Stochastic sensitivity analysis of chaotic attractors in 2D non-invertible maps

Chaos, Solitons and Fractals 126 (2019) 78–84

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Stochastic sensitivity analysis of chaotic attractors in 2D non-invertible maps I. Bashkirtseva, L. Ryashko∗ Department of Theoretical and Mathematical Physics, Ural Federal University, Lenin ave., 51, Ekaterinburg, 620000, Russian Federation

a r t i c l e

i n f o

Article history: Received 6 March 2019 Revised 18 May 2019 Accepted 28 May 2019

Keywords: Chaotic attractors Non-invertible maps Random disturbances Stochastic sensitivity Noise-induced transitions

a b s t r a c t A response of chaotic discrete-time dynamical systems on parametric random disturbances is considered. For two-dimensional stochastic systems with non-invertible maps, a dispersion of random states near chaotic attractors is studied. To analyse the dispersion near the border of the chaotic attractor, we elaborate an asymptotic approach based on the stochastic sensitivity technique. In this analysis, critical curves defining parts of this border are used. An application of this theory to the study of the dispersion of random states near borders of chaotic attractors is given on the example of the Sprott model. Constructive abilities of the elaborated approach for the analysis of noise-induced escape from the basin of the chaotic attractor are demonstrated.

1. Introduction A problem of the analysis of chaotic systems is topical for modern nonlinear science. The inevitable random noise complicates the already complex chaotic dynamics. In recent decades, stochastic phenomena in nonlinear dynamic systems attract attention of many researchers. A wide diversity of noise-induced effects has been discovered in different fields of science and engineering. Here, one should note the stochastic resonance [1–3], noise-induced transitions [4,5], stochastic excitability [6,7], noiseinduced multistability, preference and annihilation of attractors [8– 10], noise-induced synchronization [11–13], stochastic bifurcations [14,15], and noise-induced chaos-order transformations [16–20]. An influence of noise on chaotic attractors was widely studied (see, e.g. [21–24]). As a rule, all these phenomena were originally found either in experiments or by the direct numerical simulation of the corresponding stochastic mathematical models. Currently, the interest of researchers has shifted to clarifying the roots of these phenomena. For the study of probabilistic mechanisms of qualitative deformations caused by random disturbances, appropriate analytical methods should be elaborated. In the parametric analysis of the deterministic dynamics, a classical approach is based on the study of attractors and corresponding bifurcations. Most stochastic phenomena can be explained by the noise-induced transitions between coexisting attractors, or stochastic transitions between parts ∗

Corresponding author. E-mail address: [email protected] (L. Ryashko).

https://doi.org/10.1016/j.chaos.2019.05.032 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

© 2019 Elsevier Ltd. All rights reserved.

of the spatial attractors, or by the presence of so-called transient attractors. In the stochastic analysis, it is important to take into account not only a mutual arrangement of attractors and separatrices, but also a level of the sensitivity of these attractors to random noise. Nowadays, in the investigation of stochastic dynamics, there is a general mathematical approach based on the study of changes in probabilistic distributions. In general, dynamics of these distributions can be formally described for continuous- and discrete-time systems by Fokker–Planck–Kolmogorov [25,26] and Frobenius– Perron [27,28] equations, respectively. However, a solution of the FPK-equation is technically difficult even in the two-dimensional case. As for the FP-equation, the situation is even more complicated. Indeed, only for extremely specific one-dimensional examples this functional equation can be solved analytically. In these circumstances, an approach based on asymptotics and approximations is being actively developed [5,29–33]. In the framework of this direction, the constructive stochastic sensitivity function (SSF) technique is actively used in the analysis of various stochastic nonlinear phenomena (see, e.g. [34–39]). A main idea of this technique is to extract the main asymptotics from the probabilistic distribution law under weak noise. This asymptotics allows us to approximate the mean square deviation of random trajectories from the unforced deterministic attractor. Initially, SSF technique was elaborated for equilibria and cycles of both continuous [40] and discrete-time [41] systems. Later, this theory has been extended to the more complex analysis of quasiperiodic attractors having the form of closed invariant curves and tori [42,43]. Recently, this theory was built for chaotic attractors of

