Time Dependent Saddle Node Bifurcation:Breaking Time and the Point of No Return in a

Non-Autonomous Model of CriticalTransitions

Jeremiah H. Li1, Felix X.-F. Ye1, Hong Qian1, and Sui Huang2

1Department of Applied MathematicsUniversity of Washington

Seattle, WA 98195-3925, USA2Institute for Systems Biology

Seattle, WA 98109, USA

November 30, 2016

Abstract

There is a growing awareness that catastrophic natural phenomena canbe mathematically represented in terms of saddle-node bifurcations. In par-ticular, the term “tipping point” has in recent years entered the discourseof the general public in relation to ecology, medicine, and public health.The mathematical theory of saddle-node bifurcation and its associated topo-logical theory of catastrophe as put forth by Thom and Zeeman has seenapplication in a wide range of fields including molecular biophysics, meso-scopic physics, and climate science. In this paper, we investigate a simplemodel of a non-autonomous system with a time dependent parameter p(τ)and its corresponding “dynamic” (time-dependent) saddle-node bifurcation.We show that the actual point of no return for a system undergoing tip-ping can be significantly delayed in comparison to the breaking time τ atwhich the corresponding autonomous system with a time-independent pa-rameter pa = p(τ) undergoes a bifurcation. A dimensionless parameter

1

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α = λp30V−2 is introduced, in which λ is the curvature of the autonomous

saddle-node bifurcation according to parameter p(τ), which has an initialvalue of p0 and a constant rate of change V . We find that the breaking timeτ is always less than the actual point of no return τ∗ after which the criticaltransition is irreversible; specifically, the relation τ∗ − τ ' 2.338(λV )−

13

is analytically obtained. For a system with a small λV , there exists a sig-nificant window of opportunity (τ , τ∗) during which rapid reversal of theenvironment can save the system from catastrophe.

1 IntroductionThe concepts of attractors and bifurcations from the mathematical theory of non-linear dynamics [1, 2] have been essential in understanding biological systems inmechanistic terms [3], with particular relevance in fields such as cellular biochem-istry [4], developmental biology [5], and cancer medicine [6]. The most com-monly encountered one-dimensional bifurcations are of the saddle-node (fold),transcritical, and pitchfork types [1, 2]. There is, however, an essential differencein structural stability between the latter two types and the first; to wit, recall thatthe canonical forms for the latter two types of bifurcations are given by

dx

dt= x(λ− x) transcritical bifurcation (1)

dx

dt= x(λ− x2) pitchfork bifurcation (2)

where x is the state variable and λ is the bifurcation parameter. It is easy toverify that these systems are structurally unstable [2]: For the transcritical case,the two fixed points of dx

dt= x(λ − x) + ε will never collide when ε > 0. For

the pitchfork case, by employing the triple-root condition for a cubic equation wesee that if ε 6= 0, dx

dt= x(λ − x2) − ε(x2 − 1) will no longer have a pitchfork

bifurcation, but only one of the saddle-node type [7]. In mathematical terms, bothphenomena are not structurally stable; in biological terms, they are not robust.On the other hand, it is well known that saddle-node bifurcations are structurallystable, and are thus a more robust nonlinear phenomenon [8]1. It is for preciselythis reason that saddle-node bifurcations, and the associated cusp catastrophe, playan important role in biology. Indeed, they have afforded fundamental insights and

1Note that robust pitchfork bifurcations can exist if defined in particular ways, such as the typedefined via the “hidden” second-order phase transition associated with any cusp catastrophe [9].

2

conceptual understanding of a wide range of biological phenomena, from voltage-dependent channel kinetics and forced macromolecular bond rupture [10], to E.coli phenotype switching [11] and ecological insect outbreak [12].

More recently, there has been intense interest in using a dynamical systems ap-proach to the characterization of a class of phenomenon known as “critical transi-tions” or “tipping points” [13], where complex systems experience a qualitativelylarge and abrupt change very rapidly (in comparison to normal dynamics)—forexample, a critical transition of a financial market might be a financial crisis; or,in an ecosystem, a critical transition might be rapid desertification [14, 15]. Thegeneric nature of this class of phenomena means that it has broad applications;indeed, the literature on “tipping point” phenomena spans a wide variety of fields,including cancer biology [16, 17], quick-onset disease theory [18], ecology [19],environmental and climate science [20, 15], and sociology [21].

