Chaos 29 081102 (2019) httpsdoiorg10106315115348 29 081102

copy 2019 Author(s)

Predicting noise-induced critical transitionsin bistable systemsCite as Chaos 29 081102 (2019) httpsdoiorg10106315115348Submitted 17 June 2019 Accepted 24 July 2019 Published Online 20 August 2019

Jinzhong Ma Yong Xu Yongge Li Ruilan Tian and Juumlrgen Kurths

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Chaos ARTICLE scitationorgjournalcha

Predicting noise-induced critical transitions inbistable systems

Cite as Chaos 29 081102 (2019) doi 10106315115348

Submitted 17 June 2019 middot Accepted 24 July 2019 middotPublished Online 20 August 2019 View Online Export Citation CrossMark

Jinzhong Ma1 Yong Xu12a) Yongge Li3 Ruilan Tian4 and Juumlrgen Kurths56

AFFILIATIONS

1Department of Applied Mathematics Northwestern Polytechnical University Xirsquoan 710072 China2MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xirsquoan 710072 China3Center for Mathematical Sciences and School of Mathematics and Statistics Huazhong University of Science and Technology

Wuhan 430074 China4Centre for Nonlinear Dynamics Research Shijiazhuang Tiedao University Shijiazhuang 050043 China5Potsdam Institute for Climate Impact Research Potsdam 14412 Germany6Department of Physics Humboldt University at Berlin Berlin 12489 Germany

a)Email hsux3nwpueducn

ABSTRACT

Critical transitions from one dynamical state to another contrasting state are observed in many complex systems To understand the eectsof stochastic events on critical transitions and to predict their occurrence as a control parameter varies are of utmost importance in variousapplications In this paper we carry out a prediction of noise-induced critical transitions using a bistable model as a prototype class of realsystems We nd that the largest Lyapunov exponent and the Shannon entropy can act as general early warning indicators to predict noise-induced critical transitions even for an earlier transition due to strong uctuations Furthermore the concept of the parameter dependent basinof the unsafe regime is introduced via incorporating a suitable probabilistic notionWend that this is an ecient tool to approximately quantifythe range of the control parameter where noise-induced critical transitionsmay occur Ourmethodmay serve as a paradigm to understand andpredict noise-induced critical transitions in multistable systems or complex networks and evenmay be extended to a broad range of disciplinesto address the issues of resilience

Published under license by AIP Publishing httpsdoiorg10106315115348

Critical transitions are widespread in natural social and techno-

logical systems which can lead to detrimental systembreakdowns

Therefore it is important to develop eective tools to predict

noise-induced critical transitions early Many studies in recent

years have been devoted to exploring early warning indicators to

predict and characterize the onset of critical transitions However

these early warning indicators are valid only for relatively small

perturbations and they sometimes cannot provide enough time

for us to avert an impending noise-induced critical transition To

overcome these problems in this paper we verify that the largest

Lyapunov exponent and the Shannon entropy can act as general

early warning indicators Furthermore we introduce the param-

eter dependent basin of an unsafe regime to quantify the range

of the control parameter where noise-induced critical transitions

may occur

I INTRODUCTION

Complex systems in ecology economics sociologyphysiology1ndash4 and many other elds often undergo slow changesaected by stochastic events whose persistent eects may lead to asudden or irreversible change of the system such as a transition fromone stable state to another one In typical examples such a transi-tion is able to cause considerable impacts on human economies andsocieties56 or even induce some unexpected disasters78 Therefore itis crucially important to investigate noise-induced critical transitionsso as to predict such a catastrophic event

Recently some studies in critical transitions showed that a phe-nomenon known as ldquocritical slowing-down (CSD)rdquo can be used todetect a sudden transition9ndash11 CSD means that the rate of returnto equilibrium following small perturbations slows down as systemsapproach critical transitions It can cause an increase in the variance

Chaos 29 081102 (2019) doi 10106315115348 29 081102-1

Published under license by AIP Publishing

Chaos ARTICLE scitationorgjournalcha

and autocorrelation of a system prior to a transition12ndash14 ThereforeCSD-related predictions have become a hot topic and are increasinglyattracting much attention from various disciplines15ndash17 However itshould be mentioned that the CSD principle holds only for a criti-cal transition suciently approaching the deterministic bifurcationnamely perturbations are relatively small Many practical systems infact are usually intrinsically or extrinsically convoluted with varyingscales of perturbations1819 Strong uctuations may induce an earliercritical transition far before the deterministic bifurcation for whichall CSD-based early warning indicators may fail2021 A crucial ques-tion is whether these indicators can provide enough time for us toavert an imminent critical transition2223 To counter these problemsin this paper not only more general early warning indicators willbe explored but also the range of the control parameter in whichnoise-induced critical transitions may occur will be quantied

