Table Of ContentSuppression of oscillations by L´evy noise
A. I. Olemskoia,b,∗∗, S. S. Borysovb,∗, I. A. Shudab
0 aInstitute of Applied Physics, Nat. Acad. Sci. of Ukraine,
1
58 Petropavlovskaya St., 40030 Sumy, Ukraine
0
2 bSumy State University, 2 Rimskii-Korsakov St., 40007 Sumy, Ukraine
n
a
J
4
Abstract
]
h We find analytical solution of pair of stochastic equations with arbitrary
c
forces and multiplicative L´evy noises in a steady-state nonequilibrium case.
e
m This solution shows that L´evy flights suppress always a quasi-periodical mo-
- tion related to the limit cycle. We prove that such suppression is caused by
t
a that the L´evy variation ∆L ∼ (∆t)1/α with the exponent α < 2 is always
t
s negligible in comparison with the Gaussian variation ∆W ∼ (∆t)1/2 in the
.
t
a ∆t → 0 limit. Moreover, this difference is shown to remove the problem of
m
the calculus choice because related addition to the physical force is of order
d- (∆t)2/α (cid:28) ∆t.
n
o Keywords: L´evy noise; Stationary state; Limit cycle
c
PACS: 02.50.Ey, 05.40.Fb, 82.40.Bj
[
3
v
1. Introduction
8
1
0 It is known crucial changing in behavior of the systems that display noise-
2
induced [1, 2] and recurrence [3, 4] phase transitions, stochastic resonance
.
0
[5, 6], noise induced pattern formation [7, 8], noise induced transport [9, 2]
1
9 etc. is caused by interplay between noise and non-linearity (see Ref. [10],
0 for review). Noises of different origin can play a constructive role in dy-
:
v namical behavior such as hopping between multiple stable attractors [11, 12]
i
X and stabilization of the Lorenz attractor near the threshold of its formation
r
a
∗Corresponding author
∗∗Principal corresponding author
Email addresses: [email protected] (A. I. Olemskoi), [email protected] (S. S.
Borysov), [email protected] (I. A. Shuda)
Preprint submitted to Elsevier January 4, 2010
[13, 14]. This type of behavior is inherent in finite systems where exam-
ples of substantial alteration under effect of intrinsic noises give epidemics
[15]–[17], predator-prey population dynamics [18, 19], opinion dynamics [20],
biochemical clocks [21, 22], genetic networks [23], cyclic trapping reactions
[24] et cetera.
Above pointed out phase transitions present the simplest case, when joint
effectofbothnoiseandnon-linearityarrivesatnon-trivialfixedpointappear-
ance only on the phase-plane of the system states. In this consideration, we
are interested in studying much more complicated situation, when stochastic
system may display oscillatory behavior related to the limit cycle appearing
as a result of the Hopf bifurcation [25, 26]. It has long been conjectured [27]
that in some situations the influence of noise would be sufficient to produce
cyclic behavior [28]. Moreover, it has been shown that excitable [29], bistable
[30] and close to bifurcations [31] systems display oscillation behavior, whose
adjacency to ideally periodic signal depends resonantly on the noise inten-
sity [32] (due to this reason, such oscillations were been called coherence
resonance [29] or stochastic coherence [10]).
Characteristic peculiarity of above considerations is that all of them are
restrictedbystudyingtheGaussiannoiseeffect, whilesuchanoiseisaspecial
case of the L´evy stable process (the principle difference of these noises is
known [33] to consist in the form of the probability distribution that exhibits
the asymptotic power-law decay in the latter case and decays exponentially
in the former one). Nowadays, anomalous diffusion processes associated with
the L´evy stable noise are attracting much attention in a vast variety of fields
not only of natural sciences (physics, biology, earth science, and so on), but
of social sciences such as risk management, finance, etc.
In the context of physics, recent investigation [34] has shown that joint
effect of both non-linearity and L´evy noise may cause the occurrence of gen-
uine phase transitions which relates to a fixed point on the phase-plane of the
system states. In this connection, natural question arises: may be displayed
a self-organized quasi-periodical behavior related to the limit cycle by a sys-
tem driven by the L´evy stable noise? This work is devoted to the answer to
above question within analytical study of two-dimensional stochastic system.
