Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions

In real systems, the unpredictable jump changes of the random environment can induce the critical transitions (CTs) between two non-adjacent states, which are more catastrophic. Taking an asymmetric Lévy-noise-induced tri-stable model with desirable, sub-desirable, and undesirable states as a prototype class of real systems, a prediction of the noise-induced CTs from the desirable state directly to the undesirable one is carried out. We first calculate the region that the current state of the given model is absorbed into the undesirable state based on the escape probability, which is named as the absorbed region. Then, a new concept of the parameter dependent basin of the unsafe regime (PDBUR) under the asymmetric Lévy noise is introduced. It is an efficient tool for approximately quantifying the ranges of the parameters, where the noise-induced CTs from the desirable state directly to the undesirable one may occur. More importantly, it may provide theoretical guidance for us to adopt some measures to avert a noise-induced catastrophic CT.


Model description
CTs between two non-adjacent states are widespread in various real-world systems. For example, a body may shift from a healthy state to a cancer state directly. Then, it is difficult to return to the healthy state even the sub-healthy state because the best time for a treatment is missing. To warn these catastrophic CTs between non-adjacent states, a self-constructed Lévy-noise-induced tri-stable model with desirable, sub-desirable, and undesirable states is considered. The form is where f (x, ε) = −1.14x 5 − 0.4x 4 + 4.6x 3 + 0.5x 2 − 3.8x − ε, ε is a control parameter, and these special coefficient values for f (x, ε) are chosen to construct a basic model that roughly fits real systems [29] . L t is a Lévy process with the generating triplet (0, d, σv α,β ), its time derivative is the Lévy noise, v α,β is an asymmetric Lévy jump measure, and v α,β (dy) = where Γ(2 − α) = +∞ 0 t (2−α)−1 e −t dt, I {.} denotes the indicator function, α is the stability index, and β is the skewness parameter [30] . We consider α ∈ (1, 2) and β ∈ [−1, 0] throughout the paper.
In the absence of L t , the system (1) can be reduced to the corresponding deterministic system, and the geometric structure of it versus ε is shown in Fig. 1. We find that there are five different regimes which have the following characteristics. The stable branches where x S1 , x S2 , and x S3 are located represent the desirable state, the sub-desirable state, and the undesirable state, respectively. Moreover, the unstable branches where x U1 and x U2 are located mark the height of the potential barrier between two adjacent stable branches. The coordinate values of Consider the large jumps in the Lévy noise, and three possible cases that the system (1) shifts directly to the undesirable state are also given in Fig. 1. They are CTs from the desirable state to the undesirable state in ε ∈ [ε Fold 4 , ε Fold 2 ) and ε ∈ [ε Fold 2 , ε Fold 1 ], and CT from the sub-desirable state to the undesirable state in ε ∈ (ε Fold 1 , ε Fold 3 ]. We record these three cases as CT 1, CT 2, and CT 3, respectively. Although both CT 1 and CT 2 shift from the desirable state to the undesirable state, the two states are adjacent for CT 1 and non-adjacent for CT 2. Similar to CT 1, the sub-desirable state and the undesirable state are also adjacent for CT 3. From the perspective of noise-induced dynamic transitions, CT 1, CT 2, and CT 3 can correspond to three escape problems of particles, as follows: Case 1 and Case 3 corresponding to CT 1 and CT 3 have the similar escape problems between two adjacent intervals, while Case 2 corresponding to CT 2 is the escape problem of particles between two non-adjacent intervals. Although x ∈ [x U1 , x S1 ] has left the current interval, the escape fails for CT 2 if x eventually falls in [x U2 , x U1 ].
In fact, a CT occurs when the current stable state is absorbed into another state. For example, Lévy-noise-induced CT 1 occurs because almost all x ∈ [x U2 , x S1 ] are absorbed into the undesirable state [x S3 , x U 2 ]. For a fixed ε ∈ [ε Fold 4 , ε Fold 2 ), the possibility that a Lévynoise-induced CT occurs can be estimated via measuring the part of [x U2 , x S1 ] that escapes to [x S3 , x U 2 ]. Similarly, for a fixed ε ∈ [ε Fold 2 , ε Fold 1 ] or ε ∈ (ε Fold 1 , ε Fold 3 ], the part of [x U1 , x S1 ] or [x U 2 , x S2 ] that is absorbed into [x S3 , x U2 ] should be quantified. Here, we quantify the absorbed region of the current interval via analyzing the escape probability, and the different forms of the escape probability corresponding to Cases 1, 2, and 3 will be obtained below.

