Noise-induced transitions in a double-well oscillator with nonlinear dissipation

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by  Vladimir V. Semenov

Noise-induced transitions in a double-well oscillator with nonlinear dissipation
Vladimir V. Semenov,1,∗ Alexander B. Neiman,2,† Tatyana E. Vadivasova,1 and Vadim S. Anishchenko1
1Department of Physics, Saratov State University,
Astrakhanskaya str., 83, 410012, Saratov, Russia
2Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA
(Dated: March 29, 2016)
Wedevelopamodelofbistableoscillatorwithnonlineardissipation. Usinganumericalsimulation
andanelectroniccircuitrealizationofthissystemwestudyitsresponsetoadditivenoiseexcitations.
We show that depending on noise intensity the system undergoes multiple qualitative changes in
thestructureofitssteady-stateprobabilitydensityfunction(PDF).Inparticular,thePDFexhibits
two pitchfork bifurcations versus noise intensity, which we describe using an effective potential and
correspondingnormalformofthebifurcation. Thesestochasticeffectsareexplainedbythepartition
6 of the phase space by the nullclines of the deterministic oscillator.
1
0 PACSnumbers: 02.30.Ks,05.10.-a,05.40.-a,84.30.-r
2 Keywords: bistability,double-welloscillator,noise,stochasticbifurcations

r
a
M I. INTRODUCTION noise (see e.g. [19, 20]). For example, with a multiplica-
tive noise stochastic bistable oscillator shows reentrant

8 Bistable dynamics is typical for many natural systems (multiple)noise-inducedtransitionswhenthenoiseinten-
2 in physics [1–3], chemistry [4, 5], biology [6–11], ecology sity varies [21]. Noise-induced transitions were observed

[12,13],geophysics[14–16]. Thesimplestkindofbistabil- in excitable systems ranging from a single excitable neu-
]
O ity occurs when a system possesses two stable equilibria rons [22–24] to coupled excitable elements and media
A in the phase space, separated by a saddle. Adding noise [25, 26]. In many cases noise-induced transitions are
givesrisetorandomswitchingsbetweenthedeterministi- not true bifurcations [27], rather they underlie qualita-
.
n callystablestates, resultinginasteadystateprobability tive changes of stochastic dynamics when noise strength
li density with two local maxima. The Kramers oscillator isthecontrolparameter. Noise-inducedphasetransitions
n is a classical example of the stochastic bistable system were studied in spatially distributed systems perturbed
[ describing Brownian motion in a double-well potential by multiplicative noise [28] and were shown to exist for
2 [2, 4, 17], the case of additive noise [29]. In this paper we develop
v a generalized bistable oscillator with nonlinear dissipa-
8 dU(y) (cid:112) tion and report on a multiple noise-induced transitions
y˙ =v, v˙ =−γv− + 2γDn(t), (1)
9 dy due to additive noise in this system. We first develop
2 an electronic circuit of the oscillator and demonstrate
0 where γ is the (constant) drag coefficient, U(y) is a
noise-induced transitions in analog experiment. Second,
0 double-well potential and n(t) is Gaussian white noise,
weuseacorrespondingdeterministicmodelofthecircuit
. D is the noise intensity. Two-dimensional equilibrium
1 andnumericalsimulationsofstochasticmodeltoexplain
0 PDF is:
mechanisms of noise-induced transitions.
6 (cid:20) 1 (cid:18)v2 (cid:19)(cid:21)
1 P(y,v)=kexp − +U(y) , (2)
: D 2
v II. MODEL AND METHODS
i withthenormalizationconstantk,andpossesstwomax-
X
ima, corresponding to the potential wells, separated by
Fig. 1 shows a circuit diagram with two nonlinear el-
r
a saddle point of the potential. This structure does not
a ements N1 and N2 with the S- and N-type of the I-V
depend on the noise intensity D: although the peaks in
characteristic, respectively: i = F(V), V = G(i).
thePDFaresmearedout,theirpositionisinvariantwith N1 N2
The circuit is similar to Nagumo’s tunnel diode neuron
respect to increase of noise intensity.
model [30, 31], except it contains nonlinear resistor N2
External random perturbations may result in the so-
in series with the inductor, L. The circuit also includes
callednoise-inducedtransitionswherebystationaryPDF
a source of broadband Gaussian noise current i (t),
changes its structural shape, e.g. number of extrema, noise
which will be assumed white in the following. By us-
when noise intensity varies [18]. Such transitions may
ing the Kirchhoff’s current law the following differential
occur both with multiplicative noise as in the origi-
equationsforthevoltageV acrossthecapacitanceC and
nal Horsthemke-Lefever scenario [18], and with additive
the current i through the inductance L can be derived:

dV
Cdt(cid:48) +iN1+i+inoise(t(cid:48))=0, (3)
∗correspondingauthor: [email protected] di
†Electronicaddress: [email protected] V =Ldt(cid:48) +VN2.