Mathematically, systems with tipping points can be represented as dynami-cal systems with saddle-node bifurcations, where the tipping points are the cor-responding fold catastrophes. However, it is critical to understand that in real-world systems, the bifurcation parameter of a dynamical system can itself betime-dependent; thus, tipping points are fold catastrophes induced by parameterschanging in time. In other words, tipping points are induced by changes in exter-nal conditions, and systems with tipping points are described by non-autonomousdynamical systems. To be more mathematically precise, these systems can ingeneral be described by a non-autonomous dynamical system

d~x

dt= ~f(~x, p(t)) (3)

such that:

• If p(t) is a constant, the vector field ~f(~x, p) has (for some value p) an attrac-tor for p < p which loses stability for p > p.

• If p(t) depends on t, we have p(t) such that: (i) at t = t, p(t) = p, (ii)p(t) < p when t < t, and (iii) p(t) > p when t > t.

When the parameters change slowly enough, the system can be approximatedas its autonomous counterpart (that is, where p is the time-independent bifurcationparameter), or as a slow-fast system [14]. However, cases where such approxi-mations are not valid require treatments of the fully non-autonomous system, ofwhich there are fewer [22].

3

The purpose of the present paper is to present a non-autonomous treatmentof a simple system with a tipping point. We do this by introducing an explicitlytime-dependent mathematical model for systems with a generic tipping point. Inparticular, we introduce the concepts of a system’s “breaking time” and “point ofno return” alongside the well developed modern mathematical notion of “pullbackattractors”. Most notably, we discuss how a non-autonomous treatment of tippingpoint phenomena reveals possibilities for saving systems which autonomous treat-ments would deem as having already undergone an irreversible transition. To wit:consider the system described in Eq. 3: in the autonomous limit where the con-stant parameter p is varied at an infinitely slow speed, the system undergoes acritical transition precisely at the moment when p crosses p. However, we showthat a non-autonomous treatment reveals that if the parameter begins at p < pand increases at a constant rate V to lead to a disappearance of a stable steadystate at t = t (which we call the “breaking time”), the system can be rescued ifa rapid reversal of the environment is effected, even for t > t. However, thereis only a finite window of opportunity: a “point of no return” t∗ > t exists, afterwhich the system cannot be saved. This window of opportunity (t, t∗) thereforehas potential for great practical significance in managing tipping point phenom-ena. We obtain a result for the relationship between these two quantities based onthe analytic solution for the pullback attractor of our simple model (Eq. 20 below):t∗ − t ' 2.338(λV )−1/3, where λ is the curvature of the autonomous saddle-nodebifurcation according to parameter p.

The paper is structured as follows. In Sec. 2, we give a brief introduction tonon-autonomous systems and the notion of pullback attractors, followed by sev-eral simple examples to illustrate the dynamics of non-autonomous systems with:(i) a “moving fixed point”, (ii) a moving fixed point with a changing eigenvalue,and (iii) a more complex situation with a “hidden catastrophe”. This section isincluded purely for pedagogical purposes. Motivated readers are also referred toa more mathematical discussion [23, 24]. Sec. 3 introduces the main model forthe dynamic saddle-node bifurcation (Eq. 20). We explore its behavior using nu-merical computation. The notions of t and t∗ are defined, and a relation betweenthem is observed from numerical computations. We then present the analyticalsolution of the model in terms of the first Airy function, from which we obtain anexact relation between t and t∗. The paper concludes with Sec. 4 which discussesour model and the concepts of breaking time and the point of no return; also, werelate the concepts to the notion of “bifurcation-induced tipping” (B-tipping) inthe context of climate science.

4

2 Simple non-autonomous dynamics and pullbackattractors

2.1 Pullback attractors and tipping point phenomenaAn autonomous dynamical system described by the system of ordinary differentialequations

d

dt~x(t) = ~f

(~x(t), p

)~x, ~f(~x) ∈ Rn, t ∈ R (4)

has an n-dimensional vector field ~f(~x, p) that is independent of time. In the theoryof dynamical systems of this form, the concept of attractors is among the mostimportant—they characterize the long-time behavior of a system [1, 2, 7]. Inthe context of tipping point phenomena modeled by autonomous systems withsaddle node bifurcations, a system always tracks the stable equilibrium point asthe parameter is slowly increased, until the the fold catastrophe, at which point thesystem undergoes a “critical transition” (that is, it “tips”) into another attractor.