I. Bashkirtseva and L. Ryashko / Chaos, Solitons and Fractals 126 (2019) 78–84

one-dimensional discrete-time systems [44]. Nowadays, the development of analytical methods for studying noise-induced deformations of chaotic attractors for systems of higher dimensions is undoubtedly a challenging problem. Even in two-dimensional case, it is mathematically nontrivial. The present paper aims to develop the SSF technique for chaotic attractors in discrete systems with non-invertible 2D-maps. In the deterministic case, methods of the analysis of chaotic attractors in such systems were elaborated in [45,46]. One of the most important achievements of this theory is to identify a special role of the so-called critical curves. In particular, using these curves one can construct borders of the chaotic attractors. As it turned out, these curves are key objects in analyzing the influence of the random noise on discrete systems with non-invertible 2D-maps. In the present paper, we will show how the critical curves can be used in the constructive analysis of the stochastic sensitivity of chaotic attractors of such systems. The present paper is organized as follows. In Section 2, a mathematical description of the stochastic sensitivity technique of regular attractors (stable equilibria and k-cycles) is given as a short background. In Section 3, a novel theory of the stochastic sensitivity of chaotic attractors of systems with non-invertible 2D-maps is presented. Here, a general case of the parametric noise is covered. Constructive abilities of this theory are illustrated in Section 4 on the example of the stochastically forced Sprott discrete model [47]. 2. Stochastic sensitivity of attractors Consider a general discrete-time nonlinear system

xt+1 = f (xt , ηt ),

ηt = εξt ,

(1)

where xt is an n-dimensional vector, f(x, η) is an n-dimensional smooth vector-function, ξ t is an m-dimensional uncorrelated random process with parameters Eξt = 0, Eξt ξt = V, V is a covariance m × m-matrix, and ε is a scalar parameter of the noise intensity. Consider a solution x¯t of the corresponding deterministic system (1) with ε = 0. Let xtε be a solution of the stochastic system (1) with the initial condition xε0 = x¯0 + ε z0 . In the analysis of the deviations of xtε from x¯t , an important role is played by the asymptotics

 ∂ xtε  xtε − x¯t zt = = lim . ∂ε ε=0 ε→0 ε

This asymptotics defines a sensitivity of the solution x¯t to the small random disturbances. Dynamics of the pair (x¯t , zt ) is governed by the stochastic linear extension system

x¯t+1 = f (x¯t , 0 ) zt+1 = Ft zt + Gt ξt ,

Ft =

∂f ∂f (x¯ , 0 ), Gt = (x¯ , 0 ). ∂x t ∂η t

(2)

The matrix Mt = Ezt zt of the second moments satisfies the following deterministic system

x¯t+1 = f (x¯t , 0 ) Mt+1 = Ft Mt Ft + Qt ,

Qt = Gt V Gt .

(3)

For the small noise intensity ε , values Mt allow us to approximate a dispersion of the random states xtε around x¯t : E(xtε − x¯t )(xtε − x¯t ) ≈ ε 2 Mt . So, we can consider Mt as a stochastic sensitivity matrix of the state x¯t of the arbitrary deterministic solution to the random disturbances of system (1). Let us show how this asymptotic probabilistic analysis can used for the study of the stochastic sensitivity of different types of attractors. First, consider a case when the attractor is an equilibrium. Let x¯ be an exponentially stable equilibrium of the deterministic system (1) (with ε = 0). For the stationary solution x¯t ≡ x¯, the system

79

(3) gives a recurrent equation

Mt+1 = F Mt F  + Q,

(4)

with the constant matrices

F=

∂f ∂f (x¯, 0 ), Q = GV G , G = (x¯, 0 ). ∂x ∂η

Due to the exponential stability of x¯, it holds that the spectral radius ρ (F) < 1, and the Eq. (4) has a unique stable stationary solution W satisfying the following algebraic equation:

W = F W F  + Q.