2
Eqs. (6) can be written in the ”coordinate-velocity”
formwithdynamicalvariablesy, v ≡y˙ andx=v−ay+
by3:

y˙ =v,

εv˙ =−y−c (v−ay+by3)+c (v−ay+by3)3−
1 3 √
c (v−ay+by3)5+εv(a−3by2)− 2Dn(t).
5
(7)
In the oscillatory form (7) becomes,
FIG. 1: Circuit diagram of the model.
1 √
y¨+q (y,y˙)y˙+ q (y)=− 2Dn(t), (8)
1 ε 2
In the dimensionless variables x = V/v and y = i/i
0 0
with v0 = 1 V, i0 = 1 A and dimensionless time t = whereq2(y)definestheformofthepotential,andq1(y,y˙)
[(v /(i L)]t(cid:48), Eq.(3) can be re-written as, is the nonlinear dissipation,
0 0
√
(cid:26) 1
εy˙x˙==x−−yG−(yF)(,x)− 2Dn(t), (4) q1(y,y˙)=−a+3by2+ ε(c1−
3
c (cid:80) 3! y˙n−1(by2−a)3−ny3−n+
The parameter ε sets separation of slow and fast vari- 3 n!(3−n)!
n=1
ables of the system, ε = (C/L)(v0/i0)2. The first equa- +c (cid:80)5 5! y˙n−1(by2−a)5−ny5−n), (9)
tion for the voltage contains an additive source of white 5 n!(5−n)!
n=1
Gaussian noise, n(t), with the intensity D: (cid:104)n(t)(cid:105) = 0,
(cid:104)n(t)n(t+τ)(cid:105)=2Dδ(τ). Depending on the shape of the q (y)=y+c (by2−a)y−
2 1
functions F(x) and G(y) the circuit demonstrates wide
range of dynamics, including various types of bistabil- c (by2−a)3y3+c (by2−a)5y5.
3 5
ity, self-sustained oscillations and excitability. It allows
to observe a wide range of dynamical regimes: from the We note that unlike for the case of linear resistor Eq.(5),
behavior like in oscillator (1) with a double-well poten- thesystem’sdissipationnowdependsonbothcoordinate
tial to dynamics of an excitable oscillator or a bistable andvelocity,yandy˙. Asweshowbelow,itgivesthemain
self-sustained oscillators. This letter is restricted to reason of the qualitative difference between dynamics of
bistabilityoftwostableequilibriawithN-shapefunction the system (7) and behavior of the models (1) and (5).
G(y)=−ay+by3 with positive coefficients a and b. The proposed system was studied by means of ana-
We start with a linear resistor N1, F(x) = c x, with log and numerical simulations. Experimental electronic
1
positive c . The circuit is described by, setup was developed by using principles of analog mod-
1
eling of stochastic systems [33, 34]. The main part of
εy¨=−((3by2−a)ε+√c1)y˙− (5) the analog model is the operational amplifier integrator,
(1−c1a+c1by2)y− 2Dn(t). whose output voltage is proportional to the input volt-
1 (cid:82)t
The friction is nonlinear, but depends solely on the ”co- age integrated over time: Vout = −R C Vindt or
ordinate” variable, y. For sufficiently small ε, c (cid:29) 0 0 0
ε(3by2 − a) and dissipation becomes essentially lin1ear. R0C0V˙out = −Vin. Circuit diagram is shown in Fig. 2.