Escape probability
For any u(x) ∈ H 2 0 (R), the generator of the stochastic process x of the system (1) is where d is the intensity of Gaussian noise, and σ is the intensity of Lévy noise. u (x) and u (x) represent the first and second derivatives of u(x) versus x, respectively. Note that y belongs to the symmetry interval (−1, 1) in the de-singularizing term I |y|<1 (y)yu (x). However, the range of y is generally an asymmetric interval (a, b).
Suppose that x belongs to a general interval (a, b). Then, δ in I |y|<1 (y)yu (x) is no longer equal to 1. From the following derivation process, we find δ = (b − a)/2. Hence, we first replace I {|y|<1} (y) in Eq. (2) to I {|y|<(b−a)/2} (y) and get Next, the different forms of the escape probability corresponding to Cases 1, 2, and 3 will be derived based on Eq. (3).

Case 1 or Case 3
The asymmetric Lévy-noise-induced CT 1 and CT 3 have the same form of the escape problem as follows: To keep the computational domain fixed as (−1, 1), we perform a scalar conversion, where z ∈ (−1, 1). Defining we get Let y = 2 b−a y. Then, the integral term in Eq. (3) can be rewritten as where v (z) is the first derivative of v(z) versus z. Therefore, the equivalent form of Eq. (3) is where v (z) is the second derivative of v(z) versus z.
Suppose that the integral in Eq. (4) is denoted by I 0 , We decompose I 0 = C 1 I 1 + C 2 I 2 , where I 1 can be rewritten as For z > 0, we have For z < 0, we get I 2 can be rewritten as For z > 0, we have For z < 0, we get Combining the above results of I 1 and I 2 , we rewrite Eq. (4) as while Next, Eqs. (5) and (6) are discretized. We divide the computational domain [−1, 1] into 2J subintervals, i.e., z j = jh, −J j J with each subinterval having the size h = 1/J. Denote the numerical solution by the vector Then, the discretizations corresponding to Eqs. (5) and (6) are while where the summation symbol means that the terms of both end indices are multiplied by 1/2, means that only the term of the bottom index is multiplied by 1/2, and means that only the term of the upper index is multiplied by 1/2.
The truncation errors of the central difference schemes for the derivatives in Eqs. (7) and (8) are of the second-order O(h 2 ). From the error analysis of Refs. [31] and [32], the leading-order error of the quadrature rule is where ζ is the Riemann zeta function. Then, Eqs. (7) and (8) can be further written as while By taking (a, b) as (x U 2 , x S1 ) under a fixed ε ∈ [ε Fold 4 , ε Fold 2 ), V −J = 1 and V J = 0, the escape probability V P 1 (x) corresponding to CT 1 can be obtained via Eqs. (9) and (10). Similarly, the escape probability V P 3 (x) corresponding to CT 3 can be obtained when (a, b) is taken as (