Given a noise-induced complex system with a varying controlparameter there are two common features during the process of atransition near a critical value One is that the instability of the sys-tem becomes more prominent24ndash26 and the other is that the height ofthe potential barrier decreases namely the current desirable state isgradually absorbed127ndash29With these in mind we can carry out a pre-diction of noise-induced critical transitions via detecting the insta-bility of a system and quantifying the range of the control parameterwhere the desirable state may be absorbed In nonlinear stochasticdynamics the largest Lyapunov exponent can measure the intrin-sic instability of the trajectories in a system and detect qualitativechanges of the system behavior with varying control parameters30ndash32

In addition the Shannon entropy is another measure of instabilityand it can reect the change in the dynamics of a system33ndash35 In factthe likelihood that the height of the potential barrier disappears or adesirable stable state is absorbed depends on its stability against sig-nicant perturbations The traditional linearization-based approachto study stability is too local to adequately assess how stable a stateis A new measure basin stability related to the volume of the basinof attraction was recently proposed3637 It is nonlocal nonlinear andeasily applicable to complex systems The extension may provide apowerful tool for the prediction of noise-induced critical transitionsin complex systems

In this paper we analyze the instability of an imminent criti-cal transition through the largest Lyapunov exponent and Shannonentropy respectively Furthermore we extend basin stability to anew concept the ldquoparameter dependent basin of an unsafe regimerdquo(PDBUR) to predict noise-induced critical transitions We nd thatthe largest Lyapunov exponent and the Shannon entropy can act asgeneric early warning indicators of noise-induced critical transitionsand PDBUR can approximately quantify a range of control parame-ters where noise-induced critical transitions may occur A combina-tion of them provides sucient warnings for disaster managementgroups to hinder noise-induced critical transitions

The rest of the paper is organized as follows In Sect II thenoise-induced critical transitions of a givenmodel are introduced InSec III the largest Lyapunov exponent and the Shannon entropy arevalidated as early warning indicators to predict these noise-inducedcritical transitions Section IV introduces the concept of PDBUR toapproximately quantify the range of the control parameter wherenoise-induced critical transitions may occur Finally discussions arepresented in Sec V

II NOISE-INDUCED CRITICAL TRANSITIONS IN

BISTABLE SYSTEMS

To describe parameterized phosphorus dynamics in a lakeapproaching eutrophication under a regime of externalperturbations38ndash40 a stochastic bistable model can be dened as

dx = f (xα)dt +radic

σdB(t) (1)

where f (xα) = minus part

partxU(xα) = α minus x + x8

x8+1and U(xα) denotes

the potential function of x α is a control parameter B(t) is theBrownian motion and σ denotes the noise intensity

Equation (1) can be reduced to the corresponding deterministicsystem for σ = 0 and its equilibrium point xE with changing α canbe obtained through

f (xEα) = 0

The stability of xE can be determined based on the positive ornegative value of the derivative of f (xα) with respect to x

df (xα)

dx

∣

∣

∣

∣

x=xE

= minus1 +8x7

(x8 + 1)2

∣

∣

∣

∣

x=xE

Then the geometric structure of the deterministic system vsα is shown in Fig 1 Three equilibrium points xS1 xU and xS2in the bistable region (shaded part) are exhibited for a given αxS1 in the lower branch means that Eq (1) remains in the desir-able state while xS2 in the upper branch says that Eq (1) stays ina state space populated by the undesirable state xU in the mid-dle branch (dotted line) marks the height of the potential barrierbetween xS1 and xS2 The coordinate values of the two bifurcationpoints Fold 1 and Fold 2 are (αFold1 = 0660 xE_Fold1 = 0768) and(αFold2 = 0389 xE_Fold2 = 1204) respectively If Eq (1) approachesFold 1 a slight incremental change in α may bring it beyond thebifurcation and induce a transition to the undesirable state xS2

