The effect of colored noise on heteroclinic orbits
Jean-Régis Angilella




  

Université de Caen Normandie, ESIX Cherbourg, ABTE EA4651-ToxEMAC, 14000 Caen, France

  



The dynamics of a weakly dissipative Hamiltonian system submitted to stochastic perturbations has been investigated by means of asymptotic methods. The probability of noise-induced separatrix crossing, which drastically changes the fate of the system, is derived analytically in the case where noise is an additive Kubo-Anderson process. This theory shows how the geometry of the separatrix, as well as the noise intensity and correlation time, affect the statistics of crossing.
 Results can be applied to a wide variety of systems, and are valid in the limit where the noise correlation time scale is much smaller than the time scale of the undisturbed Hamiltonian dynamics.  
 



   

The effect of colored noise on heteroclinic orbits
    Jean-Régis Angilella
    
==================================================





§ INTRODUCTION


 
 
 
 Most dynamical systems in real conditions are submitted to some noise, and 
  even weak noise can significantly affect the evolution of the system. Spectacular phenomena like noise-induced chaos <cit.>, noise-induced synchronization <cit.>, and noise-induced escape
 <cit.>
received considerable attention in the last decades. These works allowed to clarify the role of noise and to derive relevant statistical laws. In the case of planar systems displaying homoclinic or heteroclinic orbits, it was shown that weak noise could induce separatrix crossing, and this effect was studied by means of stochastic Melnikov functions <cit.>. 



  Some authors considered the effect of piecewise constant noise on dynamical systems  <cit.>. These perturbations are known to have non-zero correlation times, i.e. they are colored noises 
<cit.>.  The advantage of this approach is that it can be applied to many realistic phenomena of interest, where some external forcing takes random  values over finite time intervals <cit.>. 
  When noise has a bounded amplitude,  the use of stochastic Melnikov functions allows to derive a necessary condition for the appearance of chaos near homoclinic or heteroclinic cycles. These are extremely useful criteria, since they provide a threshold  below which  noise-induced chaos is impossible.  This was done by  Sivathanu et al. <cit.>, who studied a Duffing oscillator submitted to dichotomous noise, that is on-off signals that change at random times.  
  
  However, in spite of the efforts done in last decades, no general theory for the effect of such a colored noise on Hamiltonian systems has emerged so far.
 This situation corresponds to a wide class of problems involving mechanical or electrical oscillators driven by a piecewise constant forcing, or transport of inertial particles in flows, where turbulent eddies are often modeled as a random force kept constant during the lifetime of the eddy (see for example Ref. <cit.>).  
 The aim of the present work is to derive a general expression for the appearance of noise-induced separatrix crossing, which could be useful for systems of this kind.
  
 To achieve this goal, a weakly dissipative planar Hamiltonian system perturbed by a random force will be considered.  Because of dissipation, phase-space trajectories initiated on a heteroclinic orbit will slowly drift on one side of this orbit (also called separatrix in the following). The question investigated in this paper is twofold. Firstly, we will determine under which conditions the stochastic term can force the system to reach the opposite side of the separatrix, in spite of dissipation. Secondly, we will calculate the probability of this event. 
 We will assume that the duration of time intervals, over which the random force is constant, obeys an exponential distribution. This manifests the fact that these durations are memoryless. Our noise is therefore a Kubo-Anderson process <cit.>, with a finite correlation time scale δτ which will be shown to strongly influence the probability of noise-induced crossing. The general theory is presented in section <ref>. An application is discussed in section <ref>.          
 

 

§ PROBABILITY OF SEPARATRIX CROSSING


 
 
 

 §.§ General considerations


 
 We consider a state variable 𝐫(t) = (q(t),p(t)) evolving as
 
    q̇   =   ∂ H/∂ p + Λ  U_1(𝐫) + ε_1 ξ_1(t),
    ṗ   =    - ∂ H/∂ q + Λ  U_2(𝐫) + ε_2 ξ_2(t),