For non-autonomous dynamical systems, however, which are described by asystem of ordinary differential equations

d

dt~x(t) = ~f

(~x(t), p(t)

)= ~g(~x(t), t) ~x,~g(~x, t) ∈ Rn, t ∈ R (5)

the vector field ~g is itself an explicit function of time—in physical terms, thechanging vector field reflects the changing environment of the system. Because ofthis, a system starting in a particular initial state ~x∗ ∈ Rn where ~g(~x∗, t) = 0 whent = t0 will in general not remain at ~x∗ for all future times t = t1 > t0—contrastthis to equilibrium points in autonomous systems, where trajectories starting thereremain there for all future time.

Thus, the idea of attractors from autonomous theory is insufficient to describenon-autonomous systems (and therefore general tipping phenomena)—the con-cepts must be generalized—which is where the concept of a “pullback attractor”comes in [23]. Note the fact that the solutions to non-autonomous systems explic-itly depend not only on the elapsed time t−t0 but on both t and t0. The dependenceon two variables thus means there are two types of attractors for non-autonomoussystems—in fact, it means that attractors in non-autonomous systems involve atime-dependence, unlike attractors in autonomous systems [24]. Also, the mathe-matical theory of non-autonomous dynamics has established that attraction of the

5

“pullback” type is of a more fundamental character. Indeed, an autonomous sys-tem’s attractors are the same as their pullback attractors—they are only differentfor non-autonomous systems.

Qualitatively, a pullback attractor of a non-autonomous system can be thoughtof as the attractor for a large collection of trajectories which began infinitely longago; quantitatively, it corresponds with the trajectory of the system with an initialcondition (t0, ~x0) in the case when t0 → −∞. To be more mathematically precise:let ~ϕ(t | ~x0, t0) be the solution to Eq. 5 with an initial condition ~x(t0) = ~x0. Thenthe pullback attractor of the system is given by ~ϕpb(t):2

~ϕpb(t) ≡ limt0→−∞

~ϕ(t | ~x0, t0) (6)

Note that the pullback attractor is independent of x0—in essence, it is the trajec-tory a system will take after enough time has elapsed for the “memory” of thestarting state ~x0 to be lost. For systems which start sometime in the finite past,they quickly converge to ~ϕpb(t), as illustrated in Fig. 1 and Fig. 2.

What follows are several simple examples.

2.2 A moving fixed pointThe first example is of a non-autonomous dynamical system with a single fixedpoint whose location changes smoothly in time; that is, dx

dt= −λ(x− x(t)) where

x(t) is the location of the fixed point. The exact solution to this non-autonomousODE is

ϕ(t | x0, t0) = x0e−λ(t−t0) + λ

∫ t

t0

e−λ(t−τ)x(τ) dτ (7)

= (x0 − x(t0)) e−λ(t−t0) + x(t)−∫ t−t0

0

e−λs(dx(t− s)

dt

)ds.

(8)

We obtain the pullback attractor by taking t0 → −∞:

ϕpb(t) = λ

∫ t

−∞e−λ(t−τ)x(τ) dτ = λ

∫ ∞0

e−λsx(t− s) ds. (9)

2Actually, an even more mathematically precise statement is that the limit of ~ϕ(t|B, t0) ast0 → −∞ for any subset B ⊂ Rn, is itself a set. In this paper we only consider point attractors:then we just have the simple Eq. 6.

6

It is instructive to note that ϕpb(t) differs from the function one obtains if one takest→∞ in Eq. 7:

ϕf (t0) = limt→∞

λe−λt∫ t

t0

eλτ x(τ) dτ = limτ→∞

λ

∫ τ

0

e−λsx(τ + t0 − s) ds. (10)

ϕf (t0) is called the forward attractor [23]. If x(t) ≡ x0 is independent of t, thenboth Eqs. 9 and 10 are equal to the same constant x0. If x(t) has limits x±∞when t tends to ±∞, then ϕf (t0) = x+∞, but ϕpb(t) is a smooth function of t thatconnects x−∞ to x+∞ [24]. In general, however, it is not certain that the limit inEq. 10 always exists, although the pullback attractor itself always exists.

The plots in Fig. 1 also illustrate that ϕpb(t) is a more appropriate charac-terization of the “long-time limiting behavior” than ϕf (t0): namely, for a systemwith some initial condition, the latter tells us only about its behavior in the infinitefuture. In order to observe the long-term behavior of the system without goinginto the infinite future, one has to instead start the system in the infinite past.