Figure 2 shows the average state of x in the presence of a noisewith changing α We nd that the critical point approaches Fold 1when σ is smallWhile the critical transition takes placemuch earlierwith large σ and the larger the σ the more obvious the phenomenon

FIG 1 Geometric structure of Eq (1) with respect to changing α and σ = 0

Chaos 29 081102 (2019) doi 10106315115348 29 081102-2

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Chaos ARTICLE scitationorgjournalcha

FIG 2 Noise-induced critical transitions of Eq (1) vs different σ xAve denotes theaverage state of x in the presence of noise The inset plot shows that the criticalpoint approaches Fold 1

is It is thus evident that stochastic perturbations may induce a tran-sition to the alternative stable state if they are suciently large tobring the system over the height of the potential barrier Here suchlarge σ values that destroy the bistable structure are not considered

III EARLY WARNING INDICATORS OF NOISE-INDUCED

CRITICAL TRANSITIONS

In a noise-induced bistable model the response will becomemore unstable as it approaches the critical point Then a criticaltransition is easy to occur As is well known the largest Lyapunovexponent and the Shannon entropy are appropriate measures forinstability of trajectories in a stochastic system Therefore we willfurther examine that the largest Lyapunov exponent and the Shan-non entropy may act as general early warning indicators to predictnoise-induced critical transitions

A Largest Lyapunov exponent

The largest Lyapunov exponent λ is a very powerful tool toinvestigate complex system dynamics One of the most advantageousapplications is to detect the qualitative changes of the systembehaviorwith changing control parameters4142Here it will be used to identifynoise-induced critical transitions In the present paper the methodintroduced by Benettin et al will be employed to calculate λ4344

The λ of Eq (1) for dierent σ are shown in Fig 3 We uncoverthat an obvious increase of λ can be obtained when the critical transi-tion approaches the deterministic bifurcation Moreover an increasein λ is also evident when the critical transition takes place far fromthe bifurcation point of the corresponding deterministic system dueto strong uctuations More importantly the increase in λ appears inadvance with the critical transition earlier as shown by the verticaldashed lines in Fig 3 It is thus clear that the remarkable dierenceof λ is a general feature to an impending noise-induced critical tran-sition Through a space diagram in Fig 4 we can more intuitivelywitness an earlier increase of λwith increasing σ Therefore λ can act

FIG 3 Largest Lyapunov exponent λ of Eq (1) for four different σ

as a general early warning indicator to predict noise-induced criticaltransitions

B Shannon entropy

Entropy-based measures are ecient to quantify the dynamicalcharacteristics of complex systems45 Entropy analysis with respect toa changing parameter may provide a special signal for an impendingnoise-induced critical transition46

We use the denition of the entropy for a discrete randomvariable introduced by Shannon3347 Let X be a discrete randomvariable with the possible states X1X2 Xn whose correspond-ing probability is Pi i = 1 2 n Then the Shannon entropy isdened as

Hs = minusn

sum

i=1

Pilog2(Pi) (2)

To obtain the Shannon entropy of Eq (1) three parts of α

need to be considered including α lt αFold2 α isin [αFold2αFold1] andα gt αFold1 For α isin [αFold2αFold1] the expressions to calculate theprobability of the two possible states xS1 and xS2 are

P11 =lang

1

Tt x(t) lt xU t le T

rang

and

P12 =lang

1

Tt x(t) ge xU t le T

rang

where T is the time length of a trajectory and 〈middot〉 is the operator toaverage multiple realizations

For α lt αFold2 we dene

P21 =lang

1

Tt x(t) lt xE_Fold2 t le T

rang

Chaos 29 081102 (2019) doi 10106315115348 29 081102-3

Published under license by AIP Publishing

Chaos ARTICLE scitationorgjournalcha

FIG 4 Largest Lyapunov exponent λ of Eq (1) vs σ and α (a) The space diagram of λ (b) The vertical view of λ

and

P22 =lang

1

Tt x(t) ge xE_Fold2 t le T

rang

Similarly the probability in α gt αFold1 is given as

P31 =lang

1

Tt x(t) lt xE_Fold1 t le T

rang

and

P32 =lang

1

Tt x(t) ge xE_Fold1 t le T

rang

Based on Eq (2) the Hs of Eq (1) with three dierent parts ofα is composed of

Hsi = minusPi1log2Pi1 minus Pi2log2Pi2 i = 1 2 3 (3)