(5)

The matrix W is the stochastic sensitivity matrix of the equilibrium x¯ [41]. By this matrix, a dispersion of random states xtε around x¯ can be approximated as E(xtε − x¯ )(xtε − x¯ ) ≈ ε 2W. Consider now a case when the attractor is a discrete cycle. Let A = {x¯1 , . . . , x¯k } be an exponentially stable k-cycle of the deterministic system (1) (with ε = 0 ). Consider here the k-periodic solution x¯t : x¯t+1 = f (x¯t , 0 ), x¯t+k = x¯t (t = 1, 2, . . . ). In this case, the system (3) gives a recurrent equation

Mt+1 = Ft Mt Ft + Qt ,

(6)

with k-periodic matrices

Ft =

∂f ∂f (x¯ , 0 ), Qt = Gt V Gt , Gt = (x¯ , 0 ). ∂x t ∂η t

The necessary and sufficient condition of the exponential stability of the k-cycle A is ρ (Fk · . . . · F2 · F1 ) < 1. Due to this inequality, system (6) has a unique stable k-periodic solution Wt . The set of matrices {W1 , . . . , Wk } defines the stochastic sensitivity of the elements {x¯1 , . . . , x¯k } of the cycle A [41]. A more complicated case of the quasiperiodic attractor was considered in detail in [42,48] where the corresponding stochastic sensitivity technique of closed invariant curves was elaborated. All attractors considered here (equilibria, discrete cycles, closed invariant curves) are regular. In the present paper, a main interest is to study the stochastic sensitivity of chaotic attractors. For chaotic attractors of one-dimensional maps, the stochastic sensitivity technique was developed in [44]. In the next Section, we present our results on the stochastic sensitivity analysis of 2D systems. 3. Analysis of stochastic sensitivity of chaotic attractors In what follows, we study the stochastic sensitivity of chaotic attractors of system (1) with the non-invertible map for n = 2. Let A be a chaotic attractor of system (1) with ε = 0, and L be a piecewise smooth border of A. Under the stochastic forcing, a random solution starting from A can jump over the border L and get outside A. Because of the stability of A, after some iterations, the random solution returns to the inner part of A. As a result of this stochastic hopping, some probabilistic distribution around the chaotic attractor A is formed. A stochastic dynamics inside A can be extremely complicated because of a divergence of trajectories in the chaotic case. In the present paper, we focus on the features of this steady probabilistic distribution only outside A. It is a behavior of random solutions outside the attractor that is a determining factor in the study of various stochastic phenomena caused by noise-induced transitions. In these circumstances, two key problems should be solved. The first question is how to determine the border L of the chaotic attractor A. The second problem is to estimate the dispersion of random states that extend beyond the boundary L of the chaotic attractor A. As for the first question, the problem of the description of the chaotic attractor borders is already well studied (see, e.g. [45,46] and bibliography therein). Mathematically, this problem

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can be solved with the help of so-called critical curves. These curves localize zones of the phase plane where the map is noninvertible. A constructive essence of this theory can be presented as the following algorithm. Let γ be an initial critical curve: γ = {x | detF (x ) = 0}, where ∂f F (x ) = (x, 0 ). Denote L0 = γ ∩ A. The function f(x, 0) maps the ∂x curve L0 into the next critical curve L1 = {y | y = f (x, 0 ), x ∈ L0 }. Further, in the same way, one can construct the following critical curves L2 = f (L1 , 0 ), L3 = f (L2 , 0 ), . . . . The border L of the whole chaotic attractor A is formed by these critical curves. So, a state xt of the deterministic solution localizes on the border L1 if the preimage xt−1 belongs to L0 . Then successive states xt+1 , xt+2 , . . . belong to L2 , L3 , . . . correspondingly. Let us now consider the second problem associated with the estimation of the dispersion of random states lying outside the attractor A. Here, we need some geometrical preliminaries. Consider an arbitrary point x¯ ∈ L0 and its image y¯ = f (x¯, 0 ) ∈ L1 . Let q be a tangent vector to L0 at the point x¯. Then, the vector r = F (x¯ )q is a tangent vector to L1 at the point y¯ . Due to the singularity F (x¯ ), there is a non-zero vector v such as F (x¯ )v = 0. Note that vectors v and q are linear independent. So, for any vector x there exist scalars α and β such that the following decomposition can be written: x = α q + βv, Let n be a normalized vector that is orthogonal to L1 at the point y¯ . Taking into account n F (x¯ )q = n r = 0 and F (x¯ )v = 0, we get