If λ is very large, the relaxation toward x(t) occurs on a much faster timescalethan the dynamics of the moving fixed point x(t) itself. In this case, Eq. 9 becomesϕpb(t;λ) ' x(t).

Figs. 1A and 1B illustrate two examples of x(t): a linear case, x(t) = −vt,and a nonlinear case: x(t) = µ(−t)1/2, where t ≤ 0. The corresponding solutionsare

ϕlin(t | x0, t0) = (x0 + vt)e−λ(t−t0) + x(t) +v

λ

[1− e−λ(t−t0)

], (11)

ϕnonl(t | x0, t0) =(x0 − µ

√−t0)e−λ(t−t0) + x(t) + µe−λt

∫ |t0||t|

e−λτ d√τ .

(12)

In both cases, x(t) ' x(t) + C for large (t − t0), where C is a positive constant.For the linear case, C = vλ−1; for the nonlinear case, C ' (µ/2)

√π/λ, the last

term in Eq. 12. We conclude that for a system with a moving fixed point, thelong-term dynamics follows the fixed point x(t) plus a constant, represented bythe last term in Eq. 8, which is the displacement of x(t) within the characteristictime λ−1.

To briefly summarize, there are three key features of a pullback attractor: italways exists, it is unique, and it is a close counterpart of the corresponding au-tonomous ODE attractor under a small perturbation. It is also important to keepin mind, that a pullback attractor needs not be a single curve of t (called a point

7

φpb(t) t0=-4

t0=-2 t0=1

-4 -3 -2 -1 0 1 2-1

0

1

2

3

t

x

Linear x˜(t): λ=1, v=1, x0=1

φpb(t) t0=-10

t0=-7 t0=-5

-10 -8 -6 -4 -2 00.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

t

x

Nonlinear x˜(t): λ=1, μ=1, x0=1

φpb(t) t0=-6

t0=-4.5 t0=-3

-6 -5 -4 -3 -2 -1 00.5

1.0

1.5

2.0

2.5

3.0

t

x

φpb(t) and trajectories: x0=1

Figure 1: Trajectories and pullback attractors of various non-autonomous ODEsdescribed in Secs. 2.1 and 2.2. (A) Moving fixed point with x(t) = −vt accordingto Eqs. (8) and (9); v = 1, λ = 1, x0 = 1 and several initial times t0. (B) Movingfixed point with x(t) = µ(−t)1/2, t ≤ 0; µ = 1, λ = 1, x0 = 1 and several initialtimes t0. (C) Moving fixed point with changing eigenvalue according to (13), withx(t) = 1

2r′(t) = v(−t)1/2, t ≤ 0; v = 1, x0 = 1, and several initial times t0.

attractor). It can be an entire interval. One such example is dxdt

= x(t − x), thepullback attractor of which is the set of t < x < 0 for t ≤ 0 and 0 < x < t fort ≥ 0.

2.3 Moving fixed point that vanishes with vanishing eigenvalueNow let us consider a situation in which the eigenvalue at the moving fixed pointis also changing with time: dx

dt= −r′(t)(x− x(t)) with r(0) = 0. The solution to

this is

x(t) = (x(0)− x(0)) e−r(t) + x(t)− e−r(t)∫ t

0

er(τ) dx(τ). (13)

The first term disappears if we let x(0) = x(0). Figure 1C shows an example inwhich we take x(t) = 1

2r′(t) = v(−t)1/2. In this case,

x(t) = (vt)e−r(t) + 2(vt)2e−r(t)∫ t/t

0

er(st)(1− s) ds. (14)

8

and r(t) = 4v3(−t)3/2 becomes imaginary when t > t. The fixed point vanishes at

t = 0 at the same time when the eigenvalue becomes zero. This is the breakingtime in the dynamic saddle-node bifurcation (see below).

2.4 System with a “hidden catastrophe”What happens if the vector field g(x, t) of a non-autonomous system dx

dt= g(x, t)

has multiple roots for a certain range of t? The pullback attractor in this caseresembles the bifurcation diagram of an autonomous system dx

dt= g(x, λ) with a

fold catastrophe: namely, it connects the root of g(x, t) when t → −∞ with theroot of g(x, t) when t → +∞. Fig. 2 shows the pullback attractor of an examplesystem

dx

dt= x(1− x2) + t, (15)

along with several sample trajectories for several initial conditions. The curvex(1− x2) + t = 0 is also plotted to illustrate how the pullback attractor tracks theautonomous bifurcation diagram for t→ ±∞.