Here the initial condition is set as x(0) = xS1 corresponding todierent α

Based on Eq (3) the numerical results of Hs for dierent σ areshown in Fig 5 Although an increase inHs is obtainedwhen the crit-ical transition approaches the deterministic bifurcation it appears sosuddenly that there is not enough time to avert it Thusλ ismore suit-able for predicting the critical transitions than Hs when σ is smallerTo an earlier critical transition under strong uctuations a mean-ingful change of Hs allows us to witness the possibility to warn anoise-induced critical transition Similar to the result ofλ an increasein Hs appears earlier with the critical transition occurring earlierFrom a space diagram of Hs shown in Fig 6 we nd that an ear-lier increase appears with increasing σ and it becomes easier to beidentied All our numerical results indicate that both λ and Hs canact as general early warning indicators to predict the noise-inducedcritical transitions

IV PARAMETER DEPENDENT BASIN OF THE UNSAFE

REGIME

Beyond establishing more general early warning indicators ofimpending noise-induced critical transitions we hope to furtherquantify the range of the control parameter in which noise-inducedcritical transitions may occur In this section the concept of PDBURwill be introduced by incorporating a suitable probabilistic notionand it will be an appropriate measure of the inuence of Gaussianwhite noise on critical transitions

FIG 5 Shannon entropy Hs of Eq (1) with four different σ

Chaos 29 081102 (2019) doi 10106315115348 29 081102-4

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Chaos ARTICLE scitationorgjournalcha

FIG 6 Shannon entropy Hs of Eq (1) vs σ and α (a) The space diagram of Hs (b) The vertical view of Hs

A The absorbed region within [xS1xU] under the

fixed α isin [αFold2αFold1]

Stochastic events may bring a complex system into another con-trasting and undesirable stable state ie almost all x isin [xS1 xU] areabsorbed Therefore the possibility of a noise-induced critical tran-sition occurring with a xed α isin [αFold2αFold1] can be estimated viameasuring the part of [xS1 xU] that may be absorbed

Considering three representative U(x) of Eq (1) as shown inFig 7 we nd that the occurrence of a critical transition depends onthe perturbations as well as on the barrier height ofU(x) In the caseof α1 the potential well of xS1 is deeper a transition occurs only if the

FIG 7 Potential function U(x) of Eq (1) for σ = 0 and three different α

perturbations are suciently large The ability to switch between xS1and xS2 is almost the same with α2 Nevertheless the potential well ofxS1 becomes shallower in the case of α3 and the small perturbationsmay result in a transition to a contrasting dynamical state In Table I[xS1 xU] corresponding to these three dierent cases of α are pre-sented Based on the escape probability wewill give a denition of theabsorbed region within [xS1 xU] under the xed α isin [αFold2αFold1]

In fact if Eq (1) starts from a point x in the region [xS1 xU]it will either exit the region from the left side xS1 or from theright side xU Here we focus on the escape probability exitingthrough xU denoted as P(x) which can be described by the followingequation4849

LP(x) = f (xα)dP(x)

dx+

1

2σd2P(x)

dx2= 0 (4)

where L is a generator For the boundary cases we suppose thatthe escape does not occur once the solution path of Eq (1) reachesxS1 On the contrary the escape takes place if the solution pathreaches xU Thus the boundary conditions of Eq (4) should be set asP(x)|x=xS1 = 0 and P(x)|x=xU = 1 A numerical method for solvingEq (4) can be found in Appendix

From Fig 2 we observe that noise-induced critical transitionsappear in the cases of σ = 001 and σ = 003withα3 To obtain somegeneric behavior of P(x) in noise-induced critical transitions P(x)

TABLE I The [xS1 xU ] corresponding to three different cases of α

α [xS1 xU]

α1 [0401 1139]α2 [0526 0980]α3 [0712 0821]

Chaos 29 081102 (2019) doi 10106315115348 29 081102-5

Published under license by AIP Publishing

Chaos ARTICLE scitationorgjournalcha

FIG 8 Escape probability P(x) for α3 and two different σ The solid line is y(x)The inset shows that P(x) is very close to y(x)

under these two cases are simulated in Fig 8 There is an interest-ing phenomenon that P(x) is very close to y(x) = 1

xUminusxS1x + xS1

xS1minusxU

a line connecting (xS1 0) and (xU 1) in the x vs P(x) plane Itmeans that the tangent slope of P(x) is very close to 1

xUminusxS1when

the region [xS1 xU] is almost completely absorbed That is we canapproximately obtain the possible absorbed region within [xS1 xU]via detecting the change of the tangent slope of P(x) Thereforean approximate denition of the absorbed region within [xS1 xU] isgiven as

