Stochastic Resonance in Underdamped Periodic Potential Systems with Alpha Stable Lévy Noise Ruo-Nan Liu and Yan-Mei Kang* School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Corresponding author email:
[email protected]
Abstract In this paper, we investigate the effect of alpha stable Lévy noise with alpha stability index ( 0 2 ) on stochastic resonance (SR) in underdamped periodic potential systems by the non-perturbative expansion moment method and stochastic simulation. Using the spectral amplification factor as a quantifying index, we find that SR can occur in both sinusoidal potentials and ratchet potentials when is close to 2 , while the resonant effect becomes weaker as the stability index decreases. By means of massive numerical statistics, we ascribe this trend to the typical jumps of non-Gaussian Lévy noise ( 0 2 ), which play a destructive role on the periodicity of the long time mean response. We also disclose that the skewness parameter of Lévy noise has a more notable impact on the resonant effect of the asymmetric ratchet potential than that of the symmetric sinusoidal potential because of symmetry breaking.
Keywords stochastic resonance; periodic potential; Non-perturbative expansion moment method; Lévy noise
1. Introduction Stochastic resonance (SR), as a term coined by Benzi et al. [1] in explaining the cyclical variation of the warm climate and the cold climate in paleoclimatology, has been broadly applied to describe a counterintuitive phenomenon where the presence of suitable amount of noise can optimize the output signal quality in a nonlinear system [2, 3]. Due to the wide application in a variety of fields [4-11] such as physics, biology, chemistry, economics and circuits measurement, SR has been extensively studying in various models including bistable model, threshold device, excitable and multi-stable systems from the background of thermal noise to non-thermal noise [12-19]. https://authors.elsevier.com/a/1Wzbw1LUy9JHli
When the environmental noise is modeled by thermal Gaussian noise, SR has been extensively documented [2, 20]. As far as the periodic potential systems that acts as a classical model for multi-stable SR is concerned, the existing investigations have expounded the occurrence of SR from different viewpoints [21-29]. For instance, Saikia et al. [25] numerically disclosed that SR can be observed when the driving frequency is close to the natural frequency of the sinusoidal potential. With the same sinusoidal potential, Saikia [26] also found that SR occurs when the damping coefficient is below a critical value, Reenbohn et al. [27] showed SR and ratchet effect under a biharmonic external driving, while Liu et al. [29] found that the optimal noise intensity for SR shifts to a larger value as the noise correlation time increases. These existing literatures clarify the behavior of SR by stochastic simulation in the periodic potentials to a certain extent, but the relevant investigation has not been extended to the case of non-thermal noise. Nevertheless, in recent experimental investigations, a growing evidence of the non-thermal fluctuation has been found in the periodic potential systems such as Josephson threshold detectors [30, 31, 32]. The phenomenon of SR induced by the non-thermal fluctuation has been found in bistable potentials [14, 33-35], but how the non-thermal noise affects the SR in the multistate periodic potential systems is still unclear. Theoretically, the non-thermal fluctuation in underdamped periodic potential systems has been modeled by non-Gaussian Lévy noise [36, 37, 38]. The alpha stable Lévy noise is governed by four parameters [39, 40, 41]: alpha stability index
1
0 2 ,
skewness parameter
1, mean parameter R and scale parameter 0, , thus it is more
powerful in describing general realistic fluctuations [42]. Depending on the choice of alpha stability index, it covers Gaussian noise ( 2 ) and non-Gaussian noise ( 0 2 ) which can model the thermal fluctuations and non-thermal fluctuations, respectively. In order to have a general insight into the cooperative effect of the realistic noise and the harmonic signal, we take the underdamped periodic potential systems driven by alpha stable Lévy noise into account, and explore how the alpha stability index and skewness parameters of Lévy noise affect the occurrence of SR. The paper is organized as follows. In Section 2, the model and the general numerical https://authors.elsevier.com/a/1Wzbw1LUy9JHli
scheme are introduced. In Section 3, a semi-analytic method of moments based on non-perturbative expansion is presented for observing SR when the Lévy noise is Gaussian-type ( 2 ), and the simulated results in the case of non-Gaussian Lévy noise ( 0 2 ) are presented in Section 4. Conclusion and discussion are drawn in Section 5.