Case 2
For the asymmetric L vy-noise-induced CT 2, there is an interval [x U2 , x U1 ] between the two non-adjacent intervals ( d). Then, c < d < a < b, and the corresponding dynamic transition is as follows: [33] . Based on this scalar conversion, the value corresponding to ( c, d) on the z-axis can be obtained, leading Similar to CT 1 and CT 3, the integral in Eq. (4) is denoted by I 0 , We decompose I 0 = C 1 I 1 + C 2 I 2 , where Since I 1 here is the same as I 1 of Case 1 or Case 3, we only need to analyze I 2 , For z > 0, we get For z < 0, we have Combining the above results of I 1 and I 2 , we rewrite Eq. (4) as while Using the same discretization method in Case 1 or Case 3, the discretizations corresponding to Eqs. (11) and (12) are while For a fixed ε ∈ [ε Fold 2 , ε Fold 1 ], we take (a, b) = (x U1 , x S1 ), (c, d) = [x S3 , x U2 ], V −J = V xU1 and V J = 0, and the escape probability V P 2 (x) corresponding to CT 2 can be obtained via Eqs. (13) and (14). The specific method for choosing V xU1 will be introduced below.

Absorbed region
Based on Eqs. (9) and (10), can be obtained. Furthermore, P 2 (x) of ∀x ∈ [x U1 , x S1 ] follows from Eqs. (13) and (14). However, it is impossible to determine how large the value of P i (x) (i = 1, 2, 3) is, and the corresponding x is considered to be absorbed into the undesirable state [x S3 , x U2 ]. To quantify the part of [x U 2 , x S1 ], [x U 1 , x S1 ] or [x U 2 , x S2 ] that may be absorbed into [x S3 , x U2 ], the key is to give a threshold value of the corresponding P i (x). Similar to the method in Ref. [3], the absorbed regions corresponding to CT 1, CT 2, and CT 3 are also defined here via the relationship between the tangent slope of P i (x) and the slope of a given line in the xP i (x)-plane. It should be noted that, in the following, α = 1.5, β = −1, σ = 0.8, and d = 0.1 are taken as a set of basic parameter values.

Absorbed region within
For CT 1, the given line in the xP 1 (x)-plane is x + x S1 x S1 − x U2 , and it is a line connecting (x U 2 , 1) and (x S1 , 0). Then, an approximate definition of the absorbed region within [x U 2 , x S1 ] is given as follows.
is defined as the absorbed region D 1 of [x U2 , x S1 ] under the Lévy noise.
Based on Definition 1, P 1 (x) and D 1 for ε = −2.15 ∈ [ε Fold 4 , ε Fold 2 ) and different noise parameters are shown in Fig. 2. In addition, the ranges of D 1 corresponding to different α, β, d, and σ are presented in Table 1, respectively. It is found that D 1 within [x U2 , x S1 ] = [−1.552, 1.711] increases with decreasing α. When α = 1.01, even the whole [x U1 , x S1 ] becomes D 1 as shown in Fig. 2(a). This is because a smaller α can generate larger jumps which are advantageous for x ∈ [x U 2 , x S1 ] to escape. Furthermore, as shown in Figs. 2(b)-2(d), the range of D 1 also increases with decreasing β or increasing d and σ, which is due to the change of these parameters that can increase the amplitude of the jumps in the Lévy noise.