Denition 1 Assuming that a tagged partition of [xS1 xU] is anite sequence

xS1 = x0 lt x1 lt x2 lt middot middot middot lt xnminus1 lt xn = xU

we call the set [xk xU] xS1 le xk lt xU k = 0 1 n minus 1 satisfyingP(xk+1)minusP(xk)

xk+1minusxkge ε 1

xUminusxS1 0 lt ε lt 1 as the ldquoabsorbed regionrdquo within

[xS1 xU] under σ and it is denoted as DARWithin Denition 1 the selection of ε is critical which may be

variable for dierent systems If the critical transition occurs quicklya smaller ε should be considered such as ε lt 1

2 On the contrary a

larger ε that approaches 1 should be chosen The selection of ε makesit possible that we not only obtain a more accurate result but alsohave enough time to adapt some management to avert an imminentnoise-induced critical transition Taking ε = 1

2as a case study in the

present paper DAR under dierent σ and α isin [αFold2αFold1] will beinvestigated

Based on Eq (4) and Denition 1 P(x) and DAR for dierentα and σ are shown in Fig 9 The corresponding interval length ofDAR is presented in Table II Obviously the larger σ is the closerP(x) is to y(x) and the larger the range of DAR is Especially forσ = 001 and σ = 003 shown in Fig 8(c) P(x) almost coincideswith y(x) and the whole [xS1 xU] becomes DAR It indicates that thenoise-induced critical transitions have taken place in both cases andthe same conclusion can be observed in Fig 2 Although the whole[xS1 xU] is not absorbed a noise-induced critical transition mayalso occur For example a critical transition in Eq (1) has occurred

under α3 and σ = 0001 A natural question is for how big micro(DAR)

micro([xS1 xU ])

FIG 9 Escape probability P(x) and the corresponding absorbed region DAR

(bar-type) for different σ (a) α1 = 040 (b) α2 = 052 (c) α3 = 065 The topsolid line is y(x) The inset in (c) shows that P(x) is very close to y(x)

1= microP is (here micro is a measurement of the interval length) a noise-induced critical transition may occur Next we will dene PDBURof noise-induced critical transitions

B PDBUR of noise-induced critical transitions

Through the noise-induced critical transition of Eq (1) withσ = 00001 in Fig 2 we nd that Eq (1) is about to leave the lowerbranch of xE when α = 0654 and it has left it when α = 0655 microP

in both cases are

microP|α=0654 = 0379

and

microP|α=0655 = 0428

Therefore we approximate PDBUR of α via counting the pointwhere

microP gemicroP|α=0654 + microP|α=0655

2asymp 0404

1= microP_c

Denition 2 For DAR and [xS1 xU] under σ and α isin[αFold2αFold1] we call the set of α satisfying microP ge microP_c as PDBURand it is denoted as UB(α σ)

TABLE II The interval length of DAR under different α and σ

σ

α 00001 0001 001 003

040 [11061139] [10561139] [09461139] [08531139]052 [09560980] [09220980] [08230980] [07260980]065 [07920821] [07340821] [07120821] [07120821]

Chaos 29 081102 (2019) doi 10106315115348 29 081102-6

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Chaos ARTICLE scitationorgjournalcha

FIG 10 The measurement microP and the PDBUR UB(α σ) with respect todifferent σ The bar-type denotes interval length of UB(α σ)

In Fig 10 microp and UB(α σ) for dierent σ are shown Weobserve that σ plays a crucial role to expand the interval length ofUB(α σ) the larger the σ the larger the range ofUB(α σ) When σ

is large enough such as σ = 003 nearly microP asymp 85 of the bistableregion [αFold2αFold1] becomes UB(α σ) This phenomenon is alsoassociated with an earlier noise-induced critical transition takingplace under strong uctuations

Tomake the resultsmore intuitiveUB(α σ) and xAve under fourdierent σ are simultaneously shown in Fig 11 ObviouslyUB(α σ)