2. Model and Method Without loss of generality, we consider an underdamped particle of unit mass moving in a periodic potential driven by an external periodic signal and alpha stable Lévy noise, the corresponding Langevin equation reads
d2 x dx V ( x) (t ) (t ) 2 dt dt x where V ( x) V1 sin(
(1)
2 4 x) V2 sin( x) denotes the periodic potential of period L , is the L L
damping coefficient, (t ) 0 cos t is the driving periodic signal of amplitude 0 and frequency , and (t ) stands for the alpha stable Lévy noise, which is the generalized derivative of Lévy motion (t ) [39]. Here we emphasize that the general form of V (x) ( V2 0 ) could denote the ratchet potential (see Fig. 1) appearing in molecular motor, superconducting quantum interference devices and so on [43-46], and it is reduced to a sinusoidal potential when V2 0 which usually applies to Josephson junction detector, pendulum motion and superionic conductor etc. [47, 48]. One can refer to Ref. [45, 46, 47] for a comprehensive review. Lévy motion (t ) is a jump-diffusion process obeying to a quadruple distribution
L , ( ; , ) defined through the characteristic function k d eik L , ( ; , ) with
exp{ k [1 i (tan 2 ) sgn(k )} ik}, 1; (k ) 2 exp{ k [1 i sgn(t )ln k ] ik}, 1
(2)
where the alpha stability index ( 0 2 ) determines the thickness of the tail and the Gaussian distribution can be recovered when
2 , the skewness parameter measures the
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asymmetry,
indicates the mean (or center) of the distribution which is fixed to zero
throughout the context, the dispersion coefficient characterizes the deviation of the distribution, and acts as the noise intensity (denotes as D ). As the complete specification requires four parameters, it can better fit the actual data and maintain the mechanism of noise generation and propagation.
1
V(x)
0.5 0 -0.5 -1 -6
-4
-2
0 x
2
4
6
Fig.1. Schemata of the sinusoidal potential V2 0 (black dashed line) and the ratchet potential
V2 0.2 (blue solid line) with V1 1 . From the figure, it is clear that the ratchet potential is asymmetric, while the sinusoidal potential is symmetric.
For the sake of completeness, let us give a brief introduction to the generating algorithm for alpha stable random number
[49]. If define W1 U1 1/ 2 , W2 ln U 2 , 1
0 arctan( tan( / 2)) , 1 (1 2 tan 2 ( / 2))2 with uniformly distributed random numbers U i U (0,1) ( i 1,2 ), then with
sin{ W1 0 } cosW1 (W1 0 ) X 1 [ ] {cos(W1 )}1 / W2
1
(3)
for 1 , and
X
2
{( / 2 W1 ) tan W1 ln(
/ 2W2 cosW1 )} / 2 W1
https://authors.elsevier.com/a/1Wzbw1LUy9JHli
(4)
for 1 , the alpha stable Lévy random number can be generated according to
1; X , X 2 log( ) / , 1.
(5)
Using the above random number generation algorithm to produce sample trajectories, we give the probability density function for Lévy distribution L , ( ; , ) under different stability indexes and skewness parameters, as shown in Fig.2 (a) and (b). From these pictures, it can be seen that the Lévy distribution becomes symmetric when 0 and it is reduced to the well-known Gaussian distribution when 2 . We exhibit the time series for Gaussian noise and non-Gaussian Lévy noise in Fig.2 (c) and (d), respectively. Note that the unpredictable large jump in non-Gaussian Lévy noise might induce immediate transitions between nonconsecutive two potential wells for multi-stable systems [42], thus non-Gaussian Lévy noise might result in completely different observations from the Gaussian-type Lévy noise. Therefore, it is worthy to check whether SR still occurs in the periodic potential system (1) when 0 2 and how the parameters of Lévy noise influence the resonant effect.