Thus, the value μ1 (y¯ ) of the stochastic sensitivity at the point y¯ of the border L1 can be found in the explicit form

n F (x¯ )x = α n F (x¯ )q + β n F (x¯ )v = 0.

x˜1,2 = x ± 3ε

(7)

For the parametric analysis of random states deviations from the border L outside A, we will use the stochastic sensitivity function technique. The stochastic sensitivity at the point y¯ of the border L1 of the chaotic attractor A is defined by the stochastic sensitivity of the deterministic solution x¯t passing through the point y¯ . Let x¯ be an arbitrary point of L0 and x¯t be a deterministic solution satisfying the condition x¯0 = x¯. Then x¯1 = y¯ ∈ L1 . Successive values z0 , z1 of the asymptotics of deviations of random states xε0 , xε1 from x¯0 , x¯1 are connected (see (2)) by the equation

z1 = F0 z0 + G0 ξ0 . It follows from equalities (7) and F0 = F (x¯ ) that

where y¯ = f (x¯, 0 ), the normalized vector n(y¯ ) is orthogonal to L1 at the point y¯ , and Q (x¯ ) = G(x¯ )V G(x¯ ) . The successive values μ2 , μ3 , . . . of the stochastic sensitivity at the points x¯2 ∈ L2 , x¯3 ∈ L3 , . . . lying on the border of the chaotic attractor A can be found recurrently. Indeed, let nt (t = 1, 2, . . . ) be normalised vectors that are orthogonal to the curves Lt at the points x¯t of the deterministic solution starting from x¯0 = x¯ ∈ L0 . Here, it is natural to put W1 = μ1 n1 n1 as the value of the stochastic sensitivity matrix at the point x¯1 of the curve L1 . Successive matrices W2 , W3 , . . . characterizing the stochastic sensitivity of the borders L2 , L3 , . . . at the points x¯2 , x¯3 , . . . , and scalar values μ2 , μ3 , . . . can be found by formulas

    μt+1 = nt+1 Ft Wt Ft + Qt nt+1 , Wt+1 = μt+1 nt+1 nt+1 , t = 1, 2, . . . . Note that a sequence nt of the orthogonal vectors can be found with the help of the corresponding tangent vectors qt which are calculated recurrently: qt+1 = Ft qt . Here, q0 is the vector which is tangent to L0 at the point x¯. The stochastic sensitivity function μt (x) of the border Lt of the chaotic attractor A can be used for the construction of the corresponding confidence band. This band is a family of the confidence intervals. At the point x ∈ Lt , boundaries of such interval according to 3σ -rule are calculated as



This means that the asymptotics z1 of the deviations along the vector n that is orthogonal to L1 at the point y¯ does not depend on z0 . The projection n z1 is completely defined by this normal vector, the matrix G0 , and random disturbance ξ 0 . Due to this important property, in the mean square analysis we have

μ = E(n z1 )2 = n Q0 n.