FIG 11 The PDBUR UB(α σ) (shaded parts) for four different σ

can be quantied when a critical transition impending Because thecritical transition occurs more suddenly with σ = 003 UB(α σ)

should be considered once Eq (1) leaves the desirable state Ourresults also reect this phenomenon Therefore a satisfactory resulthas been found based on Denitions 1 and 2 by taking ε = 1

2 More

accurate results may be realized in terms of the other εFigure 12 indicates themicrop with respect toα and σ Onceα and σ

enterUB(α σ) there is a strong possibility that a noise-induced crit-ical transition occurs Now some management should be adapted toavert it As the well-known concept of basin stability the introduced

FIG 12 The measurement microP vs σ and α (a) The space diagram of microP The inserted plane represents microP_c = 0404 (b) The vertical view of microP and UB(α σ) can bevisualized on the upper right

Chaos 29 081102 (2019) doi 10106315115348 29 081102-7

Published under license by AIP Publishing

Chaos ARTICLE scitationorgjournalcha

UB(α σ) is the important geometric structure that can be used topredict the noise-induced critical transitions in complex systems

V DISCUSSIONS

In this paper we have focused on a Gaussian white noise-induced eutrophication model as a concrete example to show theprediction of noise-induced critical transitions To understand therole of stochastic events with respect to the emergence of a criticaltransition the existing results on stochastic dynamic theory shouldbe used for its prediction With this in mind we carry out a measureof the instability to an impending critical transition Our results showthat the largest Lyapunov exponent and the Shannon entropy canserve as generic early warning indicators to predict noise-inducedcritical transitions Considering some practical diculties in avert-ing an imminent noise-induced critical transition for example thelargest Lyapunov exponent or the Shannon entropy sometimes istoo late to warn a transition Therefore PDBUR is further intro-duced to approximately quantify the range of the control parameterwhere noise-induced critical transitions may occur from a nonlocalperspective

Furthermore some problems should be addressed For exam-ple what will happen for these catastrophic transitions in multistableor high-dimensional systems or the eects of dierent kinds ofrandom noises like Leacutevy noise still deserves more concentrationsIn addition noises dierent from the Gaussian one and the cou-pling of multistable or high-dimensional systems may induce richdynamical behaviors with unexpected critical transitions Whetherthe denitions and methods proposed in this paper can be extendedto warn more complex critical transitions should be explored infuture

ACKNOWLEDGMENTS

This work was supported by the National Natural ScienceFoundation of China under Grant No 11772255 the Fundamen-tal Research Funds for the Central Universities the Shaanxi ScienceFund for Distinguished Young Scholars and Innovation Foundationfor Doctor Dissertation of Northwestern Polytechnical UniversityY Li thanks the China Postdoctoral Science Foundation fundedproject

APPENDIX

Let xS1 = x0 lt x1 lt x2 lt middot middot middot lt xnminus1 lt xn = xU be a tagged partition of [xS1 xU] and 1x be the step length Based on the second-orderdierence Eq (4) can be rewritten as

σ

2

P(xjminus1) minus 2P(xj) + P(xj+1)

1x2+ f (xjα)

P(xj+1) minus P(xjminus1)

21x= 0 j = 1 n minus 1

Then we get

diag(σ

2

)

AP(x) + a + diag(f (xα))BP(x) + b =(

diag(σ

2

)

A + diag(f (xα))B)

P(x) + (a + b) = 0

where

diag(σ

2

)

=1

1x2

σ

2σ

2

σ

2σ

2

a =1

1x2

σ

2P(x0)

0

0

σ

2P(xn)

diag(f (xα)) =1

21x

f (x1α)

f (x2α)

f (xnminus2α)

f (xnminus1α)

Chaos 29 081102 (2019) doi 10106315115348 29 081102-8

Published under license by AIP Publishing

Chaos ARTICLE scitationorgjournalcha

b =1

21x

minusf (x1α)P(x0)

0

0

f (xnminus1α)P(xn)

A =

minus2 1

1 minus2 1

1 minus2 1

1 minus2

P(x) =

P(x1)

P(x2)

P(xnminus2)

P(xnminus1)

and

B =

0 1

minus1 0 1

minus1 0 1

minus1 0

Finally P(x) can be obtained as

P(x) =(

diag(σ

2

)

A + diag(f (xα))B)minus1

(minus(a + b))

= minus(

diag(σ

2

)

A + diag(f (xα))B)minus1

(a + b)