=2.0 =1.5 =1.0 =0.5
(a)
L( ;=1,=0)
0.5 0.4 0.3 0.2
0.2 0.15 0.1 0.05
0.1 0
=1 =0.5 =0 =-0.5 =-1
(b) 0.25 L( ;=1,=0)
0.6
-5
-2.5
0
2.5
5
0
-5
-2.5
0
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2.5
5
6
1.5
4
0.5
Lévy noise
Gaussian noise
(d)
(c)
1
0 -0.5
2 0 -2
-1 -1.5 0
10
20
30
40
50
60
-4 0
10
20
30
40
50
60
t
t Fig.2 Probability density functions (a)
0 and (b) 1.5 for Lévy distribution L , ( ; , ) ;
Time series of Lévy noise (c) 2 and (d) 1.7 with 0.5 , D 0.25 . It is easy to see that the non-Gaussian Lévy distribution shows a property of spike peak and heavy tail, and the non-Gaussian Lévy noise has large sudden jumps.
In general, the spatial fractional-order Fokker-Planck (FP) equation governing the non-Gaussian Lévy noise [13] is not easy to be solved, so we exclusively attempt to seek a semi-analytic probability description when
2 , but seek numerical solutions using the
stochastic second-order Heun’s method [27] when 0 2 . The following difference scheme is adopted all through the context.
~ xn1 xn vn t , 2 2 4 4 v~n1 vn [vn V1 cos( xn ) V2 cos( xn ) 0 cos(tn )]t t 1/ n , L L L L t xn1 xn (vn v~n1 ) , 2 t 2 2 4 4 vn1 vn [vn V1 cos( xn ) V2 cos( xn ) 0 cos t 2 L L L L 2 2 4 4 v~n1 V1 cos( ~ xn1 ) V2 cos( ~ xn1 ) 0 cos((t n t ))] t 1/ n L L L L where n is the alpha stable Lévy distributed random number in the nth time interval. In this paper, we focus on the response of the velocity v (t ) to the periodic forcing, since v(t ) x (t ) leading to v( ) ix( ) in Fourier transform domain should behave similar to x (t ) when noise is changed [23]. Since the heavy-tailed, discontinuous and irregular jumps in Lévy noise https://authors.elsevier.com/a/1Wzbw1LUy9JHli
might cause the sample paths to reach infinity rapidly for the small stability index, we adopt a truncation scheme [50, 51] on v (t ) such that v(t ) 50 sign(v(t )) whenever v(t ) 50 . Here we stress that according to our simulation experiences, whether adopting the above truncation technique depends on the size of the alpha stability index. When the alpha stability index is large such as 1.5 , one can skip over this treatment because the simulated resonant curves are consistent with or without truncation. But when the alpha stability index is quite small such as 1.0 , the large sudden jumps tend to bring the simulated trajectories to infinity, and one needs to wait a very long time for the system to come back if there is no compulsory truncation, so in this case an appropriate truncation scheme should not be omitted. Considering the physical and computer saturation effects, a suitable truncation scheme should be adoptable. This should be the reason why a truncation scheme is employed in many relevant references [50, 51].