where H(𝐫) is a differentiable scalar function, 𝐔 = (U_1,U_2) is a steady vector field playing the role of a deterministic perturbation, Λ and ε_i's are
positive constants. We will assume that the deterministic perturbation is dissipative (∇ . 𝐔 < 0), even though the calculations presented here are valid for any vector field 𝐔.
The last terms ε_i ξ_i are the components of an additive noise.
Each function ξ_i(t) is a Kubo-Anderson process, as defined by Brissaud & Frisch <cit.>. It is
piecewise constant and equal to some ξ_ik in the time interval τ_k < t < τ_k+1, where τ_k is an increasing series. Both the noise values ξ_ik and the times τ_k are random variables. The former are assumed to be non-dimensional, with  zero average and  unit  variance. The latter are such that the durations δτ_k = τ_k+1 - τ_k are memoryless and are therefore taken to have an exponential distribution with mean value δτ and variance δτ^2. The number of discontinuity points τ_k within a duration Δ t  obeys a Poisson distribution with parameter Δ t/δτ. Under these hypotheses the autocorrelation function ⟨ξ_i(t) ξ_i(t+Δ t) ⟩ is equal to
exp(-  Δ t/δτ).

Throughout the paper we assume that the undisturbed system (i.e. Eqs. (<ref>)-(<ref>) with Λ = 0 = ε_i) has hyperbolic saddle stagnation points A and B, related by a heteroclinic trajectory of equation H(q,p) = H(A) = H(B)  (Fig. <ref>). Such a trajectory is generally called separatrix, as it separates well-defined portions of the phase space. In the undisturbed dynamics, the state variable 𝐫(t) cannot cross the separatrix. In contrast, in the perturbed system, i.e. Λ and ε_i's close to zero but non-zero, crossing is possible in the sense that the sign of H(𝐫(t)) - H(A) can vary with time.  
 This means that if 𝐫(t) travels close to A at some time t=τ_a, then 𝐫(t) will reach the vicinity of B for some time t=τ_b, but on the opposite side of separatrix AB. 
To follow the location of the state variable 𝐫(t) in the disturbed system, we will make use of the Hamiltonian H and study the variations of H(𝐫(t)). 



Defining τ_a and τ_b as the the times when 𝐫(t) passes nearest to A and B respectively, the phenomenon of separatrix crossing is defined by the event Δ H ≠ 0, where

    Δ H ≡ H(𝐫(τ_b))-H(𝐫(τ_a)) = ∫_τ_a^τ_b∇ H . 𝐫̇  dt,

and ∇ = (     ∂ / ∂ q ,  ∂ / ∂ p )^T.
By using Eqs. (<ref>)-(<ref>), the variation of undisturbed Hamiltonian reads

    Δ H  = Δ H^0 + ε_1 ∫_τ_a^τ_b∂ H/∂ q ξ_1(t)   dt
    + ε_2 ∫_τ_a^τ_b∂ H/∂ p ξ_2(t)   dt,

where Δ H^0 is the jump of Hamiltonian in the deterministic case :

    Δ H^0 = Λ∫_τ_a^τ_b∇ H . 𝐔  dt

which can be calculated, either analytically or numerically, as soon as H and 𝐔 are known. The integrals in Eq. (<ref>) can be approximated by considering that 𝐫(t) travels close to the separatrix. 
Let t_0 be the time when 𝐫(t) passes nearest to C, the middle point of separatrix AB. 
Then we assume that
𝐫(t) ≃𝐫_0(t-t_0), where 𝐫_0(t)=(q_0(t),p_0(t)) is a solution of the Hamiltonian system running on AB, that is H(q_0(t),p_0(t)) = H(A) for t ∈ ]-∞,+∞[, with an initial position  𝐫_0(0) equal to point C. 
This is a classical approximation, widely used in the calculation of Melnikov functions <cit.>. Under these hypotheses, the jump of Hamiltonian can be approximated as

    Δ H  = Δ H^0 - ε_1 ∫_τ_a^τ_bṗ_0(t-t_0)  ξ_1(t)   dt
    + ε_2 ∫_τ_a^τ_bq̇_0 (t-t_0)  ξ_2(t)   dt.