It contains two integrators, A1 and A10, whose output
Then the system is closely akin to Kramers oscillator,
voltages are taken as the dynamical variables, x and
Eq.(1), with linear friction. Our analog and numer- ∗
y , respectively. Then the signals x and y are trans-
ical simulations indicated no qualitative differences in ∗ ∗ ∗
formed in order to match expressions of the right hand
dynamics of the Kramers oscillator and the circuit (5)
side of Eqs. (6). The necessary signal transformations
with linear resistor N1, i.e. no noise-induced qualitative
arecarriedoutbyusinganalogmultipliersAD633JNand
change in the stationary PDF.
theoperationalamplifiersTL072CPconnectedinthein-
Next, we consider the case of nonlinear resistor N1
with F(x) = c x−c x3 +c x5 and with fixed positive verting and non-inverting amplifier configurations. Fi-
1 3 5
nally, transformed signals come to the input of the in-
coefficients c = 1,c = 9,c = 22. A variety of the
1 3 5
tegrators as V . The experimental setup allows to ob-
electronic elements and circuits have I-V characteristic in
tain the instantaneous values of the variables x ,y and
like that. For example, N1 can be realized by the so- ∗ ∗
v = y˙ = x +ay −by3. Time series were recorded
calledlambda-diodecircuit[32]. Inthiscasethefollowing ∗ ∗ ∗ ∗ ∗
from corresponding outputs (marked in Fig. 2) using an
equations describes the system under study,
acquisition board (National Instruments NI-PCI 6133).
All signals were digitized at the sampling frequency of
√
(cid:26)
εx˙ =−y−c x+c x3−c x5− 2Dn(t), 50 kHz. 150 s long realizations were used for further of-
1 3 5 (6)
y˙ =x+ay−by3. fline processing. A noise generator G2-59 was used to
3
FIG. 2: Scheme of the experimental setup.
FIG. 3: Noise-induced transitions in analog simulations. (a):
Time traces of state variables for various values of noise in-
tensity: 1 – D = 1.51×10−4, 2 – D = 3.78×10−4, 3 –
produce broadband Gaussian noise, whose spectral den- D = 3.00×10−3. (b): Stationary probability density func-
sity was almost constant in the frequency range 0 – 100 tions (PDF) P(y,v) corresponding to traces on (a). Other
kHz. In this frequency range noise can be approximated parameters are: ε=0.01,c =1,c =9,c =22,a=1.2,b=
1 3 5
by white Gaussian. 100.
ThecircuitinFig. 2isdescribedbythefollowingequa-
tions:
control parameter in analog experiments and numerical
 dx
RxCxdt∗ =−y∗−c1x∗+c3x3∗− simulations.
∗ (10)
c x5−ξ(t ),
R5 C∗ y˙ =∗x +ay −by3,
y y ∗ ∗ ∗ ∗ III. NOISE-INDUCED TRANSITIONS AND
EFFECTIVE POTENTIAL
where C = 30 nF, C = 300 nF, R =1 KΩ is the re-
x y x
sistance at the integrator A1 (R = R = R = R =
1 2 13 20
R = 1 KΩ), R = 10 KΩ is the resistance at the inte- A. Analog Experiment
x y
grator A10 (R = R = R = 10 KΩ). The param-
14 19 y
eter a is equal to the input value of the voltage Va at Experimentswiththeanalogcircuitshowedthatnoise
(cid:18) (cid:19)
the analog multiplier A14, b = 10 1+ R17 ; c = 1, strength is true control parameter of the system. For
R 1 weak noise the circuit exhibits bistable dynamics with
18
(cid:18) (cid:19)
R R R C a typical hopping between two metastable states and
c =4 1+ 5 , c =0.4 7; ε= x x. Transition to
3 R 5 R R C two peaks in the PDF shown in Fig. 3(a1,b1). In-
6 8 y y
dimensionless equations (7) is then carried out by sub- crease of noise intensity leads to qualitative change in
stitution t = t /τ , x = x /v , y = y /v ,v = v /v , the stochastic dynamics: switching between two states
∗ 0 ∗ 0 ∗ 0 ∗ 0
where τ = R C = R C = 3 ms is the circuit’s time disappearsandsothePDFhasasingleglobalmaximum
0 0 0 y y
constant and v = 1 V. The intensity D(cid:48) of noise gener- [Fig. 3(a2,b2)]. Furthermore, larger noise results in yet
0
atedbynoisegeneratorisrelatedtothedimensionlessD another noise-induced transition whereby dynamics be-
of Eq.(7) by D =D(cid:48)/τ . comesagainbistablewithtwo-statehoppinganddouble-
0
Numericalsimulationswerecarriedoutbytheintegra- peaked stationary PDF [Fig. 3(a3,b3)]. These multi-
tion of Eq. (7) using the Heun method [35] with time ple noise-induced transitions are shown in Fig. 4 for the
step ∆t=0.0001. marginal PDFs of the coordinate and velocity variables,
(cid:82)∞ (cid:82)∞
Inthefollowingtheparametersofdeterministicsystem P(y) = P(y,v)dv, P(v) = P(y,v)dy. While
−∞ −∞
were set to ε = 0.01, a = 1.2, b = 100, c = 1, c = 9, the PDF of the coordinate, P(y), shows noise-controlled
1 3
c = 22. For this set of parameters the deterministic changes of its modality from bimodal to unimodal and