3. Stochastic resonance under Gaussian Lévy noise As mentioned in Section 1, Lévy noise of four parameters is a big cluster of noise and it fell into Gaussian noise when 2 . When 2 , the system (1) can be rewritten into the following Langevin equation
d2x dx V ( x) (t ) (t ) 2 dt x dt where white Gaussian noise
(6)
(t ) connected with a noise intensity D via the
fluctuation-dissipation theorem satisfies
(t ) 0 and (t ) (t ) 2D ( )
with
D k BT . Here T denotes temperature and k B represents for Boltzmann constant. We remark that SR is a phenomenon observed in the statistical average response of nonlinear stochastic systems, the abstraction of its typical characteristics from massive observations always consumes much calculation time, thus the direct simulation is a straightforward method but not our best choice. This motivates us to develop a semi-analytic technique, i.e. the method of moments to explore the phenomenon of SR in the system (1). This task is generally not practical, but as shown below it is feasible when Lévy noise is https://authors.elsevier.com/a/1Wzbw1LUy9JHli
reduced to Gaussian noise. In this case, the following FP equation (7) is just the Kramers equation and for this Kramers equation SR has been studied for the bistable potential with Gaussian white noise [12, 52] and dichotomic noise [53]. Let P( x, v, t )
denote the time-dependent coordinate-velocity probability density
function, and then the governing FP equation [47] reads
P( x, v, t ) 2P V ( x) P D 2 {[v (t )] P} v t v v x x
(7)
where P( x, v, t ) obeys the periodic boundary condition [54], i.e. Px, v, t Px L, v, t . In the absence of (t ) , it is easy to verify from Eq. (7) that as the time tends to infinity the system (6) attains a steady state where the time-independent probability density function is given by P0 ( x, v) Z 1 exp( (V ( x)
v2 ) / D) with normalization constant Z. 2
Let F ( x, v) be an arbitrary function of x and v , and assume the time-dependent moment
F ( x, v) (t )
L
0
F ( x, v) P( x, v, t )dvdx exists, then multiplying the both sides of
Eq.(7) by F ( x, v) and integrating lead to (after integration by parts)
F ( x, v) (t ) 2F V ( x) F F . D [v (t )] v 2 t v x v x
(8)
With the help of Floquet’s theorem [2], the asymptotic solution of FP equation (7) is a time-periodic function i.e. P( x, v, t
2 ) P( x, v, t ) , thus the solution can be decomposed into
a Fourier series [52, 55]
P( x, v, t )
P ( x, v)e
n
int
n
where the unknown Pn ( x, v) ( n 0 ) should satisfy
L
0
(9)
Pn ( x, v)dvdx 0 according to the
conservation of probability. Since the expansion coefficients Pn ( x, v) are not explicitly dependent on the driving amplitude, Eq. (9) is a kind of non-perturbative series expansion and can be regarded as an extension of the previous method of moments [54]. Substitution of Eq. (9) into Eq. (8) and using the orthogonality of trigonometric functions https://authors.elsevier.com/a/1Wzbw1LUy9JHli
yield
in F
{D n
2F v 2
[v V ( x)] n
F v
v n
F x
} n
0 F ( 2 v
n 1
F v
(10)
) n 1
where ... n means taking average with respect to the weight function Pn ( x, v) . Inspired by [52], we make an ansatz Pn ( x, v) P0 ( x, v) p n ( x, v) and adopt the easily verifiable identities
vF( x, v) 0
in Fpn
D F v
0
D
0
，
V ( x) F (v, x) 0
F p n v v
0
D F x
to turn Eq.(10) into 0
D F p n F p n x v v x
0
0 F ( p n 1 2 v
0
F p n 1 v
) . (11) 0
Considering that the asymptotic solution of FP equation (7) satisfies periodic boundary with respect to x and natural boundary with respect to v [52, 54], we can expand
pn(x,v) cn; j,k H j(v/vt )e
ik
2π x L
(12)
j 0 k
where H j () is the jth Hermite polynomial with the thermal velocity vt 2D / taken as a scale factor. Substitution of expansion (12) into Eq. (11) with F ( x, v) H s (v / vt )e
il
2 x L
( s 0,1, 2, ... , l 0, 1, 2,... ) yields an infinite-dimensional set of linear algebraic equations. For the aim of numerical calculation, we truncate the Fourier series (9) at N and Eq. (12) at J and K to obtain a block-tridiagonal system
Dn cn E (cn 1 cn 1 ) 0, n 1
(13)
where the matrices Dn and E are given by
A0( n ) C Dn
B0 A1( n ) C
B1 A2( n )
, BJ 1 C AJ( n )
0 G 0 E G 0 G 0
with the entries of the matrices As(n ) , Bs ( 0 s J ), C and G being
( As )l , k (s in) e (n)
i (l k )
2
2 x L
, ( Bs )l ,k 0
2 i (l k ) L x i 2 D / ( s 1)l e L
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, 0
(C ) l ,k i D /(2 ) k
2
2 i (l k ) L x e L
, (G ) l , k 0
0
e
2 2D /
i (l k )
2 x L
. 0
We adopt the Gaussian block-elimination method to solve the system (13). After the solution has been found, the nth dynamical susceptibility v( n )
L
2
0
0
(
) n vPn ( x, v)dvdx which
measures the scaled strength of nth harmonic in the output signal [52], can be obtained as
v( n )
K
k K
cn;1,k (
2
0
)n 2D / e
ik
2 x L
(14)
. 0
And then the long time ensemble mean response reads
v
N
0
n 1
2
(t ) 2 ( as
) n Re vn eint .