4. Example In [47], Sprott has suggested a simple discrete 2D-system with the Neimark–Sacker bifurcation [49,50] and transition to chaos. This system

(9)

can be written as the single equation with delay

x1,t+1 = 1 − ax21,t−1 + bx1,t . If to swap x1,t−1 and x1,t on the right side of this equation, then one get a well-known Henon equation [51,52]. In Fig. 1, a bifurcation diagram and largest Lyapunov exponent  are shown for fixed b = 0.3 versus parameter a. Here, it can be

Λ

1

0.1

0.5

0.05

0

0

−0.5

−0.05

−1

(8)

where nt (x) is a normalized vector that is orthogonal to Lt at the point x. In the next section, we apply these theoretical results to the analysis of the dispersion of random states around the chaotic attractor of the Sprott model.

x1

a)

 μt (x )nt (x ),

x1,t+1 = 1 − ax22,t + bx1,t x2,t+1 = x1,t

n z 1 = n G 0 ξ0 . 

μ1 (y¯ ) = n (y¯ )Q (x¯ )n(y¯ ),

0.6

0.7

0.8

0.9

1

1.1

a

b)

−0.1

0.6

0.7

0.8

0.9

Fig. 1. Bifurcation diagram (a) and largest Lyapunov exponents (b) for system (9) with b = 0.3.

1

1.1

a

I. Bashkirtseva and L. Ryashko / Chaos, Solitons and Fractals 126 (2019) 78–84

x2

x2

1

1

0.5

0

0

a = 0.5 a = 0.61 a = 0.7 a = 1.01

−0.5

a)

81

−0.5

−1

0

0.5

1

x1

b)

−2

−1

0

1

x1

Fig. 2. Attractors of system (1): (a) regular attractors; (b) chaotic attractor (blue) inside the basin of attraction (white) for a = 1.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

x2

x2

L2

L3

1

1

L1 L0

0

0

L4 L5

−1 −1

0

−1

x1

1

−1

0

1

x1

Fig. 3. Chaotic attractor (gray) and a sequence of critical curves for a = 1.1: L0 (black), L1 (red), L2 (green), L3 (blue), L4 (pink), L5 (brown). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Chaotic attractor (gray), its borders (blue) and borders of confidence domains for ε = 0.01 (red dashed), ε = 0.02 (green dashed). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

seen how for increasing parameter a the stable equilibrium loses its stability, the system undergoes the Neimark–Sacker bifurcation, and quasiperiodic attractors (closed invariant curves) appear. Further, these attractors are destroyed and an alternation of regular and chaotic zones is observed. In Fig. 2, examples of regular and chaotic attractors are shown. The map f of system (9) with Jacobi matrix F and its determinant are as follows:

Here, ξ t is an uncorrelated Gaussian sequence with parameters ξt = 0, ξt2 = 1, and ε is the noise intensity. Under the influence of random disturbances, solutions of stochastic system (10) leaving the deterministic chaotic attractor form some probabilistic distribution around it. In Fig. 5, random states of stochastic system (10) are plotted by green color for two values of the noise intensity. It is clear that the stronger noise causes the larger deviations from the deterministic attractor A. Here, it should be noted that these deviations are quite nonuniform along the border L. This non-uniformity can be described analytically with the help of the stochastic sensitivity function technique presented in Section 2. For considered system (10),



f ( x1 , x2 ) =



1 − ax22 + bx1 , F= x1



b 1



−2ax2 , detF = 2ax2 . 0

First, let us find borders of the chaotic attractor. A singularity condition detF = 0 of the Jacobi matrix gives us the initial critical line γ = {(x1 , x2 ) | x2 = 0} which plays an important role in the determining these borders. Here and further we fix a = 1.1, b = 0.3. For these values, the chaotic attractor A is shown in Fig. 3 by gray color. Inside A, consider a segment L0 = γ ∩ A shown by black. Using the map f we get a sequence L1 = f (L0 ), L2 = f (L1 ), L3 = f (L2 ), L4 = f (L3 ), L5 = f (L4 ). Critical curves L1 , . . . , L5 define borders of the chaotic attractor A (see Fig. 3). Here, for example, the curve L1 belongs to the line x2 = (x1 − 1 )/b, and the curve L2 belongs to the parabola x1 = 1 − a(x2 − 1 )2 /b2 + bx2 . Using these critical curves, one can construct a whole border L of the deterministic chaotic attractor A (see blue curve in Fig. 4). Stochastic sensitivity analysis Consider the system (1) forced by additive noise

x1,t+1 = 1 − ax22,t + bx1,t + εξt x2,t+1 = x1,t .

