5 0 5 10 15 T=100

- 2 - 1 0 1 2 3

- 8 - 6 - 4 - 2 0 2 4 6 8 P C( E ,M R)

- 20 - 15 - 10 - 5 0 5 - 50 0 50 100 150 200 250

- 15 - 10 - 5 0 5

x 10137 - 1 0 1 2 3 4 5 6 7x 10

- 2 - 1 0 1 2 3 - 8 - 6 - 4 - 2 0 2 4 6 8 10 T 1=17.5

- 3 - 2 - 1 0 1 2 3 - 15 - 10 - 5 0 5 10 T 2=19.37

- 3 - 2 - 1 0 1 2 3 - 15 - 10 - 5 0 5 10 15 T

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