(15)
According to Floquet’s theorem [2], the long-time probability density of the system (6) is time periodic, thus it is natural to obtain the convergent Fourier series expansion (9). Since we are only interested in observing the phenomenon of SR at the first three harmonics as usual, the truncation order N 3 is fixed. Our numerical experiences show that within a large parameter range the method of moments converges well as the truncation orders increase (see Fig.3). For example, the dynamical susceptibilities obtained at J 6 , K 13 are almost coincident with those obtained at J 3 , K 7 . That is to say, the increase of the truncation orders does not change the results by more than 0.1%. In the following calculations, we adopt J 6 and K 13 to exhibit the theoretical results.
1.2
1.6
(a)
1.5
0.8 0.6
1.3
) Im((1) v
) Re((1) v
1.4
1.2 1.1
0.4 0.2 0
1
-0.2
0.9
-0.4
0.8 0
(b)
1
0.1
0.2
D
0.3
0.4
-0.6 0
0.1
0.2
D
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0.3
0.4
1
0.6
(d)
0.8
0.4
0.6
0.2
Im(v(2))
Re(v(2))
(c)
0.4 0.2 0 0
0 -0.2
0.1
0.2
0.3
-0.4 0
0.4
0.1
0.2
D
0.3
0.4
D 0.7
(e)
0.2
0.5
0
Im(v(3))
Re(v(3))
0.1
-0.1 -0.2
0.4 0.3 0.2
-0.3
0.1
-0.4 -0.5 0
(f)
0.6
0.1
0.2
0.3
0.4
0 0
0.1
D
0.2
0.3
0.4
D
Fig.3. Dependence of the real part ((a), (c) and (e)) and imaginary part ((b), (d) and (f)) of the first three dynamics susceptibilities on noise intensity under different truncation orders: J 3 , K 7 (black solid line), J 4 , K 9 (green dashed line) and J 6 , K 13 (red asterisk). The system parameters are taken as 0.4 , V1 1 , V 2 0.1 , L 2 , 0 0.2 , and 0.25 .
Now we turn to observe the first three harmonic SRs in the system (6). With the nth 2
order spectral amplification factor v(n ) ( 1 n 3 ) [2, 20, 52, 56] as a measurement, we show the curves of the spectral amplification factors via noise intensity for different V 2 in Fig.4. It is clear that the non-monotonic dependence of these curves reveals SR of underdamped periodic potential systems for given parameters. It also can be seen that as the asymmetry coefficient V 2 increases from zero, the peaks of v(1)
2
( 3) 2
and v
decrease, but
2
the peak of v( 2 ) appears due to the symmetric breaking. Additionally, one can see that the first order harmonic spectral amplification factor always plays a dominant role in the SR behavior even in the case of asymmetric potentials ( V2 0 ) as the peak of the first harmonic https://authors.elsevier.com/a/1Wzbw1LUy9JHli
is much higher than that of the other harmonics.