5
system possesses three equilibria: two stable nodes and back to bimodal, the velocity PDF shows no modality
a saddle at the origin. Noise intensity, D, was used as a change. In this respect, the velocity distribution is sim-
4
D (D) ≡ var[v], while the parameters α, β are esti-
eff
mated from the least square fit of the experimentally
measured marginal PDF, P(y). The effective potential
can be re-written in the form, U (y) = 4β(−µy2/2+
eff
y4/4), with µ = α/(2β). The shape of the effective po-
tential is determined by the effective bifurcation param-
eter,µ(D). Thus,noise-inducedtransitionsofthecircuit
can be effectively described by the normal form of the
pitchfork bifurcation perturbed by white noise, [36],
(cid:112)
y˙ =4β(µy−y3)+ 2D ξ(t). (12)
eff
With positive β, the dynamics of Eq.(12) is bistable
for µ(D) > 0, monostable for µ(D) < 0 and criti-
cal at µ(D) = 0. Fig. 5(b) shows the dependence of
the effective bifurcation parameter vs noise intensity,
µ(D), and clearly indicates two pitchfork bifurcations at
D = 2.6 × 10−4 and D = 1.34 × 10−3 which match
the bifurcation diagram, Fig. 5(a), obtained from the
marginal PDF, P(y;D). Furthermore, as indicated in
Fig.5(c),whilethevelocityvariance,σ2 =var[v](andso
v
theeffectivenoiseintensity,D )increasesmonotonously
eff
with D, the coordinate variance, σ2 = var[y], shows a
y
non-monotonous dependence, reflecting transitions from
bistable to monostable regimes.
FIG.4: Noise-inducedbifurcationsinanalogsimulations. (a):
Stochasticbistableoscillatorsarecharacterizedbytwo
Marginal PDF of the coordinate, P(y), (red circles) and its
fitusingtheeffectivepotentialEq.(11)(lines). (b): Marginal time scales: fast intrawell fluctuations and slower inter-
velocityPDF,P(v),(redcircles)andaGaussiandistribution wellswitching. Themeanfrequencyofbistableoscillator
with the same mean and variance σ2 = D (lines). Noise can be quantified by the Rice frequency [17, 37], which
eff
intensity, parameters of the effective potential and the effec- istherateofzero-crossingsbytheoscillator’scoordinate
tive noise intensity are: 1 – D = 1.51×10−4, α = 14.35, with positive velocity, ω = 2π(cid:82)∞vP(y = 0,v)dv. For
β = 3193.5, Deff = 1.15×10−2; 2 – D = 3.78×10−4, α = the Kramers oscillator (R1) with l0inear friction the Rice
−9.71, β =2532.4, D =4.07×10−2; 3 – D=3.00×10−3,
eff frequency reads [37],
α=76.12,β =7707.1,D =2.35×10−1. Otherparameters
eff
are the same as in the previous figure.
√ (cid:104) (cid:105)
2πDexp −U(0)
D
ω = . (13)
R ∞ (cid:104) (cid:105)
ilar to one of the Kramers oscillator. Although, indeed (cid:82) exp −U(y) dy
D
we do not expect P(v) to follow a Gaussian distribution −∞
as for the Kramers oscillator.
Noise-induced transitions are most apparent in a dia- It increases with noise intensity [37], reflecting the
gram of the marginal PDF of coordinate, P(y), plotted increase of the Kramers rate of transitions between
vsnoiseintensity,D,inFig.5(a). HerethePDFP(y;D) metastable states: the longer is the residence in
isshownasfilledcontourlinesallowingtotrackpositions metastable states, the smaller is the Kramers’s rate
ofPDF’sextremavsnoiseintensityandclearlyindicates and the Rice frequency. The dependence of the Rice
two pitchfork bifurcations. frequency on noise intensity for our circuit is non-
The described stochastic dynamics can be represented monotonous [Fig. 5(d)]: ω is low for weak and strong
R
in terms of an effective potential U (y), such that the noise, where the system is bistable, and attains its max-
eff
marginal coordinate PDF is P(y) ∝ exp[−U (y)/D ], imal value for intermediate noise, corresponding to ef-
eff eff
with an effective noise intensity, D . Fig. 4(a) shows fective monostable dynamics with the minimal value of
eff
that the marginal coordinate PDF, P(y), can be nicely thebifurcationparameter, µ. UnlikefortheKramersos-
fitted by cillator (1) which is characterized by Gaussian velocity
distribution, the velocity PDF, P(v), for our bistable os-
(cid:20) (cid:21)
1 cillatorisnon-Gaussian[Fig.4(b)]. Neverthelessthenon-
P(y)=k exp − U (y) , U (y)=−αy2+βy4,
1 D eff eff monotonous dependence ω (D) is qualitatively approxi-
eff R
(11) mated by Eq.(13) with the effective potential and noise
where k is the normalization constant. The effective intensity, i.e. with the substitution U → U, D → D
1 eff eff
noise intensity is calculated as the variance of velocity, in (13).