(10)



Qt = Q =



1 0

0 , F0 = 0





b 1



0 , q0 = 0

 

1 , q1 = 0

 

b , 1



1 1 n1 = √ . 1 + b2 −b For example, it can be calculated that μ1 = 1/(1 + b2 ), so the stochastic sensitivity of L1 is constant. A plot of the stochastic sensitivity function μ along the border L of the chaotic attractor A is plotted in Fig. 6 by red dashed line. Note that the function μ essentially changes on the L: max μ = 11.4, min μ = 0.9. Such a variation of the stochastic sensitivity results in the change of the width of the confidence band (8). It is clearly seen in Fig. 4 where borders of confidence bands are plotted by red for ε = 0.01 and by green for ε = 0.02.

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Fig. 5. Deterministic chaotic attractor (black), random states (green), and borders of confidence domains (red dashed) for a) ε = 0.01, b) ε = 0.02. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

m

10

μ

a = 1.1 a = 1.01 a = 0.5

8

10 5

6

10 0

4

10

1

x2

0 −1

−1

0

x1

1

Fig. 6. Chaotic attractor (black), its border (blue), and stochastic sensitivity function μ at the points of this border (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In Fig. 5, one can compare a mutual spatial arrangement of these borders and the dispersion of random states calculated numerically. It is worth noting that the borders of confidence bands well agree with results of the direct numerical simulation. We emphasize that presented here technique allows us to estimate the sensitivity of the chaotic attractor to the random disturbances in the direction of the normal to border. It is interesting to compare this stochastic sensitivity with the sensitivity of trajectories lying on the attractor A. The stochastic sensitivity of the deterministic trajectory (x¯1,t , x¯2,t ) is defined by the matrix function m (t ) m12 (t ) Mt = ( 11 ) governed by (4). m21 (t ) m22 (t ) In Fig. 7, the plot of the function m(t ) = m11 (t ) is shown by red color for a = 1.1, b = 0.3. Here, we take x¯1,0 = x¯2,0 = 0 and m11 (0 ) = m12 (0 ) = m21 (0 ) = m22 (0 ) = 0. As can be seen, the stochastic sensitivity of this chaotic solution exhibits an exponential growth whereas the sensitivity of the chaotic attractor to the random disturbances in the direction of the normal to border is bounded: μ < 12 (see Fig. 6). Note that for the function m(t) the Lyapunov exponent m ≈ 2 = 0.22, where  is the largest Lyapunov exponent of the deterministic chaotic attractor A. Such a growth of m(t) is quite expected because of the high sensitivity of chaotic solutions to any disturbances. In Fig. 7, one can compare dynamics of the function m(t) of the stochastic sensitivity for regular attractors: equilibrium (a = 0.5, blue color) and 9-cycle (a = 1.01, green color). As can be seen, these functions are quickly stabilized: to the stationary level or to 9-periodic cycle, correspondingly. The stochastic sensitivity of the 9-cycle (a = 1.01, green color) are shown by red in Fig. 8. Note

2

10

0

10

0

20

40

60

80

t

Fig. 7. Stochastic sensitivity of solutions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

m a = 1.1 a = 1.01 a = 0.5

8

10

6

10

4

10

2

10

0

10

0

20

40

60

80

t

Fig. 8. Stochastic sensitivity of 9-cycle for a = 1.01. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

that this cycle has max m(t ) = 919, and this is essentially greater than the max x ∈ L μ(x) of the chaotic attractor with a = 1.1 (compare Fig. 6 and Fig. 8).