1.4
5
(a)
(b)
1.2
4 3
(2) 2 | v |
|v(1)|2
1
2
0.8 0.6 0.4
1
0.2
0 0
0.1
0.2
0.3
0.4
0 0
0.1
0.2
D
0.3
0.4
D
1.2
(c) 1
|v(3)|2
0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
D Fig.4. Dependence of the spectral amplification factor of the first harmonic (a), the second harmonic (b) and the third harmonic (c) on noise intensity: direct simulation (black dotted line) and theoretical method V2 0 (blue solid line)， V2 0.1 (green dash line) and V2 0.2 (red dash dot line). The other parameters are the same as Fig. 3. Noting the scale differences on the y-axis, it is clear that the peak of the first harmonic is higher than the peaks of the second and three harmonics.
4. Stochastic resonance under non-Gaussian Lévy noise In underdamped periodic potential systems such as Josephson junctions and molecular motors, the general realistic noise might be caused by thermal or non-thermal fluctuations. Usually, Gaussian noise is suitable for the thermal fluctuation and the non-Gaussian Lévy noise is more appropriate for describing the non-thermal fluctuations of heavy-tailed distribution. Therefore, it is natural to wonder whether SR still occurs in the periodic potential systems driven by non-Gaussian Lévy noise and how the non-Gaussian Lévy noise affects the https://authors.elsevier.com/a/1Wzbw1LUy9JHli
phenomenon. Noting that when 2 the FP equation corresponding to the system (1) is spatial fractional-order [13] and it is difficult to obtain a semi-analytic solution similar to the case of
2 , so for every set of fixed parameters, we only resort to the stochastic second-order Heun’s method to obtain 100000 sample trajectories with time step-length t 0.01. Then, we make simple arithmetic to acquire the long time ensemble mean response and thus the spectral amplification factor after discarding the transient evolution. Suppose the long time ensemble mean response v
as
(t ) has been obtained, then the nth dynamic susceptibility can
be calculated according to ( n) (an ibn ) / 0 / 2 where n
with t m
an
1 L v L 0
bn
1 L v L 0
2nt t M (t ) cos( )dt v as L L m1
as
(t ) sin(
2nt t M )dt v L L m1
as
2nt m (t m ) cos( ), L
(t ) sin( as m
2nt m ), L
2mt L and M . Considering that the spectral amplification factor of the first M t
harmonic plays a dominant role in the SR behavior, we only show the non-monotonic evolution curves of the spectral amplification factor of the first harmonic via noise intensity in this section in Figs.5-6. Comparing Fig.5 (a) and Fig.6 (a), we can see that as the stability index decreases, the peak of the first harmonic becomes lower both for the symmetric sinusoidal potential and for the asymmetrical ratchet potential. In fact, our observations that are not shown in these figures exhibit that for given parameters the dependence of the spectral amplification factor via noise intensity monotonically decreases if is sufficiently small such as 1. This signifies that SR occurs only when the non-Gaussian Lévy noise is close to Gaussian. In other words, the resonant effect is notable for the non-Gaussian Lévy noise of large stability index, and it can be concluded that Gaussian noise is optimal for observing SR among the whole cluster of Lévy noise. This observation is contrary to that observed in the case of the 1 / f noise [57] where 1 / f noise with the characteristic of the long-range dependence has been argued more effective for the occurrence of the low-frequency SR than Gaussian noise. With the alpha stability index fixed, we see that the skewness parameter has limited https://authors.elsevier.com/a/1Wzbw1LUy9JHli
influence on the resonant effect for the sinusoidal potential (Figs.5 (b)), but it has the dominant effect for the ratchet potential (Figs.6 (b)). Namely, in the ratchet potential, the smaller skewness parameter leads to a better SR performance. The difference in the sinusoidal potential and the ratchet potential should be a result of symmetry breaking, since the skewness parameter has a noticeable impact on the asymmetry of the potential in the system (1).