5
FIG. 5: Noise-induced transitions in analog simulations. (a): FIG. 6: Noise-induced bifurcations in numerical simulations
Marginal coordinate PDF vs noise intensity, P(y;D). For of Eqs. (7). (a): Normalized coordinate PDF, P(y)/P vs
max
each value of noise intensity, D, the PDF P(y), was normal- noise intensity. (b): Effective bifurcation parameter µ(D).
izedtoitsmaximalvalue,i.e. P (y)=P(y)/P . Bluedots Vertical dashed lines indicate positions of two pitchfork bi-
n max
show minima of the corresponding effective potential (11). furcations. (c): Variances of coordinate (open black cir-
(b): Effective bifurcation parameter µ(D). Vertical dashed cles), σ2, and velocity (filled red circles), σ2 vs noise in-
v v
linesindicatepositionsoftwopitchforkbifurcations. (c): Co- tensity. (d): Rice frequency, ω , vs noise intensity. Red
R
ordinate and velocity variance vs noise intensity. The coor- circles show values numerically obtained values; solid line
dinate variance, σ2 is shown by the open black circles; the shows ω calculated from Eq. (13) with the effective po-
y R
velocity variance which equals the effective noise intensity, tential U and noise intensity, D . Other parameters are:
eff eff
D =σ2, is shown by filled red circles. (d): Rice frequency, ε=0.01,c =1,c =9,c =22,a=1.2,b=100.
eff v 1 3 5
ω , vs noise intensity. Open red circles show results of ana-
R
log simulations; solid line shows Eq.(13) with the effective
potential U and effective noise D .
eff eff IV. MECHANISM OF NOISE-INDUCED
TRANSITIONS
Noise-inducedchangesintheoscillator’sdynamicscan
B. Numerical simulations
be understood by studying the structure of the phase
spaceofthecorrespingdeterministicsystemdescribedby
Numerical simulations confirmed the existence of Eqs.(7) with D = 0. Fig. 7(a) shows two stable nodes,
multiple noise-induced transitions in the mathematical separated by the saddle at the origin, and nullclines of
model of the circuit Eqs.(7) as shown in Fig. 6. Double- the system. The intrinsic feature of the system under
peaked stationary coordinate PDF, P(y), becomes uni- study is an unusual structure of the nullcline v˙ =0. Be-
modal with the increase of noise intensity. Further in- sides the conventional N-shape branch passing through
crease of noise gives rise to the backward transition equilibria [inset in Fig. 7(a)], the nullcline includes two
from mono- to bistable dynamics with doubly peaked symmetricseparateclosed-loopbranches. Letusconsider
coordinate PDF shown in Fig. 6(a). As for analog aloopattheupperleftquadrantinFig.7(a). Theupper
simulations, noise-induced transitions are well described side of the loop is attractive and the lower side is repul-
by the effective bifurcation parameter, µ(D), shown in sive. When a phase trajectory approaches the loop from
Fig.6(b). Similartoanalogexperiment,thevelocityvari- above, it slows down and moves on the attractive side
ance, σ2, which determines the effective noise intensity, untilitapproachestheseparatrixofthesaddle,andthen
v
showsmonotonousincreasewithD,whilethecoordinate eventually falls onto vicinity of the saddle equilibrium at
variance possesses a minimum, reflecting the first noise- theorigin. Repulsivesideoftheclose-loopbranchdirects
induced transition from bi- to unimodal structure of the phase trajectories towards the stable equilibrium. Sym-
coordinatePDF[compareFigs.5(c)and6(c)]. Similarly, metrical behavior occurs for the loop at the right lower
the Rice frequency exhibit non-monotonous dependence quadrant. We note, that for the linear resistor N1 the
on noise intensity attaining its maximum at D corre- nullcline v˙ =0 has a single N-shape branch only, i.e. no
sponding to the minimal value of effective bifurcation closed-loop segments.
parameter, µ(D) [compare Figs. 5(d) and 6(d)]. Weak noise results in conventional stochastic hoping
6
indicates, that phase trajectories frequently visit closed-
loopbranches[areasaremarkedbythegreendashedline
on Fig. 7(c)] which results in deflection towards the ori-
gin, as described for deterministic case on Fig. 7(a).