2.5 Saddle-node ghosts and bottleneckingAfter the “breaking time” at which a stable fixed point (a node) and an unstablefixed point (a saddle) collide, a dynamical system exhibits “bottlenecking” causedby “saddle-node ghosts” in an autonomous ODE. In fact, it is a general feature ofsystems with saddle-node bifurcations: for example, in Figs. 2A and 2B, whichdepict Eq. 15, it can be observed near x = −1 when t > 0. This behavior is dueto the fact that systems near a saddle-node bifurcation canonically behave like

dx

dt= −x2 − a2. (16)

For systems of this type, the trajectory of the dynamics is slowed down or “bottle-necked” by a “saddle node ghost” [1, 25]. This is represented in Eq. 16 by a smallbut positive value of a2.

Since a2 ≥ 0, Eq. 16 has no fixed points — all trajectories starting with x < 0will pass the x = 0 and continue to go to −∞. The effect of the saddle nodeghost is to make the amount of time it takes for them to pass the x = 0 criticallydependent upon the value of a. If we set x(0) = x0, the amount time tb it takes

9

φpb(t)x(1-x2)+t=0

-4 -2 0 2 4-3

-2

-1

0

1

2

3

t

x

φpb(t) and trajectories: x0=-3

φpb(t)x(1-x2)+t=0

-4 -2 0 2 4-3

-2

-1

0

1

2

3

t

x

φpb(t) and trajectories: x0=3

Figure 2: The pullback attractor ϕpb(t) of dxdt

= x(1 − x2) + t as the bold solidline, along with various trajectories and the bifurcation diagram of the underlyingautonomous system (dashed line). Note how ϕpb(t) does not jump at t = 0, butrather changes smoothly to connect the two roots. (A) Trajectories with x0 = −3for several different t0. The bifurcation diagram of the underlying autonomoussystem is t = −x(1− x2). (B) Trajectories with x0 = 3 for several different t0.

10

for the system to go from x0 to x(t) is found by∫ x(t)

x0

1

−s2 − a2ds =

∫ t

0

dt (17)

=⇒ t =1

a

[arctan

(x0a

)− arctan

(x(t)

a

)], (18)

and taking x(t)→ −∞, we get

tb ≡ limx(t)→−∞

t =1

a

[arctan

(x0a

)+π

2

]. (19)

Near a saddle-node bifurcation, where a is small, tb can get to be quite significant:what this means is that even if you start a trajectory of a system that has no fixedpoints at t = 0, it takes a nonzero amount of time to pass a bottleneck, and thentakes nearly no time to blow up. We shall later identify a similar blow-up time inour non-autonomous system.

3 Dynamic saddle-node bifurcationTipping point phenomena are particularly well-represented by one-dimensionaldynamical systems with saddle-node bifurcations due to their structural stabilityand robustness. In the context of non-autonomous dynamics, the problems dealwith a moving fixed point which can also change its character (e.g., eigenvalue), aswell as disappear altogether. In Sec. 2, we established simple descriptions of threedifferent mathematical phenomena: (i) relaxation of trajectories towards a movingstable fixed point; (ii) relaxation of trajectories towards a moving stable fixed pointwith vanishing eigenvalue; (iii) the effect of a saddle-node ghost on trajectories(Fig. 2, where trajectories do not immediately jump to the other branch). We shallnow focus on a simple non-autonomous equation that exhibits these three types ofbehavior, as well as the time-dependent equivalent of a saddle-node bifurcation—we shall say that such a system exhibits a dynamic saddle-node bifurcation.

3.1 Model for a dynamic saddle-node bifurcationWe now introduce a non-autonomous dynamical system with a dynamic saddle-node bifurcation. Consider the differential equation for a generic saddle-node

11

φpb(t)

x 2=α(1-t)

0.0 0.5 1.0 1.5 2.0-4

-2

0

2

4

6

t

x

φpb(t) and sample trajectories; α=4

α=4 α=16

α=36

0.0 0.5 1.0 1.5 2.0 2.5-6

-4

-2

0

2

4

6

t

x

φpb(t;α) for various α

Figure 3: (A) Plot of ϕpb(t) and trajectories with α = 4, x0 = 10, and x0 = 4for several different t0. The corresponding “autonomous bifurcation diagram” isalso plotted in think dashed line: x2 = α(1− t). (B) Plot of the pullback attractorϕpb(t) for different values of α. Note how t∗ increases with α—this correspondswith larger values of p0 or smaller values of V in the original equation (20). Thindashed lines are again the corresponding “autonomous bifurcation diagrams”.