I. Bashkirtseva and L. Ryashko / Chaos, Solitons and Fractals 126 (2019) 78–84

a)

83

b) p

1

0.5

0

c)

0

0.05

0.1

0.15

ε

Fig. 9. Noise-induced escape from the basin of attraction (white) of the chaotic attractor (gray) for system (2) with a = 1.1: (a) no exit for ε = 0.02; (b) escape for ε = 0.1. Random trajectories are shown by blue, and the borders of the confidence domains are plotted by red dashed. In (c), the exit probability p(ε ) is shown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Consider how the stochastic sensitivity function technique and method of confidence domains presented above for the chaotic attractors can be used in the analysis of noise-induced transitions. For a = 1.1, the chaotic attractor A of system (1) has not so wide basin of attraction BA shown by white in Fig. 2(b). Outside BA , one can see (light blue) a set of initial points B∞ of system (1) solutions which go to infinity. For weak noise random trajectories starting from A are concentrated near A and arranged inside BA (see Fig. 9(a) for ε = 0.02). As noise intensity increases, a dispersion of trajectories which jump over the border L grows, and moreover, trajectories can escape the basin BA and fall into B∞ . Such random trajectories go to infinity with a high probability (see Fig. 9(b) for ε = 0.1). To study such noise-induced transitions parametrically, one has to analyse a mutual arrangement of the borders of the confidence domains and the separatrix dividing BA and B∞ . In Fig. 9(a), the confidence domain entirely belongs to BA . In Fig. 9(b), the extended confidence domain partially occupies B∞ . This signals about the noise-induced escape. Details of such noise-induced transitions can be seen in Fig. 9(c) where the exit probability p is plotted versus noise intensity ε . As can be seen, the transient part of this plot is narrow, and the threshold noise intensity lies between ε = 0.05 and ε = 0.1. This fact agrees with our prediction based on the stochastic sensitivity function technique. 5. Conclusion In our paper, we considered two-dimensional discrete-time systems with the non-invertible maps forced by general parametric noise. We focused on the study of the dispersion of random states near borders of chaotic attractors. For this study, an asymptotic approach based on the stochastic sensitivity technique was elaborated. It was shown how the stochastic sensitivity of the border

of the chaotic attractor can be constructively found with the help of critical curves which define parts of this border. Explicit recurrent formulas for the stochastic sensitivity of these parts are derived. Constructive abilities of this new mathematical theory were illustrated in the analysis of the dispersion of random states near the chaotic attractor of Sprott model. It was shown how this analytical approach can be effectively used in the analysis of noiseinduced transitions from the chaotic attractor to another coexisting dynamic regimes. It should be noted that these theoretical results well agree with data of the direct numerical simulation. Acknowledgments The work was supported by Russian Science Foundation (N 1611-10098). References [1] Pikovsky AS, Kurths J. Coherence resonance in a noise-driven excitable system. Phys Rev Lett 1997;78:775. [2] Gammaitoni L, Hänggi P, Jung P, Marchesoni F. Stochastic resonance. Rev Mod Phys 1998;70:223–88. [3] McDonnell MD, Stocks NG, Pearce CEM, Abbott D. Stochastic resonance: from suprathreshold stochastic resonance to stochastic signal quantization. Cambridge University Press; 2008. [4] Horsthemke W, Lefever R. Noise-induced transitions. Berlin: Springer; 1984. [5] Anishchenko VS, Astakhov VV, Neiman AB, Vadivasova TE, Schimansky-Geier L. Nonlinear dynamics of chaotic and stochastic systems. Tutorial and modern development. Berlin, Heidelberg: Springer-Verlag; 2007. [6] Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Phys Rep 2004;392:321–424. [7] Muratov CB, Vanden-Eijnden E. Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle. Chaos 2008;18:015111. [8] Kraut S, Feudel U, Grebogi C. Preference of attractors in noisy multistable systems. Phys Rev E 1999;59:5253. [9] Pisarchik AN, Feudel U. Control of multistability. Phys Rep 2014;540:167–218. [10] Pisarchik AN, Goswami BK. Annihilation of one of the coexisting attractors in a bistable system. Phys Rev Lett 20 0 0;84:1423.

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