The paper is organized as follows. In Section 2, we consider pair of
stochastic equations with arbitrary forces and multiplicative L´evy noises to
obtain their analytical solution in a steady-state nonequilibrium case. This
allows us to conclude in Section 3 that opposite to the Gaussian noises the
L´evy flights suppress always a quasi-periodical motion related to the limit
2
cycle. Since equation, governing behavior of stochastic system driven by
multiplicative L´evy stable noise, are very complicated [35] and moreover
their derivation is now in progress [36], we complete our consideration with
Appendix A containing details of derivation of the Fokker-Planck equation.
Moreover, to demonstrate that a closed consideration of the L´evy processes is
achieved only within the Fourier representation we set forth a scheme related
to the appropriate stochastic space in Appendix B.
2. Statistical picture of limit cycle
According to the theorem of central manifold [25], to achieve a closed
description of a limit cycle it is enough to use only two degrees of freedom
related to some stochastic variables X , i = 1,2. In this way, stochastic evo-
i
lution of the system under investigation is defined by the Langevin equations
[37]
dX = f dt+g dL , i = 1,2 (1)
i i i i
with arbitrary forces f = f (x ,x ) and noise amplitudes g = g (x ,x )
i i 1 2 i i 1 2
being functions of both variables x , i = 1,2; stochastic terms are related to
i
ˆ
the L´evy stable processes L = L (t). Within the Ito calculus, these processes
i i
are determined by the elementary characteristic function
(cid:10) (cid:11)
eikidXi := eLidt (2)
with increments L = L (k ,k ;x ,x ) whose expression [35]
i i 1 2 1 2
2
Li = iki(fi +γigi)−|migiki|α2e−iϕi(α2)(cid:88)|mjgjkj|α2 e−iϕj(α2) (3)
j=1
follows from Eq.(A.25). Hereafter, we use asymmetry angles ϕ and moduli
i
m defined by the equalities
i
tan[ϕ (α)] = β sgn(g k )tan(πα/2),
i i i i
(cid:113) (4)
mα = 1+β2tan2(πα/2);
i i
everywhere, the L´evy index α ∈ (0,2) characterizes the asymptotic tail
x−(α+1) of theL´evy stable distribution at 1 (cid:54)= α < 2 (thecaseα = 2relates to
i
the Gaussian distribution), parameters β ∈ [−1,+1] define the distribution
i
3
asymmetry, location parameters −∞ < γ < +∞ denote the mean values of
i
stochastic variables X at α > 1, and the angular brackets denote averaging
i
over L´evy noises.
As is shown in Appendix A, the Fourier transformed probability distri-
bution function
+∞
(cid:90)(cid:90)
P(cid:101)(k ,k ;t) ≡ F{P(x ,x )}(k ,k ;t) := dx dx P(x ,x ;t)ei(k1x1+k2x2)
1 2 1 2 1 2 1 2 1 2
−∞
(5)
is governed by the Fokker-Planck equation
(cid:34) (cid:35)
2 2
∂P(cid:101) = (cid:88) i(fi +γigi)ki −|migiki|α2e−iϕi(α2)(cid:88)|mjgjkj|α2 e−iϕj(α2) P(cid:101). (6)
∂t
i=1 j=1
Characteristically, being Fourier transformed, r.h.s. of this equation depends
on the wave vector components k and k , while both forces f = f (x ,x )
1 2 i i 1 2
and multiplicative noise amplitudes g = g (x ,x ) are dependent on the
i i 1 2
coordinate components x and x .
1 2
According to the continuity equation (A.23), components of the steady-
(cid:80)
state probability flux are obeyed to the condition ∂J /∂x = 0 which
i i i
means the first component J = J (x ) is a function of the only variable
1 1 2
x , and vice-versa for the second component J = J (x ). Then, within the
2 2 2 1
Fourier representation, the system behaviour is defined by the equations
(cid:110)
(f1 +g1γ1)+i|m1g1|α2e−iϕ1(α2)|k1|α2−2k1
(7)
(cid:104) (cid:105)(cid:111)
× |m1g1k1|α2 e−iϕ1(α2) +|m2g2k2|α2 e−iϕ2(α2) P(cid:101) = 2πJ1(k2)δ(k1),
(cid:110)
(f2 +g2γ2)+i|m2g2|α2e−iϕ2(α2)|k2|α2−2k2
(8)
(cid:104) (cid:105)(cid:111)
× |m1g1k1|α2 e−iϕ1(α2) +|m2g2k2|α2 e−iϕ2(α2) P(cid:101) = 2πJ2(k1)δ(k2).