Then, by making use of the fact that ξ_i(t) are constant and equal to ξ_ik over intervals [τ_k,τ_k+1], with τ_0=τ_a and τ_N=τ_b, we are led to

    Δ H  = Δ H^0 + ∑_k=0^N-1ε ξ_k . 𝐧(τ_k-t_0)  δ s_k,

where

    ε = (ε_1     0 
    
    0    ε_2 
    ) ,

  ξ_k = (ξ_1k,ξ_2k)^T, and 𝐧 is the unit vector perpendicular to separatrix AB and pointing to the left-hand-side of AB. 
  In the following we write 𝐧_k in place of 𝐧(τ_k-t_0).
  Because δτ_k is small compared to the typical time scale of 𝐫(t), we made use of the approximation:

    𝐧_k  = 1/δ s_k( 
     -δ p_0k
    δ q_0k)

where δ p_0k = p_0(τ_k+1-t_0) - p_0(τ_k-t_0),   δ q_0k = q_0(τ_k+1-t_0) - q_0(τ_k-t_0) and 

    δ s_k = √(δ q_0k ^2+δ p_0k ^2) = | 𝐫̇_0(τ_k-t_0) |  δτ_k

is the discrete arc length along the path 𝐫_0.
Under these hypotheses the jump of Hamiltonian reads

    Δ H = Δ H^0 + ∑_k=0^N-1 X_k

where X_k is the random variable

    X_k = ε ξ_k . 𝐧_k    | 𝐫̇_0(τ_k-t_0) |  δτ_k.





 §.§ A sufficient condition for the absence of noise-induced crossing


   Sivathanu et al. <cit.> showed that, in the case of dynamical systems submitted to an on-off noise, there exists a threshold in the noise intensity below which noise-induced crossing cannot occur. The existence of such a threshold requires that  the noise amplitude be bounded. This means that, if the Hamiltonian drift |Δ H^0| induced by the dissipative term Λ𝐔 is too large, or if the ε_i's are too small, noise-induced crossing will not occur. Such a sufficient condition being very useful in practice, we will generalize the results of Ref. <cit.>, which had been obtained for the Duffing equation, to the present problem. 
 
First, we assume that the noise intensities are bounded, i.e. there exists positive numbers ξ_1^max and ξ_2^max such that:

    |ξ_1k| ≤ξ_1^max     |ξ_2k| ≤ξ_2^max

for all k. Then, from
Eq. (<ref>) we get:

    | Δ H/Δ H^0-1 | ≤1/|Δ H^0|∑_i=1,2∑_k=0^N-1ε_i ξ_i^max |n_ik| u_0k δτ_k,

where n_ik = n_i(τ_k-t_0) and u_0k=| 𝐫̇_0 (τ_k-t_0) |.  
Whatever the realization of the τ_k's, the sum over k in Eq. (<ref>) is a Riemann sum on the interval [τ_a,τ_b]. If δτ is much smaller than the typical time scale of the undisturbed system, 
 this sum may be approximated by an integral:

    ∑_k=0^N-1ε_i ξ_i^max |n_ik| u_0k δτ_k 
       ≃   ε_i ξ_i^max∫_τ_a^τ_b |n_i(t-t_0)| u_0(t-t_0) dt
       ≃   ε_i ξ_i^max∫_-∞^+∞ |n_i(t')| u_0(t') dt'
        =    ε_i ξ_i^max∫_A^B |n_i(s) | ds,

where u_0(t)=| 𝐫̇_0 (t) | and we made use of the change of variable t'=t-t_0 and replaced the boundaries [τ_a-t_0,τ_b-t_0] by ]-∞,+∞[. The last equality has been obtained by introducing the curvilinear coordinate s along AB, such that ds = u_0(t') dt'. 
Let ⟨ . ⟩_AB denote the average over arc AB:

    ⟨ f ⟩_AB = 1/ L_AB∫_A^B  f(s)    ds,

where L_AB is the length of arc AB.
Then, we get

    | Δ H/Δ H^0-1 | ≤1/|Δ H^0|∑_i=1,2ε_i ξ_i^max L_AB⟨ | n_i | ⟩_AB .

We define noise-induced crossing as the occurrence of the event Δ H/Δ H^0 < 0, i.e. Δ H and Δ H^0 have opposite signs. Indeed, as discussed in the introduction, if the dissipative term Λ𝐔 drives the particle 𝐫(t) on one side of the separatrix, noise-induced crossing is the fact that the particle eventually reaches the opposite side.  The probability of this event is denoted as P_n.i.c. ("noise-induced crossing") in the following.
The inequality Δ H/Δ H^0 < 0 implies

    | Δ H/Δ H^0-1 | > 1

According to inequality (<ref>), condition (<ref>) cannot be fulfilled if 

    ε_1 ξ_1^max⟨ | n_1 | ⟩_AB
     + ε_2 ξ_2^max⟨ | n_2 | ⟩_AB <  |Δ H^0|/ L_AB .