REFERENCES1M Scheer S Carpenter J A Foley C Folke and BWalker ldquoCatastrophic shiftsin ecosystemsrdquo Nature 413 591ndash596 (2001)2R M May S A Levin and G Sugihara ldquoEcology for bankersrdquo Nature 451893ndash895 (2008)3W A Brock Tipping Points Abrupt Opinion Changes and Punctuated PolicyChange (Department of Economics UWMadison 2004)4J G Venegas et al ldquoSelf-organized patchiness in asthma as a prelude to catas-trophic shiftsrdquo Nature 434 777ndash782 (2005)5F Berkes C Folke and J Colding Linking Social and Ecological Systems Manage-ment Practices and Social Mechanisms for Building Resilience (Cambridge Univer-sity Press Cambridge 1998)6M ScheerCritical Transitions in Nature and Society (PrincetonUniversity PressPrinceton NJ 2009)7A D Barnosky et al ldquoHas the earthrsquos sixthmass extinction already arrivedrdquoNature 471 51ndash57 (2011)8A D Barnosky et al ldquoApproaching a state shift in earthrsquos biosphererdquo Nature 48652ndash58 (2012)9C Wissel ldquoA universal law of the characteristic return time near thresholdsrdquoOecologia 65 101ndash107 (1984)10K Wiesenfeld and B McNamara ldquoSmall-signal amplication in bifurcatingdynamical systemsrdquo Phys Rev A 33 629ndash642 (1986)11M Scheer et al ldquoEarly-warning signals for critical transitionsrdquo Nature 46153ndash59 (2009)12S R Carpenter and W A Brock ldquoRising variance A leading indicator ofecological transitionrdquo Ecol Lett 9 311ndash318 (2006)13V Dakos et al ldquoSlowing down as an early warning signal for abrupt climatechangerdquo Proc Natl Acad Sci USA 105 14308ndash14312 (2008)14V Dakos et al ldquoMethods for detecting early warnings of critical transitionsin time series illustrated using simulated ecological datardquo PLoS ONE 7 e41010(2012)

15E A Gopalakrishnan Y Sharma T John P S Dutta and R I Sujith ldquoEarlywarning signals for critical transitions in a thermoacoustic systemrdquo Sci Rep 635310 (2016)16X Zhang C Kuehn and S Hallerberg ldquoPredictability of critical transitionsrdquoPhys Rev E 92 052905 (2015)17C F Clements M A McCarthy and J L Blanchard ldquoEarly warning signals ofrecovery in complex systemsrdquo Nat Commun 10 1681 (2019)18Z Wang Y Xu and H Yang ldquoLeacutevy noise induced stochastic resonance in anFHN modelrdquo Sci China Tech Sci 59 371ndash375 (2016)19Y Xu H Li H Wang W Jia X Yue and J Kurths ldquoThe estimates of the meanrst exit time of a bi-stable system excited by Poisson white noiserdquo J Appl Mech84 091004 (2017)20R Liu P Chen K Aihara and L Chen ldquoIdentifying early-warning signals ofcritical transitions with strong noise by dynamical network markersrdquo Sci Rep 517501 (2015)21J Ma Y Xu J Kurths H Wang andW Xu ldquoDetecting early-warning signals inperiodically forced systems with noiserdquo Chaos 28 113601 (2018)22R Biggs S R Carpenter andW A Brock ldquoTurning back from the brink detect-ing an impending regime shift in time to avert itrdquo Proc Natl Acad Sci USA 106826ndash831 (2009)23P D Ditlevsen and S J Johnsen ldquoTipping points Early warning and wishfulthinkingrdquo Geophys Res Lett 37 L19703 (2010)24L Moreau and E Sontag ldquoBalancing at the border of instabilityrdquo Phys Rev E68 020901 (2003)25R V Soleacute and S Valverde ldquoInformation transfer and phase transitions in amodelof internet tracrdquo Physica A 289 595ndash605 (2001)26F Nazarimehr S Jafari S M R H Golpayegani M Perc and J C Sprott ldquoPre-dicting tipping points of dynamical systems during a period-doubling route tochaosrdquo Chaos 28 073102 (2018)27V N Livina F Kwasniok and T M Lenton ldquoPotential analysis reveals changingnumber of climate states during the last 60 kyrrdquo Clim Past 6 77ndash82 (2010)