This results in shifting of the PDF’s maxima towards
theorigin. Thereisacriticalnoiseintensitywhichcorre-
sponds to maximal influence of the closed-loop nullcline
branches, resulting in the PDF with single peak at the
origin. For larger values of noise intensity phase trajec-
tories begin to pass through the repulsive sides of the
closed-loop nullcline branches. They are then slow down
on attractive branches of the nullcline [areas marked by
violet dashed line on Fig. 7(d)] and then deflected to-
wardstheorigin. However,becauseoflargernoise,phase
trajectoriescannowovercometheseparatrixtowardsan-
otherstableequilibrium, ratherthanfallontotheorigin.
Asaresult,theoriginisvisitedlessfrequentlythansym-
metrical areas on the left and right of the separatrix. In
this way the central peak of the PDF becomes divided
into two and bistability is restored.
Analysis of the oscillator model has shown that S-
shapedI-VcharacteristicofnonlinearelementN1(Fig.1)
with fifth-order nonlinearity in the function F(x) in
Eq. (4) is necessary for specific nullclines arrangements
shown in Fig. 7(a) and so for the observed noise-induced
transitions. Addition of higher order nonlinearities (e.g.
seventh-order term) to F(x) does not change qualita-
tively the dynamics of the system.
V. CONCLUSIONS
We have developed a generic model of the bistable os-
cillator with nonlinear dissipation. Using analog circuit
experimentandnumericalsimulationweshowedthatthis
system demonstrates multiple noise-induced transitions,
registered as changes of extrema in the stationary PDF.
Using the effective potential approach we showed that
theobservednoise-inducedtransitionsaredescribedbya
FIG. 7: Mechanism of noise-induced pitchfork bifurcations.
On all panels: equilibrium points are shown by blue circles; normalformofpitchforkbifurcation. Wenotethatqual-
blue dashed line indicates the nullcline y˙ = 0; orange solid itative structural changes in the stationary PDF are not
line shows the nullcline v˙ = 0; the separatrix of the sad- pure noise-induced transitions [18], as the bistability is
dle at the origin is shown by blue dotted line. (a): De- inherent for the model. Nevertheless, the observed tran-
terministic dynamics of Eq.(7). Phase trajectories started sitions from bimodal and unimodal and back to bimodal
from various initial conditions are shown by black arrowed cannotberealizedinthesystembychangingasinglepa-
lines. Inset shows expanded region near equilibria. Pan-
rameter of nonlinear or linear components of the circuit.
els (b)–(d) also show contour maps of the stationary PDF,
Thus, noise intensity is an independent control parame-
P(y,v), obtained numerically. (b): D = 2 × 10−5; (c):
ter which determines qualitative nature of the system’s
D = 6 × 10−5; (d): D = 2.4 × 10−3. Other parameters
dynamics. We provided a clear explanation of the mech-
are: ε=0.01,c =1,c =9,c =22,a=1.2,b=100,i.e. the
1 3 5
anism of the effect based on partition of the phase space
same as in analog experiments.
of the system by nullclines and manifolds of the saddle
equilibrium.
between two metastable states with the double-peaked
PDF [Fig. 7(b)]. Note, that probability to cross closed-
loop branches of the nullcline is rather low and so they Acknowledgements
have no effect on the position of the PDF’s maxima.
With the increase of noise intensity the PDF P(y,v) This work was supported by the Russian Foundation
smearsvertically,i.e. withrespecttovelocity,v. Fig.7(c) for Basic Research (RFBR) (grant No. 15-02-02288)
7
and by the Russian Ministry of Education and Science ABN gratefully acknowledges the support of NVIDIA
(project code 1008). We are very grateful to E. Sch¨oll, Corp. with the donation of the Tesla K40 GPU used
A. Zakharova, and G. Strelkova for helpful discussions. for this research.
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