If condition (<ref>) is satisfied, noise-induced crossing will not happen and P_n.i.c. = 0. This condition corresponds to a triangular zone in the (ε_1,ε_2) plane (Fig. <ref>), inside which the system is sheltered from stochastic forcing. 
Outside this zone, P_n.i.c. can take non-zero values. This probability is calculated in the next section.




 §.§ Asymptotic expression for the jump of Hamiltonian


We assume that noise-induced crossing occurs, and calculate the probability of this event.
Assuming that the noise intensity (that is the components of vector ξ_k) and the times  τ_k are independent, we have

    ⟨ X_k ⟩ = ε ⟨ξ_k ⟩ . ⟨𝐧_k    | 𝐫̇_0(τ_k-t_0) |  δτ_k ⟩,
 
so that ⟨ X_k ⟩ =0. According to the generalized central limit theorem, if X_k satisfies the Lyapunov condition, i.e. if there exists d > 0 such that

    1/S_N^2+d∑_k=0^N-1⟨ |X_k|^2+d⟩→  0          N →∞,

where 
S_N^2 = ∑_k⟨ X_k^2 ⟩,
then

    1/S_N∑_k=0^N-1 X_k →  Z          N →∞,

where Z is a centred Gaussian random variable with unit variance.  
Exploiting further the fact that ξ_ik, τ_k and δτ_k are independent random variables we have:

    S_N^2 = ∑_k=0^N-1∑_i=1,2ε_i^2  ⟨ξ_ik^2 ⟩ ⟨ n_ik^2
    | 𝐫̇_0(τ_k-t_0) |^2 ⟩ ⟨ δτ_k^2 ⟩,

with 𝐧_k = (n_1k,n_2k). 
The variances ⟨ξ_ik^2 ⟩ are equal to unity. Also, as discussed above, we assume that δτ_k has an exponential distribution with average δτ and variance δτ^2. Therefore,
⟨δτ_k^2 ⟩ = 2 δτ^2 and:

    S_N^2 = 2δτ ⟨∑_k=0^N-1∑_i=1,2ε_i^2    n_ik^2 
    | 𝐫̇_0(τ_k-t_0) |^2 δτ⟩ .

The sum in this last equation is of the form ⟨∑_k f(τ_k) δτ⟩, and can be approximated as follows. We have

    ∑_k f(τ_k) δτ = ∑_k f(τ_k) δτ_k - ∑_k f(τ_k) (δτ_k - δτ).

As N →∞, the first sum on the right-hand-side converges to an integral:

    ∑_k=0^N-1 f(τ_k) δτ_k →∫_τ_a^τ_b f(t) dt.

Indeed, whatever the realization of the τ_k's, this integral is a Riemann sum over the interval τ_0 = τ_a ≤ t ≤τ_b = τ_N. 
This approximation is valid if δτ is much smaller than the typical time scale of the unperturbed dynamics.
The last sum in Eq. (<ref>) has a zero average, provided τ_k and δτ_k are independent:

    ⟨∑_k=0^N-1 f(τ_k) (δτ_k - δτ) ⟩ =
    ∑_k=0^N-1⟨ f(τ_k) ⟩  (⟨δτ_k ⟩ - δτ  ) = 0.

Therefore, in the limit where N ≫ 1, we have

    ⟨∑_k f(τ_k) δτ⟩≃∫_τ_a^τ_b f(t) dt.

Finally, the sum S_N^2 can be approximated by

    S_N^2 = 2δτ ∑_i=1,2ε_i^2 ∫_τ_a^τ_b  n_i^2(t-t_0)   |𝐫̇_0(t-t_0) |^2   dt,

in the limit where δτ is small (N ≫ 1).
Again, by using the change of variable t'=t-t_0 and replacing the boundaries [τ_a-t_0,τ_b-t_0] by ]-∞,+∞[, as usually done in the calculation of Melnikov functions <cit.>, we get

    S_N^2 = 2δτ ∑_i=1,2ε_i^2 ∫_-∞^+∞  n_i^2(t')   |𝐫̇_0(t') |^2   dt'
    = 2δτ ∑_i=1,2ε_i^2 ∫_A^B  n_i^2(s)   u_0(s)    ds.