Chaos 29 081102 (2019) doi 10106315115348 29 081102-9

Published under license by AIP Publishing

Chaos ARTICLE scitationorgjournalcha

28Y Xu J Li J Feng H ZhangW Xu and J Duan ldquoLeacutevy noise-induced stochasticresonance in a bistable systemrdquo Eur Phys J B 86 198 (2013)29Y Li Y Xu and J Kurths ldquoFirst-passage-time distribution in amoving parabolicpotential with spatial roughnessrdquo Phys Rev E 99 052203 (2019)30S P Kuznetsov andD I Trubetskov ldquoChaos and hyperchaos in a backward-waveoscillatorrdquo Radiophys Quant Eletron 47 341ndash355 (2004)31S P Kuznetsov ldquoExample of a physical system with a hyperbolic attractor of theSmale-Williams typerdquo Phys Rev Lett 95 144101 (2005)32C Matsuoka and K Hiraide ldquoComputation of entropy and Lyapunov exponentby a shift transformrdquo Chaos 25 103110 (2015)33C E Shannon and W Weaver The Mathematical Theory of Communication(University of Illinois Press Urbana 1998)34R Yan and R X Gao ldquoApproximate entropy as a diagnostic tool for machinehealth monitoringrdquo Mech Syst Signal Pr 21 824ndash839 (2007)35G Poveda and H D Salas ldquoStatistical scaling Shannon entropy and general-ized space-time q-entropy of rainfall elds in tropical South Americardquo Chaos 25075409 (2015)36P J Menck J Heitzig N Marwan and J Kurths ldquoHow basin stability comple-ments the linear-stability paradigmrdquo Nat Phys 9 89ndash92 (2013)37Y Zheng L Serdukova J Duan and J Kurths ldquoTransitions in a genetic tran-scriptional regulatory system under Leacutevy motionrdquo Sci Rep 6 29274 (2016)38S R Carpenter D Ludwig andW A Brock ldquoManagement of eutrophication forlakes subject to potentially irreversible changerdquo Ecol Appl 9 751ndash771 (1999)39R Wang et al ldquoFlickering gives early warning signals of a critical transition to aeutrophic lake staterdquo Nature 492 419ndash422 (2012)40S R Carpenter ldquoEutrophication of aquatic ecosystems Bistability and soilphosphorusrdquo Proc Natl Acad Sci U S A 102 10002ndash10005 (2005)

41A E Hramov A A Koronovskii and M K Kurovskaya ldquoZero Lyapunov expo-nent in the vicinity of the saddle-node bifurcation point in the presence of noiserdquoPhys Rev E 78 036212 (2008)42F Nazarimehr S Jafari S M R H Golpayegani and J C Sprott ldquoCan Lyapunovexponent predict critical transitions in biological systemsrdquo Nonlinear Dyn 881493ndash1500 (2017)43G Benettin L Galgani A Giorgilli and J M Strelcyn ldquoLyapunov exponents forsmooth dynamical systems and Hamiltonian systems a method for computing allof them Part I Theoryrdquo Meccanica 15 9ndash20 (1980)44G Benettin L Galgani A Giorgilli and J M Strelcyn ldquoLyapunov expo-nents for smooth dynamical systems and Hamiltonian systems a method forcomputing all of them Part II Numerical applicationrdquo Meccanica 15 21ndash30(1980)45J Venkatramani S Sarkar and S Gupta ldquoInvestigations on precursor measuresfor aeroelastic utterrdquo J Sound Vib 419 318ndash336 (2018)46F Nazarimehr K Rajagopal A JM Khalaf A Alsaedi V T Pham and THayatldquoInvestigation of dynamical properties in a chaotic ow with one unstable equi-librium Circuit design and entropy analysisrdquo Chaos Solitons Fractals 115 7ndash13(2018)47J Wu Y Xu H Wang and J Kurths ldquoInformation-based measures for logicalstochastic resonance in a synthetic gene network under leacutevy ight superdiusionrdquoChaos 27 063105 (2017)48T Gao J Duan X Li and R Song ldquoMean exit time and escape probability fordynamical systems driven by leacutevy noisesrdquo SIAM J Sci Comput 36 A887ndashA906(2014)49L Serdukova Y Zheng J Duan and J Kurths ldquoStochastic basins of attractionfor metastable statesrdquo Chaos 26 073117 (2016)

Chaos 29 081102 (2019) doi 10106315115348 29 081102-10

Published under license by AIP Publishing