Since the pair of these equations determines a single distribution function
P(cid:101)(k ,k ), the consistency condition
1 2
(cid:2) (cid:3)
(f +g γ )+ie−iϕ1(α)|m g |α|k |α−2k δ(k )J (k )
1 1 1 1 1 1 1 2 2 1
(9)
(cid:2) (cid:3)
= (f +g γ )+ie−iϕ2(α)|m g |α|k |α−2k δ(k )J (k )
2 2 2 2 2 2 2 1 1 2
4
shouldbekepttorestrictthechoiceoftheprobabilityfluxcomponentsJ (k )
1 2
and J (k ).
2 1
MultiplyingEq.(7)bythefactor|m2g2|α2e−iϕ2(α2) andEq.(8)by|m1g1|α2e−iϕ1(α2)
and then subtracting results, one obtains
(cid:110)
F +i|m1m2g1g2|α2e−i[ϕ1(α2)+ϕ2(α2)]
(cid:104) (cid:105) (cid:111)
× |m1g1k1|α2 e−iϕ1(α2) +|m2g2k2|α2 e−iϕ2(α2) (cid:0)|k1|α2−2k1 −|k2|α2−2k2(cid:1) P(cid:101)
(cid:104) (cid:105)
= 2π J1(k2)δ(k1)|m2g2|α2e−iϕ2(α2) −J2(k1)δ(k2)|m1g1|α2e−iϕ1(α2)
(10)
where one denotes
F ≡ (f1 +γ1g1)|m2g2|α2e−iϕ2(α2) −(f2 +γ2g2)|m1g1|α2e−iϕ1(α2). (11)
The equation (10) yields the explicit form of the probability distribution
function
+∞
(cid:90) dk2 J1(k2)|m2g2|α2e−i[k2x2+ϕ2(α2)]
P (x ,x ) =
1 2 2π F2 −i|g1|α2|m2g2|αe−iϕ2(α)|k2|α−2k2
−∞
(12)
+∞
(cid:90) dk1 J2(k1)|m1g1|α2e−i[k1x1+ϕ1(α2)]
−
2π F1 +i|g2|α2|m1g1|αe−iϕ1(α)|k1|α−2k1
−∞
whereeffectiveforcesF aredeterminedbyEq.(11)atm = 1andϕ = 0.
1,2 2,1 2,1
In the case of constant values of the probability flux within the state
space x ,x , the Fourier transforms related are J (k ) = 2πJ(0)δ(k ) and
1 2 1 2 1 2
J (k ) = 2πJ(0)δ(k ) with J(0) = const. Then, the consistency condition (9)
2 1 2 1 i
takes the form (f +g γ )J(0) = (f +g γ )J(0), the effective force (11) is
1 1 1 2 2 2 2 1
F = (f +γ g )|g |α − (f +γ g )|g |α, and the probability density (12)
0 1 1 1 2 2 2 2 2 1 2
reads
J(0)|g |α −J(0)|g |α
P = 1 2 2 2 1 2 . (13)
F
0
To create a limit cycle this distribution function should diverges on a closed
curve, sothattheeffectiveforceequalsF = 0. Togetherwiththeconsistency
0
condition, this equation gives
J1(0) = f1 +γ1g1 = (cid:12)(cid:12)(cid:12)g1(cid:12)(cid:12)(cid:12)α2 . (14)
J(0) f2 +γ2g2 (cid:12)g2(cid:12)
2
5
But these equalities mean that the numerator of the probability density (13)
disappearsalso. Asaresult, weconcludethelimitcyclecreationisimpossible
for a stationary non-equilibrium state with both probability flux components
J (x ,x ) and J (x ,x ) being constant.
1 1 2 2 1 2
To calculate integrals in Eq.(12) for arbitrary dependencies J (k ) and
1 2
J (k ) it is convenient to write |k| = sgn(k)k = eiπθ(−k)k where θ(k) denotes
2 1
the Heaviside step function. Then, one has |k|α−2k = e−iπθ(−k)(2−α)kα−1, and
the pole points of integrands in Eq.(12) are expressed with the equality
(cid:18) (cid:19) 1
F α−1
1,2
K =
1,2 |m g |α|g |α
1,2 1,2 2,1 2 (15)
(cid:26) (cid:27)
ϕ (α)+(2−α)πθ(−(cid:60)K )+(π/2)sgn((cid:61)K )
1,2 1,2 1,2
×exp i .