Finally, the sum of the variances reads

    S_N^2 = 2δτ  L_AB∑_i=1,2ε_i^2 ⟨ n_i^2  u_0  ⟩_AB,

where u_0 = |𝐫̇_0| = |∇ H |.



 §.§ Probability of noise-induced separatrix crossing

 
The jump of Hamiltonian therefore reads, in the limit where N ≫ 1,

    Δ H ≃Δ H^0 + S_N   Z.

It is a Gaussian random variable with average Δ H^0 and standard deviation S_N given by Eq. (<ref>). This result will be used now to calculate the probability of noise-induced crossing.
Suppose that dissipation alone tends to drive the system towards the right-hand-side of AB, that is Δ H^0 < 0. Then, noise-induced separatrix crossing corresponds to the event Δ H > 0, i.e. noise drives the system towards the opposite side of AB. The probability of this event is

    P_n. i. c. = P(Z > -Δ H^0 / S_N)
    =∫_|Δ H^0| / S_N^+∞ g(z)   dz

where g(z) = exp(-z^2/2)/√(2 π). This leads to:

    P_n. i. c.  = 1/2(  |Δ H^0|/2 (δτ L_AB)^1/2  (
     ε_1^2 ⟨ n_1^2   |∇ H | ⟩_AB +  
     ε_2^2 ⟨ n_2^2   |∇ H | ⟩_AB )^1/2).

which can be used whatever the sign of Δ H^0. 

Equation (<ref>) shows how the deterministic perturbation (manifested by the numerator |Δ H^0|) and the stochastic perturbation (appearing at the denominator), affect the probability. In the case of isotropic noise (ε_1=ε_2), the denominator no longer depends on 𝐧, i.e. on the shape of the separatrix. Note that the numerator |Δ H^0 | depends, in general, on the shape of AB.
 We also observe that the correlation time scale δτ has a significant effect on separatrix crossing, and that P_n. i. c. decreases when δτ decreases.  
 
 






§ APPLICATION TO A MECHANICAL OSCILLATOR


 

To illustrate the results of the previous section we consider the classical example of a mechanical pendulum. Indeed, oscillators  received considerable attention with and without noise (see for example San Juan et al. <cit.>,  Li et al. <cit.>, 
Estevez et al. <cit.>), as they provide very useful test-cases.
Here, a pendulum will be used to check results (<ref>) and (<ref>).



  
Hamiltonian jump and probability.
The motion equation of a weakly damped pendulum with angle θ(t) with respect to the vertical axis, when submitted to a random torque εξ(t), where ε > 0 is a constant, reads

    θ̈= - g/lsinθ - λ/m l^2θ̇+ ε/m l^2ξ(t)

where g is the gravitational acceleration, l is the length of the pendulum, m is its mass, and λ is the friction coefficient. This system corresponds to Eqs. (<ref>)-(<ref>) with q=θ, p=θ̇, Λ=λ/(m l^2),  ε_1 = 0, ε_2=ε/(m l^2) and

    H(q,p) = 1/2 p^2 - ω_0^2 cos q,

where we have set ω_0  = √(g/l), and 𝐔(q,p) = (0, -p)^T. In order to apply the theory developed in the previous sections, the angular frequency ω_0 of the undisturbed pendulum and the noise correlation time scale δτ
are assumed to satisfy the asymptotic condition:

    δτ≪2 π/ω_0.

It will be shown below that, in this case, noise-induced crossing only appears when ε_2 is above some threshold ε_2^c. In this paragraph we assume that this condition is satisfied.

The phase portrait H=constant (see inset (i) of Fig. <ref>) has the classical form of periodic cat's eyes, with hyperbolic saddle points A=(-π,0) and B=(π,0) related by heteroclinic trajectories. The equation of the upper
separatrix is p_0 = 2 ω_0 cos(q_0/2). By injecting this into Eq. (<ref>) we obtain the variation of Hamiltonian in the absence of noise:

    Δ H^0 = -Λ∫_-∞^∞ p_0^2(t)   dt = -Λ∫_-π^π
    p_0   dq_0 = -8 Λω_0.