(a)
=1.5 =1.6 =1.7 =1.8 =1.9 =2
=0
4
(1) 2 | | v
3.5 3
3.2 3
(b)
2.5
2.6
=-1 =-0.5 =0 =0.5 =1
2.4 2.2
2
2
1.5 0
0.05
0.1
0.15
0.2
=1.7
2.8 (1) 2 | v |
4.5
1.8 0
0.25
0.05
0.1
D 4
=1.7,=0
3.5
(1) 2 | | v
0.2
0.25
4.5
0=0.1
3
(d)
0=0.2
4
0=0.3
3.5
0=0.4
2.5
(1) 2 | | v
(c)
0.15
D =0.3 =0.4 =0.5
=1.7,=0
3 2.5
2 1.5 0
2
0.05
0.1
0.15
D
0.2
0.25
1.5 0
0.05
0.1
0.15
0.2
0.25
D
Fig.5. Dependence of the spectral amplification factor on noise intensity. The parameters are
V1 1 , V2 0 , 0.25 , L 2 , in (a) , (b) and (c) 0.4 ; in (a) , (b) and (d) 0 0.2 . As seen from (a), the resonant effect in the sinusoidal potential system becomes weakened as the stability index decreases, while from (c) and (d) it is clear that the larger driving amplitude and the smaller damping coefficient are beneficial for exhibiting SR.
In Figs.5-6 (c) we show the effect of the driving signal amplitude on SR in the system (1). Clearly, as the signal amplitude decreases, the resonant peak becomes weakened and the https://authors.elsevier.com/a/1Wzbw1LUy9JHli
peak location shifts towards a larger noise level. This observation is attributed to that when a resonant particle transits from a weak signal modulated potential, more noise energy will be needed to overcome the effect of potential barrier, thus the optimal noise intensity becomes larger and the maximum spectral amplification factor turns smaller. This observation can be found in periodic potential systems under Gaussian thermal noise as shown in Ref. [28]. It means that this trend is not special to non-Gaussian Lévy noise. Furthermore, in Figs.5-6 (d), we show the effect of the damping coefficient on SR. It is clear that as the damping coefficient increases, the height of resonant peak drops and the optimal noise intensity becomes larger, and this observation could intuitively owe to the fact that less energy will be dissipated when the damping is lighter. In fact, this trend is also not special to non-Gaussian Lévy noise, as it has been found in Duffing oscillator [12] and underdamped periodic potential systems [26] under Gaussian thermal noise.
(a)
=1.5 =1.6 =1.7 =1.8 =1.9 =2
=0
1.8 (1) 2 | | v
1.6 1.4
2 (b)
1.6 1.4
1.2
1.2
1
1
0.8 0
0.1
0.2
0.3
0.8 0
0.4
0.1
D
(c)
(d) 1.8
1.6
0=0.3
1.6
1.4
0=0.4
|2 |(1) v
|2 |(1) v
=1.7,=0
1.2
1
1
0.2
D
0.4
0.3
0.4
=0.3 =0.4 =0.5
=1.7,=0
1.4
1.2
0.1
0.3
2
0=0.1 0=0.2
0.8 0
0.2
D
2 1.8
=-1 =-0.5 =0 =0.5 =1
=1.7
1.8 (1) 2 | | v
2
0.8 0
0.1
0.2
0.3
0.4
D
Fig.6. Dependence of the spectral amplification factor on noise intensity. Here V2 0.2 and the other parameters are the same as Fig. 5. In the ratchet potential system, again, we see that the smaller
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stability index, the larger driving amplitude and the smaller damping coefficient are beneficial for exhibiting SR. Nevertheless, due to the symmetry breaking, the obvious effect of the skewness parameter on SR (b) in this case is different from that in Fig. 5(b).
0.4
0.4
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Fig.7.The long time ensemble mean response under different alpha stability indexes: (a) 2 ; (b)
1.7 ; (c) 1.3; (d) 1.0 . The other parameters are D 0.24 , 0.4 , 0.25 , L 2 , V1 1 , V2 0 and 0 0.2 . In the case of Gaussian noise, the long time ensemble mean response is periodic (a), but it is clear from (b), (c) and (d) that as the stability index decreases, the periodicity disappears because of the presence of irregular jumps.