α−1
Due to sign-changing term (π/2)sgn((cid:61)K ) in the exponent the K poles
1,2 1,2
are located on opposite half-planes of complex variables k . Making use of
1,2
the power series expansion
(cid:18) k −K(cid:19)α−1
kα−1 = Kα−1 1+ ≈ Kα−1 +(α−1)Kα−2(k −K) (16)
K
allowsustoreducetheintegrandsinEq.(12)toapoleform. However, wecan
notclosetheintegrationcontoursaroundbothupperandlowercomplexhalf-
planesofthek variable since integrandsrelatedcontainabsolutemagnitudes.
To find the integrals needed let us specify a contribution that gives the
polelocatedontheupperhalf-planeofthecomplexnumberk. Withthisaim,
we divide this half-plane into two parts related to the positive and negative
values of the real part of k. As shows Figure 1, integrals in Eq.(12) can be
6
Figure 1: To calculation of integrals standing in equations (17) and (18)
rewritten as follows:
+∞
(cid:90) (cid:90) (cid:90)
f(k) f(k) f(k)
dk ≡ dk + dk
k −K k −K k −K
−∞ AB DE
(cid:73) (cid:90) (cid:90)
f(k) f(k) f(k)
= dk − dk + dk
k −K k −K k −K
ABC BC CA
(cid:73) (cid:90) (cid:90)
f(k) f(k) f(k)
(17)
+ dk − dk + dk
k −K k −K k −K
DEF EF FD
(cid:73) (cid:73)
f(k) f(k)
= dk + dk
k −K k −K
ABC DEF
(cid:90) (cid:90) (cid:90) (cid:90)
f(k) f(k) f(k) f(k)
− dk + dk− dk + dk.
k −K k −K k −K k −K
BC FD CA EF
With tending radiuses of the arcs CA and EF to infinity, both integrals in
the last square brackets disappear. On the other hand, when both half-axes
(cid:82) (cid:82)
BC and FD tend one to another, one has = − , so that terms in the
BC FD
square brackets standing before are cancelled also. Moreover, the integral
7
over the contour DEF equals zero because this contour does not envelop
any pole. As a result, we obtain
+∞
(cid:90) (cid:73)
f(k) f(k)
dk = dk = sgn((cid:61)K)2πif(K) (18)
k −K k −K
−∞ ABC
where the last equality is due to the residue theorem.
Finally, making use of the Cauchy integral (18) yields the probability
distribution (12) in the form
2−α 2−α
P (x ,x ) = Fα−1P e−i(K1x1−φ1) +Fα−1P e−i(K2x2−φ2) (19)
1 2 1 1 2 2
where one denotes
J (K )
2,1 1,2
P ≡ ,
1,2 α α(3−α)
(α−1)|g2,1|2(α−1)|m1,2g1,2|2(α−1) (20)
3−α (cid:16)α(cid:17) π2−α
φ ≡ ϕ + [sgn((cid:61)K )+2θ(−(cid:60)K )].
1,2 1,2 1,2 1,2
α−1 2 2α−1
3. Discussion
Analytical consideration developed in previous Section allowed us to ob-
tain the probability distribution function (19) that describes behaviour of
nonequilibrium steady-state stochastic system driven by the L´evy multiplica-
tive noise with two degrees of freedom. Recently, we have studied conditions
of the limit cycle creation in stochastic Lorenz-type systems driven by Gaus-
sian noises [38]. Noise induced resonance was found analytically to appear
in non-equilibrium steady state if the fastest variations displays a principle
variable which is coupled with two different degrees of freedom or more. The
condition of this resonance appearance is expressed formally in divergence of
the probability distribution function being inverse proportional to an effec-
tive force type of (11) – when this force vanishes on a closed curve of phase
plane, the system evolves along this cycle with diverging probability density.
In opposite to such a dependence, the distribution function (19) contains
the effective force (11) in positive power (2−α)/(α−1) only. To this end, we
can conclude the L´evy flights suppress always a quasi-periodical motion re-
lated to the limit cycle. That is main result of our consideration. The corner
stone of the difference between stochastic systems driven by the L´evy and
8
Gaussian noises is that the L´evy variation ∆L ∼ (∆t)1/α with the exponent
α < 2 is negligible in comparison with the Gaussian variation ∆W ∼ (∆t)1/2
in the ∆t → 0 limit.