Also, by integrating n_2^2 u_0 along AB we get

    L_AB ⟨ n_2^2 u_0 ⟩_AB = 8 ω_0.

To ensure that the generalized central limit theorem applies, we must check that the Lyapunov condition (<ref>) is satisfied. Firstly, from (<ref>) we get:

    S_N^2 = K_0  δτ,

where K_0 is non-zero and independent of N as N ≫ 1. Therefore, since δτ = (τ_b-τ_a)/N, S_N^2 is proportional to 1/N (note that this result is general). Secondly, for the pendulum we have

    ⟨ | X_k |^2+d⟩ =  ε_2^2+d⟨ξ_2 k^2+d⟩⟨ |n_2k u_0 |^2+d⟩⟨δτ_k^2+d⟩,

provided that the random variables ξ_2k and τ_k and δτ_k are independent. For the exponential distribution of δτ_k considered here, and d > 0, we have

    ⟨δτ_k^2+d⟩ = δτ^2+d Γ(d+3),

where Γ is Euler's gamma function. Also, the other terms appearing in expression (<ref>) are all bounded for any k=0,...,N-1, so that there exists m > 0 independent of N such that, for all k

    ε_2^2+d⟨ξ_2 k^2+d⟩⟨ |n_2k u_0 |^2+d⟩  < m.

We then have:

    0 ≤1/S_N^2+d∑_k=0^N-1⟨ |X_k|^2+d⟩ < N   m Γ(d+3)/K_0^1+d/2 δτ^1+d/2 = K_1/N^d/2→ 0        N →∞,

where K_1 > 0 is a constant. 
The Lyapunov condition is therefore satisfied, and the Hamiltonian jump for the pendulum is the Gaussian random variable

    Δ H = -8 Λ ω_0 + 16  δτ ε_2^2  ω
    _0   Z.

The probability of noise-induced crossing for the pendulum is therefore

    P_n. i. c.  = 1/2(  Λ√(2 ω_0)/ε_2 √(δτ)),       ε_2 > ε_2^c

and 0 otherwise.



  
Comparison with numerical solutions.
Figure <ref> shows the histograms of ξ_k  and of the resulting Hamiltonian jump Δ H, obtained by solving numerically Eq. (<ref>). Graphs (a)-(b) correspond to a uniform distribution of ξ_k in the range [-√(3),+√(3)], and (c)-(d) correspond to a dichotomous noise ± 1. The parameters are  ω_0 = 1, δτ = 0.005,  Λ = 0.002, and ε_2 = 0.05. For both kinds of noises,  20000 runs have been conducted, with initial positions very close to A. They correspond to pendulums released from rest at their upper equilibrium position.
The histograms of Δ H clearly indicate that this random variable is Gaussian in both cases, with an average and a variance corresponding to the theoretical result of Eq. (<ref>) (solid lines in graphs (b) and (d)).




 
Figure <ref> shows the theoretical  probability of Eq. (<ref>) (solid line), together with probabilities obtained from numerical solutions of Eq. (<ref>) (circles and triangles).  Circles correspond to uniformly distributed noise amplitudes,  and triangles correspond to the dichotomous noise. Here, the parameters are  ω_0 = 1, δτ = 0.005, five values for Λ have been chosen between 0.002 and 0.01, and ε_2 varies between 0 and 0.1. 
Circles and triangles are the percentage of pendulums which start another turn once at B, i.e. those for which θ(t) > π for some time t. We observe that the agreement is good, as symbols closely follow the theoretical result (<ref>). As expected, when data is plotted in terms of the renormalized variable  X=ε_2  √(δτ) / Λ√(ω)_0,
all points collect along the probability   1/2   (√(2)/X) (inset (ii)).