12

bifurcation dynamical system in the case when the bifurcation parameter p is anexplicit linear function of time: p(τ) = p0−V τ where p0, V > 0, so that we have

dX

dτ= −λX2 + (p0 − V τ), (20)

where V is the “speed” with which the parameter is changing. One can introducetransformed variables x and t alongside the corresponding parameter α:

x =λp0X

V, t =

V τ

p0, and α = λp30V

−2. (21)

Then we arrive at a “non-dimensionalized” version of Eq. 20

dx(t)

dt= −x2 + α(1− t). (22)

Denoting the right-hand-side of Eq. 22 as f(x, t) ≡ −x2+α(1−t), it is easy to seethat for any t < 1, the vector field f(x, t) has two roots—at x+(t) =

√α(1− t)

and x−(t) = −√α(1− t). These two roots approach each other as t increases,

and annihilate at t = 1, and for all t > 1, f(x, t) is always negative: all trajectorieseventually go to −∞.

Fig. 3 shows several solutions to Eq. 22 as well as the bifurcation diagramfor the corresponding autonomous system. Different initial conditions convergeto a single “master curve” — this phenomenon is called forward attraction [26].Here we pause to note that the trajectories of these dynamics describes systemswith tipping points—they follow a stable equilibrum until a sudden and abrupttransition into a state very different from the original because of some change inits surrounding environment; this is caused by the sudden disappearance of thestable fixed point in f(x, t) at t = t. We shall call t the breaking time.

However, it is important to note that while its environment has already fun-damentally changed, a dynamical system does not have an immediate drastic re-sponse at exactly the time t, when f(x, t) goes from having two fixed points (fort < t) to none (for t > t). Rather, there is a delay as the system traverses the bot-tleneck caused by the saddle-node ghost present in f(x, t), until it finally reaches−∞ at a time t∗ > t (see Fig. 3A): we shall call this time the “point of noreturn”—this choice of naming will become clearer shortly. In the case of aninfinitely-slowly changing parameter, the autonomous case from Eq. 19 is recov-ered: t∗ − t = tb.

Fig. 3 clearly shows that the time interval between the breaking time and pointof no return (i.e., (t, t∗) ) for a system can be quantified in terms of the system’s

13

pullback attractor. Indeed, since tipping point problems are concerned with sys-tems which have in the past followed a stable equilibrium, their dynamics aresuitably captured by the pullback attractor of the system.

An analytical solution can be found for Eq. 22, as we show in Sec. 3.3. Whatmakes a non-autonomous treatment of such systems useful compared to the stan-dard, autonomous models, is that it allows an assessment of the point of no returnfor a system with parameters that change on a similar time scale to the phase spacedynamics. On the other hand, an autonomous treatment can provide only provideuseful values of t∗ for systems where the parameter evolves on a timescale slowenough to be considered decoupled from the dynamics.

Note that trajectories that followed the stable equilibrium in the past behaveexactly as the pullback attractor, and experience the same difference between tand t∗—it is precisely this fact that makes the pullback attractor so vital in thestudy of tipping point phenomena, since systems which have not yet tipped areassumed to have been following some equilibrium in the past. Fig. 3B illustratesthe dependence of t∗ as determined by ϕpb(t;α) on α; or, in the untransformedvariables, the dependence on the velocity V and starting value p0 of the parameter.

3.2 Delayed catastrophe and the point of no returnFig. 3B illustrates the relationship between the breaking time t and the point of noreturn t∗; namely, that it is always true that t∗ > t = (p0/V ). Thus we see that thecritical transition is delayed due to the system being bottlenecked, and that thisbottlenecking depends on α.

Fig. 4 shows a log-log plot for the system with t = 1 and several values ofα. Note that the speed with which the parameter is changing (V ), and the initial“distance” from the bifurcation (p0), contribute to α differently: α ∼ p30V

−2.Furthermore, note the difference in behavior for the regimes α > 1 and α < 1.