Absorbed region within
For CT 2, the given line in the xP 2 (x)-plane is a line connecting (x U1 , y 2 (x U1 )) and (x S1 , 0), where x + x S1 x S1 − x U2 , and y 2 (x U 1 ) = xU1−xS1 xU2−xS1 is the value when x = x U1 , as shown in Fig. 3. x U 2 −x S1 x + x S1 x S1 −x U 2 is a line connecting (xU2, 1) and (xS1, 0), and the horizontal dotted line is the value of y2(x) with x = xU1 (color online) Obviously, the given line is a part of y 2 (x), and they have the same slope 1/(x U2 − x S1 ). It should be noted that y 2 (x U 1 ) is the boundary condition V −J of P 2 (x), namely, V −J = V xU1 = y 2 (x U 1 ). Therefore, an approximate definition of the absorbed region within [x U1 , x S1 ] is given as follows.
is defined as the absorbed region D 2 of [x U1 , x S1 ] under the Lévy noise.
Based on Definition 2, P 2 (x) and D 2 for ε = −1.26 ∈ [ε Fold 2 , ε Fold 1 ] and different noise parameters are shown in Fig. 4. In addition, the ranges of D 2 corresponding to different α, β, d, and σ are presented in Table 2, respectively. In Fig. 4(a), D 2 decreases first and then increases with increasing α, and it is larger especially when α is close to 2. Furthermore, as shown in Figs. 4(b)-4(d), the range of D 2 decreases with decreasing β or increasing d and σ. Obviously, all results here are different from the results in Fig. 2.
In fact, these differences appear because there is a sub-desirable state between the desirable state and the undesirable state. From the energy landscape, a potential well is added between the two potential wells, as shown in Fig. 5. This means that a particle needs to cross two barriers to escape from the desirable state to the undesirable state. When α is smaller and very close to 1, large jumps in the Lévy noise can directly induce a particle to escape to the undesirable state. When α continues to increase, such as α = 1.5, the jumps in the Lévy noise cannot directly induce the particle to escape to the undesirable state, while they can induce it to switch between the desirable state and the sub-desirable state. In this case, only the particle stays in the sub-desirable state, namely, it does not go back to the ideal state. Hence, the probability that it escapes to the undesirable state in the next step will increase. Although the decrease of β or increase of d and σ can promote the switching of the particles between the desirable state and the sub-desirable state, it is not sufficient to cause the particles to escape to the undesirable state. On the contrary, the changes of these parameters increase the likelihood that the particles return to the desirable state, and reduce the probability that the particles escape to the undesirable state, as shown in Figs. 4(b)-4(d). When α is further increased to  be close to 2, the jumps in the Lévy noise gradually decrease. Once a particle escapes to the sub-desirable state, in this case, it becomes difficult to return to the desirable state, which in turn increases the possibility that the particle escapes to the non-ideal state, as shown in Fig. 4(a). As stated above, small perturbations benefit more from the intermediate potential function (the sub-desirable state), while large jumps benefit less [19] .

Absorbed region within
] under a fixed ε ∈(ε Fold 1 , ε Fold 3 ] For CT 3, the given line in the xP 3 (x)-plane is and it is a line connecting (x U 2 , 1) and (x S2 , 0). Then, an approximate definition of the absorbed region within [x U 2 , x S2 ] is given as follows.
Based on Definition 3, P 3 (x) and D 3 for ε = −0.66 ∈ (ε Fold 1 , ε Fold 3 ] and different noise parameters are shown in Fig. 6. In addition, the ranges of D 3 corresponding to different α,  Table 3, respectively. The results show that the whole [x U2 , x S2 ] becomes D 3 for different α, β, and d, as shown in Figs. 6(a)-6(c). Furthermore, the whole [x U 2 , x S2 ] also becomes D 3 when σ is larger, while a part of it becomes D 3 when σ = 0, 2, as shown in Fig. 6(d). It indicates that the intensity of Lévy noise plays a key role in the occurrence of CT 3. Although Lévy-noise-induced CT 1 and CT 3 have similar escape problems between two adjacent intervals, CT 3 is more likely to occur because the stability of the current state of the system (1) with ε ∈ (ε Fold 1 , ε Fold 3 ] is weaker than the undesirable state.

PDBUR
Supposing that μ is a measurement of the length of an interval, a natural question is how large is for asymmetric Lévynoise-induced CT 1, CT 2, or CT 3 to occur. This question will be answered here.

PDBUR of CT 1
Definition 4 For D 1 and [x U 2 , x S1 ] under ε ∈ (ε F old 4 , ε F old 2 ], the set of ε satisfying μ 1 0.5 is defined as the PDBUR U 1 (ε) of CT 1 under the Lévy noise. Based on Definition 4, Fig. 7 shows the space diagrams of U 1 (ε) of CT 1 corresponding to different parameters of Lévy noise. It is found that, the closer ε is to Fold 2, the larger the range of U 1 (ε) is, which means that CT 1 is more likely to occur. Furthermore, the smaller α, the smaller β, and the larger σ, which are related to the amplitude or the number of the large jumps, can increase the possibility of CT 1 taking place, as shown in Figs. 7(a), 7(b), and 7(d). While d, which describes the intensity of Gaussian, has almost no effect on U 1 (ε), as shown in Fig. 7(c). Anyway, once ε and the noise parameters belong to U 1 (ε), there is a high possibility that CT 1 is impending.