Here we wish to explain the effect of the alpha stability index of Lévy noise on SR by means of the long time ensemble mean response. Fig. 7 shows that as the alpha stability index decreases, the irregular large jump in the average response appears and becomes more and more protuberant, as a result the periodicity of the long time ensemble mean response existing https://authors.elsevier.com/a/1Wzbw1LUy9JHli
in the case of Gaussian noise is destroyed by the presence of non-Gaussian Lévy noise to a more severe extent. Thus, when 2 , we cannot look forwards to the long time ensemble mean response to be periodic, but have to expect an obvious aperiodicity especially when the stability index is sufficiently small. This should be why the SR effect is weakened or does not occur in the case of non-Gaussian Lévy noise. At the end of this section, let us have some remarks on the physical mechanism underlying SR. Noting that there is a match between the mean first passage time and one-half of the driving period when SR occurs in the standard stochastic resonant system [2, 58], so we have tried to explain the phenomenon in this way but failed. Nevertheless, as is known, this failure is not a bit queer, since the exact matching relation cannot be obtained in general. Therefore, we turn to explain it in a nonconventional way to check whether there is coincidence between the noise-tuned coherent frequency peak [12, 59] and the driving frequency. In fact, our numerical experiments show that the phenomenon of SR cannot occur in the periodic potential system for all driving frequencies. That is to say, there is a suitable resonant frequency range, which is close to but below the natural frequency of the undamped linearized periodic oscillators at the periodic potential well bottoms. This demonstrates that SR in the periodic potential systems has certain connection with the nonconventional SR, although only a very rough match between the driving frequency and the noise-tuned frequency peak (with figure omitted) has been observed. This feature distinguishing from the nonconventional SR in the underdamped monostable and bistable oscillator systems [12, 59, 60] should reflect the complexity of the periodic potentials. In fact, this complexity has been reported from the viewpoint of two dynamical states identified by the phase difference of the driving force and the trajectory of the random moving particle [25, 28].
5. Conclusions In this paper, the effect of alpha stable Lévy noise with alpha stability index (0,2] on the phenomenon of SR in underdamped periodic potential systems driven by an external harmonic signal has been investigated. In the case of 2 , we present a non-perturbative moment method for calculating the first three harmonic susceptibilities. The first harmonic SR plays a dominant role among the first three harmonic SRs, and the second harmonic SR https://authors.elsevier.com/a/1Wzbw1LUy9JHli
can only occur in the ratchet potential due to symmetry breaking. In the case of 0 2 , we reveal that the effect of Lévy noise on the first harmonic SR by the second-order Heun’s method. From our investigations, we conclude that the phenomenon of SR can occur in both the sinusoidal potential and the ratchet potential only when Lévy noise is close to Gaussian if not exactly Gaussian, and the Gaussian noise is optimal among the big cluster of Lévy noise for exhibiting SR behavior. This is a result evidently different from that observed with the
1 f noise of long-range dependence [57]. This novel effect of the non-Gaussian Lévy noise is due to the aperiodicity of long time mean response caused by large sudden jumps. We also find that as a result of symmetry breaking, the skewness parameter of non-Gaussian Lévy noise has no obvious effect in a sinusoidal potential, but it plays an apparent role in the ratchet potential. Furthermore, we explore the influence of the damping coefficient and the driving amplitude on SR. From the viewpoint of the energy conversion of noise and signals, we have explained why the peak of the resonant curves decreases and shifts to a larger noise level with the decreasing amplitude or the increasing damping coefficient. This trend is not special to non-Gaussian Lévy noise as it has been found in periodic potential systems under Gaussian thermal noise [26, 28]. The investigation should be helpful to understand the cooperative behavior of the realistic noise and the signal in various systems such as molecular motor, Josephson junction detector and pendulum motion.
Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 11372233and 11772241).
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