 



  
Absence of noise-induced crossing for the pendulum.  
The threshold ε_2^c below which noise-induced crossing cannot occur can be readily obtained from Eq. (<ref>). By noticing that n_2(s) > 0 all along the upper separatrix AB and that n_2   ds = dq_0, we get L_AB⟨ |n_2| ⟩ = 2π. The sufficient condition (<ref>) leads to:

    ε_2 ≤ε_2^c = 4/πΛω_0/ξ^max,

where ξ^max = 1 for the dichotomous noise and √(3) for the uniform noise.
For the various runs of Fig. <ref>, ε_2^c is smaller than 0.015, so that the total absence of noise-induced crossing is not visible on this figure. To observe this phenomenon, we have conducted another series of runs by choosing  Λ = 0.002, ε_2 = 0.06, δτ = 0.05, and ω_0 in the range [1,30]. In terms of ω_0,  condition (<ref>) corresponds to ω_0 ≥ω_0^c = πε_2 ξ^max/ (4 Λ) ≈ 23.6 for the dichotomous noise and 40.8 for the uniform noise. Figure <ref> confirms that some violent phenomenon 
happens in the case of the dichotomous noise,  as the probability (triangles) drops to zero when ω_0 approaches ω_0^c ≈ 23.6. Condition (<ref>) being only sufficient, the probability does not vanish at exactly ω_0^c, but prior to it. As expected, the numerical probabilities shown in Fig. <ref> quit the theoretical law (<ref>)(solid line), when ω_0 is above 10, since the asymptotic condition (<ref>) is not strictly fulfilled there. Note that this might also affect the accuracy of the numerical estimation of ω_0^c.  Concerning the uniform noise, we have checked that the circles of Fig. <ref> also abruptly drop to zero as ω_0 approaches ω_0^c ≈ 40.8. 









§ CONCLUSION


Noise can affect the dynamics of planar weakly dissipative Hamiltonian systems. We have studied this phenomenon in the vicinity of heteroclinic trajectories of the unperturbed Hamiltonian system, in the case where noise is piecewise constant, with a finite correlation time scale δτ. Such a colored noise corresponds to a Kubo-Anderson process.
The analytical calculations presented here show how δτ affects the statistics of the Hamiltonian jump Δ H and the probability of noise-induced crossing.  Our noise having a bounded amplitude, we could also derive a sufficient condition for the impossibility of noise-induced crossing.

Our results have been illustrated by using the mechanical pendulum, submitted to weak damping and noise. Numerical simulations agree with the theoretical expressions of the Hamiltonian jump and of the probability of noise-induced crossing. A large number of complex systems can be predicted similarly, by means of formula (<ref>). For example, it is well known that the advection of weakly inertial particles in a two-dimensional flow can be modeled by setting (q,p) = (x,y), (the spatial coordinates of the particle), and H = ψ (the streamfunction of the flow). The effect of particle inertia is manifested by the term 𝐔, which now corresponds to the acceleration of the carrying fluid, and which has a negative divergence (see for example Refs. <cit.>). 
Such particles are often affected by random forces, like the action of turbulent eddies, or the force due to some random electric field acting on aerosols.
If necessary, our results can be generalized to the case where the noise intensity ε_i depends on the spatial position of the particle. This is a perspective of this work, which would allow to study many realistic situations 
of interest in natural or industrial dynamical systems where heteroclinic or homoclinic trajectories are present.

Finally, the phenomenon of separatrix crossing under deterministic perturbation of hamiltonian systems is well-known and has been the subject of many studies <cit.>. In practice, both deterministic and stochastic perturbations are present. The generalization of the present analysis to this case, and the derivation of a probability that would account for both effects, will be the next step of this study.

   
  
























































10


urlstyle

frey1992
M. Frey, E. Simiu, in Engineering Mechanics (ASCE, 1992), pp.
  660–663 (1992)

Frey1993
M. Frey, E. Simiu, Physica D: Nonlinear Phenomena 63(3), 321 (1993)

Simiu1996
E. Simiu, M. Frey, in Fluctuations and Order (Springer, 1996), pp.
  81–90 (1996)

Liu2002
Z. Liu, Y. Lai, L. Billings, I.B. Schwartz, Physical Review Letters
  88(12), 124101 (2002)

Tel2008
T. Tél, Y.C. Lai, M. Gruiz, International Journal of Bifurcation and Chaos
  18(02), 509 (2008)

Wang2009
Y. Wang, Y. Lai, Z. Zheng, Physical Review E 79(5), 056210 (2009)