PDBUR of CT 2
Definition 5 For D 2 and [x U 1 , x S1 ] under ε ∈ (ε Fold 2 , ε Fold 1 ], the set of ε satisfying μ 2 0.5 is defined as the PDBUR U 2 (ε) of CT 2 under the Lévy noise. Based on Definition 5, μ 2 and U 2 (ε) corresponding to asymmetric Lévy-noise-induced CT 2 are shown in Fig. 8. Similarly, CT 2 is more likely to occur when the system (1) is close to Fold 1. Compared with U 1 (ε) in Fig. 7 above or U 3 (ε) in Fig. 9 below, however, the range of U 2 (ε) is very small, especially for β and σ that are related to the amplitude or the number of the large jumps. On the contrary, the range of U 2 (ε) is larger when α is close to 2, namely, there are few jumps in the Lévy noise. Moreover, larger d describing the intensity of the Gaussian also increases the possibility of CT 2 occurring. All results indicate that small perturbations in the Lévy noise are conducive to the occurrence of a CT between non-adjacent states. Because V xU1 < 1, U 2 (ε) is different from the PDBUR of CT 1 or CT 3. It does not mean that CT 2 must occur in its region, while it can show that x ∈ [x U1 , x S1 ] under ε ∈ [ε Fold 2 , ε Fold 1 ] has a high probability of escaping to the undesirable state relative to the boundary condition V xU1 . However, the quantization of the PDBUR of CT 2 can guide us in preventing a catastrophic CT.

PDBUR of CT 3 Definition 6 For
0.5 is defined as the PDBUR U 3 (ε) of CT 3 under the Lévy noise. Based on Definition 6, the space diagrams of U 3 (ε) of asymmetric Lévy-noise-induced CT 3 are shown in Fig. 9. Obviously, all the regions of ε versus α, ε versus β, ε versus d, and ε versus σ become U 3 (ε) under the given set of basic parameter values. This means that the Lévy noise can easily induce CT 3 to occur in ε ∈ (ε Fold 1 , ε Fold 3 ]. Moreover, our results show that μ 3 is less than 1 when σ is small, that is, [x U 2 , x S2 ] does not all become the absorbed region. Therefore, σ plays an important role in this case. Our goal is to avoid ε and the noise parameters entering U 3 (ε) in practical systems.

Conclusions
In this paper, we have focused on an asymmetric Lévy-noise-induced tri-stable model as a concrete example to quantify the ranges of the parameters where CTs from the desirable state directly to the undesirable state may occur. Based on the escape probability, the absorbed regions that the current state of the given model is absorbed into the undesirable state corresponding to CT 1, CT 2, and CT 3 are first defined. Then, the concept of the PDBUR under the asymmetric Lévy noise is introduced. Once the control parameter and the noise parameters enter the PDBUR of CT 1, CT 2, or CT 3, there is a high possibility that a catastrophic CT is impending. Now, some managements should be adopted to avert it.
However, in the definitions of the absorbed region and the PDBUR, the scale factor is taken as 1/2. More accurate results may be realized in terms of other values, which need to be further explored and developed. Moreover, we take α = 1.5, β = −1, σ = 0.8, and d = 0.1 as a set of basic parameter values in the calculation. Figures 6 and 9 have shown that the current state of the given model is not all absorbed into the undesirable state when σ is smaller, such as σ = 0.2. Therefore, more interesting phenomena may be obtained for other basic parameter values. Our method may be regarded as a complement to existing early warning indicators, and more general methods need to be established.

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