3.3 Analytical solution in terms of Airy functionsEq. 22,

dx(z)

dz= −x2 + α(1− z), (23)

is a Riccati equation, dxdt

= a0(t) + a1(t)x+ a2(t)x2, with a2(t) = −1, a1(t) = 0,

and a0 = α(1 − t). According to the standard method of attack, this 1st ordernonlinear ODE can be transformed into a 2nd order linear ODE via x = 1

u

(dudt

)so

14

●

●

●●

●

●

●

●●●●●●●●●●●●●●●●●●● ●●

-5 0 5

0

1

2

3

4

Ln[α]

Ln[t* ]

Log-Log Plot of α and t*

Figure 4: The time of the point of no return, t∗, as a function of α, in a log-logplot, according to Eq. 22. The points are obtained from numerical solutions tothe differential equation, e.g., from Fig. 3B. There are two distinct regimes withα < 1 and α > 1. The thin solid curve t∗ = 1 + 2.338α−

13 is obtained from

the analytical solution of the pullback attractor. For a very slowly changing p(t),α� 1 and t∗ = t = 1. This is the autonomous system with a tipping point.

that we haved2u

dt2= α(1− t)u. (24)

Eq. 24 is related to the Airy equation, and can be solved in terms of Airy functions.We are interested in its solution with initial values u(t0) = 1 and du(t0)

dt= x0.

The general solution to Eq. 24 is

u(t) =[C1Ai(z) + C2Bi(z)

]z= 3√α(1−t)

, (25)

where C1,2 are two constants, and

Ai(z) =1

π

∫ ∞0

cos

(ξ3

3+ zξ

)dξ, (26a)

Bi(z) =1

π

∫ ∞0

[exp

(−ξ

3

3+ zξ

)+ sin

(ξ3

3+ zξ

)]dξ. (26b)

15

Then,du(t)

dt= − 3√α[C1

dAi(z)dz

+ C2dBi(z)dz

]z= 3√α(1−t)

, (27)

and the solution to Eq. 23 is

x(t) = − 3√α

C1dAi(z)dz

+ C2dBi(z)dz

C1Ai(z) + C2Bi(z)

z= 3√α(1−t)

, (28)

with the constants

C1(t0) =

[ dBi(z)dz

+ x03√αBi(z)

Ai(z)dBi(z)dz− Bi(z)dAi(z)

dz

]z= 3√α(1−t0)

, (29a)

C2(t0) =

[− x0

3√αAi(z)− dAi(z)dz

Ai(z)dBi(z)dz− Bi(z)dAi(z)

dz

]z= 3√α(1−t0)

. (29b)

The pullback attractor is obtained when t0 → −∞. Since Ai(z) = 0 and Bi(z)→+∞ as z → +∞, we have

limt0→−∞

(C2(t0)

C1(t0)

)= lim

z→+∞

(− x0

3√αAi(z)− dAi(z)dz

dBi(z)dz

+ x03√αBi(z)

)= 0. (30)

ϕpb(t) = −[

3√α

Ai(z)dAi(z)dz

]z= 3√α(1−t)

. (31)

The denominator Ai(z) vanishes when z ≈ −2.338, corresponding to the asymp-tote where ϕpb → −∞. Hence the time of no return is given by t∗ ≈ 1+2.338α−

13 :

in other words, once the system breaks at t = 1, the amount of time it takes for thesystem to reach the point of no return and fully tip (i.e., t∗ − t) scales accordingto α (see Fig. 5).

If V is very small, then α is very large, and t∗ ' 1, and the correspondingautonomous system is recovered.

4 DiscussionThe importance of the relation t∗ > t is that it reveals that for a changing environ-ment which has just undergone a fundamental change, there remains an interval

16

-Ai ' (-z)

Ai(-z)

-3 -2 -1 0 1 2 3

-10

-5

0

5

10

t

x

Behavior of Ai and Ai'

Figure 5: The left dashed curve is − 1Ai(−z)

dAi(−z)dz

as function of z. The solidcurve crosses the horizontal axis at z = 1.02 since Ai(−1.02) is a local maxi-mum of the Airy function. It has an asymptote (vertical dashed line) at z = 2.34since Ai(−2.34) = 0. The dashed parabola represents ±

√−z. Note that the

solid line asymptotically approaches to the vertical dashed line since Ai(x) ∼12π−

12x−

14 e−

23x3/2 as x → ∞ [27]. Thus, the solid line goes as

√−z − (4z)−1

when z → −∞.

of time (t, t∗) during which the system can be saved from reaching the asymp-totically different state if the changes to the environment are rapidly reversed. Inother words, as long as the point of no return has not yet been reached, the systemis still in theory salvageable. This has potential significance in the management oftipping point phenomena, since it provides a way to quantify and estimate the win-dow of time after a significant environment change during which it is still possibleto prevent the system from fully tipping.