It is interesting to note that above difference removes the problem of the
calculus choice [1, 37]. This problem is known to be caused by irregularity of
the time dependence X(t) of stochastic variable (for the sake of simplicity,
we consider one-dimensional case again). Hence, in the integral of equation
of motion (1)
t L(t)
(cid:90) (cid:90)
X(t) = f(cid:0)x(t(cid:48))(cid:1)dt(cid:48) + g(cid:0)x(t˜(cid:48))(cid:1)dL(t(cid:48)) (21)
0 L(0)
we should take the noise amplitude g(cid:0)x(t˜(cid:48))(cid:1) at the time moment
t˜(cid:48) = t(cid:48) +λ∆t(cid:48); λ ∈ [0,1], ∆t(cid:48) → 0 (22)
that does not coincide with the integration time t(cid:48) due to a parameter λ ∈
[0,1] whose value fixes calculus choice (for example, the magnitude λ = 1/2
relates to the Stratonovich case) [1, 37]. With accounting equations (22) and
(1), one obtains
g(cid:0)x(t˜)(cid:1) (cid:39) g(cid:0)x(t)(cid:1)+λg(cid:48)(cid:0)x(t)(cid:1)∆X(t)
(23)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(cid:39) g x(t) +λg(cid:48) x(t) f x(t) ∆t+λg(cid:48) x(t) g x(t) ∆L(t)
where primes denote differentiation over argument x. Being inserted into
Eq.(21), the first term in the last line of Eq.(23) relates to usual case of the
ˆ
Ito calculus. Corresponding insertion of the second term gives an addition
whose order ∆L·∆t ∼ (∆t)1+(1/α) (cid:28) ∆t is higher than one for the previous
term (such a situation is inherent in the Gaussian case as well). Finally, after
insertion of the last term of Eq.(23) the last integrand in Eq.(21) obtains an
addition of order (∆L)2 ∼ (∆t)2/α. In special case of the Gaussian noise
(α = 2), the order 2/α of this addition coincides with the same in the first
integrand of Eq.(21), that is resulted in addition λg(x)g(cid:48)(x) to the physical
force f(x). Principally different situation is realized for the L´evy stable
process, when the index α < 2 and above addition should be suppressed in
comparison with physical force because (∆t)2/α (cid:28) ∆t.
9
Appendix A. Derivation of Fokker-Planck equation for the L´evy
multiplicative noises
Following to the line of Ref. [35], we start with consideration of one-
dimensional L´evy process X(t) whose Chapman-Kolmogorov equation
(cid:90)
p(x,t+dt|x ,t ) = dy p(x,t+dt|y,t)p(y,t|x ,t ) (A.1)
0 0 0 0
connects transition probabilities taken in intermediate positions y related to
time t. According to the definition
p(k,t+dt|y,t) := edKX(k,dt|y,t), (A.2)
the inverse Fourier transform
(cid:90)
dk
p(x,t+dt|y,t) = e−ik(x−y)p(k,t+dt|y,t) (A.3)
2π
is expressed in terms of the elementary cumulant dK (k,dt|y,t) of the char-
X
acteristic function of the stochastic process X(t). Then, with using the
dt → 0 limit and the identity
(cid:90) (cid:90)
dk
p(x,t|x ,t ) = dy p(y,t|x ,t ) e−ik(x−y), (A.4)
0 0 0 0
2π
the equation (A.1) arrives at the chain of equalities:
p(x,t+dt|x ,t )−p(x,t|x ,t )
0 0 0 0
(cid:90) (cid:90)
dk
= dy p(y,t|x ,t ) e−ik(x−y)[edKX(k,dt|y,t) −1]
0 0
2π
(cid:90) (cid:90) dk (A.5)
(cid:39) dy p(y,t|x ,t ) e−ik(x−y)dK (k,dt|y,t)
0 0 X
2π
(cid:90)
= dy dK (x−y,t)p(y,t|x ,t ) ≡ dK (x,t)(cid:63)p(x,t|x ,t ).
X 0 0 X 0 0
Here, (cid:63) denotes the convolution of the inverse Fourier transform
(cid:90)
dk
dK (x−y,t) = dK (k,dt|y,t)e−ik(x−y). (A.6)
X X
2π
As a result, with accounting the definition
dK (x,t) := L(x)dt, (A.7)
X
10