Kramers1940
H.A. Kramers, Physica 7(4), 284 (1940)

Kautz1988
R.L. Kautz, Physical Review A 38(4), 2066 (1988)

Beale1989
P.D. Beale, Physical Review A 40(7), 3998 (1989)

Grassberger1989
P. Grassberger, Journal of Physics A: Mathematical and General 22(16),
  3283 (1989)

Bulsara1990
A. Bulsara, W. Schieve, E. Jacobs, Physical Review A 41(2), 668 (1990)

Simiu1996Melnikov
E. Simiu, M.R. Frey, Journal of Engineering Mechanics 122(3), 263
  (1996)

Franaszek1996
M. Franaszek, E. Simiu, Physical Review E 54(2), 1298 (1996)

Franaszek1998
M. Franaszek, M. Frey, E. Simiu, in Stochastically excited nonlinear
  ocean structures. Chapter 7. (World Scientific. M.F. Shlesinger and T. Swean
  Eds., 1998), pp. 187–212

soskin2001
S. Soskin, R. Mannella, M. Arrayás, A. Silchenko, Physical Review E
  63(5), 051111 (2001)

Khovanov2003
I.A. Khovanov, D.G. Luchinsky, R. Mannella, P.V.E. McClintock, A.N. Silchenko,
  in AIP Conference Proceedings, vol. 665 (AIP, 2003), vol. 665, pp.
  435–442

Khovanov2008
I. Khovanov, D.G. Luchinsky, P.V.E. McClintock, A. Silchenko, International
  Journal of Bifurcation and Chaos 18(06), 1727 (2008)

SuMei2008
Y. Su, D. Mei, International Journal of Theoretical Physics 47, 2409
  (2008)

Anderson1954
P. Anderson, Journal of the Physical Society of Japan 9(3), 316 (1954)

Kubo1954
R. Kubo, Journal of the Physical Society of Japan 9(6), 935 (1954)

Sivathanu1995
Y.R. Sivathanu, C. Hagwood, E. Simiu, Physical Review E 52(5), 4669
  (1995)

Hanggi1989
P. Hänggi, Noise in nonlinear dynamical systems 1, 307 (1989)

Kitahara1980
K. Kitahara, W. Horsthemke, R. Lefever, Y. Inaba, Progr. Theor. Phys
  64, 1233 (1980)

Irwin1990
A.J. Irwin, S.J. Fraser, R. Kapral, Physical Review Letters 64(20),
  2343 (1990)

Kapral1993
R. Kapral, S. Fraser, J. Stat. Phys. 70, 61 (1993)

Kallio1989
G. Kallio, M. Reeks, International Journal of Multiphase Flow 15(3),
  433 (1989)

Brissaud1974
A. Brissaud, U. Frisch, Journal of Mathematical Physics 15(5), 524
  (1974)

GH83
J. Guckenheimer, P. Holmes,   (1983).
Springer, New York, NY.

Sanjuan1996
M.A.F. Sanjuán, International Journal of Theoretical Physics
  35(8), 1745 (1996)

Li2011
L.S. Li, X. Yu, W. Li, International Journal of Theoretical Physics
  50(4), 1255 (2011)

Estevez2011
P.G. Estévez, Ş. Kuru, J. Negro, L.M. Nieto, International Journal of
  Theoretical Physics 50(7), 2046 (2011)

Maxey1987jfm
M.R. Maxey, J. Fluid Mech. 174, 441 (1987)

Cartwright2010
J. Cartwright, U. Feudel, G. Karolyi, A. De Moura, O. Piro, T. Tel,  pp. 51–87
  (2010).
Non-linear Dynamics and Chaos: Advances and Perspectives. Thiel, M.
  Ed. Springer-Verlag Berlin Heidelberg.

Cary1986
J.R. Cary, D. Escande, J.L. Tennyson, Physical Review A 34(5), 4256
  (1986)

Neishtadt1991
A. Neishtadt, Chaos: An Interdisciplinary Journal of Nonlinear Science
  1(1), 42 (1991)

Itin2002
A. Itin, R. de La Llave, A. Neishtadt, A. Vasiliev, Chaos: An Interdisciplinary
  Journal of Nonlinear Science 12(4), 1043 (2002)