A simple example in biology is given by the relationship between many animalpopulation densities and the per capita growth rate of the population—specifically,the strong Allee effect describes how the per capita growth rate of some popula-tions is maximal at some intermediate density and negative at low densities. Thus,for a laboratory experiment involving, for example, yeast, one could map the bi-

17

furcation diagram of per capita growth rate (the state variable) vs. a changingpopulation density (the parameter) by performing regular dilutions on the popu-lations (as done experimentally by Dai et. al [19]). Then as the dilution factor isincreased, the system passes its breaking point at the exact time when the popula-tion density decreases beyond the critical point where the instantaneous per capitagrowth rate is no longer sufficient to maintain the population, and the system be-gins to crash.

However, it is still salvageable, as the point of no return (the time at which theentire population has died off) has not yet been reached. Indeed, if the environ-ment change is rapidly reversed by adding a sufficient number of additional yeastcells into the system, then the population can be saved, and the per capita growthrate will recover. It is easy to imagine using similar models to track and estimatethis time frame of salvageability for phenomena such as collapsing fish stocks dueto overfishing.

With regard to tipping points in climate science, it has been established thattipping points in general can be further classified into several categories. In par-ticular, in [28], Ashwin et. al suggested that critical transitions associated withtipping points can be classified into the following categories:

• “bifurcation-induced tipping” (B-tipping), in which the critical transition isdirectly associated with slow passage through a bifurcation [29],

• “noise-induced tipping” (N-tipping), in which stochastic effects result in thesystem leaving basins of attraction,

• and “rate-dependent tipping” (R-tipping), in which the system changes tooslowly to track a changing attractor.

In the context of climate science terminology, then, our model is an simpleexample of B-tipping where the parameter change is not necessarily slow, and isone which affords an exact solution to the amount of time it takes for a system tofully tip from a previously tracked equilibrium to a new, asymptotically locatedstate. A similar class of systems involving “parameter shifts” in non-autonomoussystems is explored in a more mathematical formal way in [22]. What follows isan example application of our model to a simple climate model.

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4.1 Breaking time and the point of no return in a simple cli-mate model with B-Tipping

We end with a brief example of the application of our model to the Fraedrichenergy-balance model [30] from climate science (see [28] for an excellent discus-sion of examples of other types of tipping which can occur in this system, or [22]for a description of parameter shifts and R-tipping in this system).

The model introduced by Fraedrich describes the evolution of the global meansurface temperature of an ocean-covered spherical planet subjected to radiativeheating; it also takes into account ice-albedo and greenhouse feedback. Thismodel, where T , the mean temperature, is the state variable, is given by

cdT

dt= R ↓ −R ↑, (32)

where R ↓ and R ↑ represent the incoming and outgoing radiation respectivelyand are given in terms of physical parameters:

R ↓ = 1

4µI0(1− αp) R ↑= εsaσT

4, (33)

where I0 is the solar constant, c > 0 is the thermal inertia of the ocean, αp isthe planetary albedo, εsa is the effective emissivity, and µ is an external param-eter which allows for variations in the solar constant or for long-term variationsin planetary orbit. Now, consider a planet whose albedo-temperature relation’sslope is negligible, and whose albedo increases (due to some unrelated cause) at aconstant rate γ in time. Then the planetary albedo is given by

αp(t) = γt (34)

Then, Eq. 32 can be written as

dT

dt= −aT 4 + b− bγt, (35)

where

a ≡ εsaσ

cb ≡ I0µ

4c. (36)

Note that a > 0. Then by introducing a transformed variable W ≡ T 2, we have

dW

dt= 2T

dT

dt= 2(−aW 2 + b− bγt)

√W. (37)

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Presuming that we are bounded away from domains that might cause existenceproblems, Eq. 37 is bifurcation-equivalent to

dW

dt= −aW 2 + b− bγt, (38)

which is in the form of our model, Eq. 20, from which the breaking time and thepoint of no return for the system can be calculated. Physically, the breaking timecorresponds to the point at which the planet’s ocean’s environment can no longersupport a stable, “interglacial” state, and the point of no return corresponds to thepoint at which the ocean reaches an irreversible “deep freeze”.

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