Rethinking the Definition of Rate-Induced Tipping
RRethinking the Definition of Rate-Induced Tipping
Rethinking the Definition of Rate-Induced Tipping
Alanna Hoyer-Leitzel a) and Alice N Nadeau b) Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075 USA Department of Mathematics, Cornell University, Ithaca, NY 14853, USA (Dated: 4 February 2021)
The current definition of rate-induced tipping is tied to the idea of a pullback attractor limiting in forward and backwardtime to a stable quasi-static equilibrium. Here we propose a new definition that encompasses the standard definitionin the literature for certain scalar systems and includes previously excluded N -dimensional systems that exhibit rate-dependent critical transitions. Tipping points and critical transitions, characterized byqualitative changes in a system due to small changes in itsparameters , have been topics of interest to both the sci-entific community and the public, especially in the last twodecades. Reasons for this interest perhaps stems from thewide variety of chemical, biological, physical, and societalsystems which exhibit such behavior. Examples includeorgan failure, desertification, and runaway ice cover .In mathematical models, tipping can be caused by differ-ent mechanisms, including the rate at which parametersare changing , called rate-induced tipping . Here we dis-cuss several examples of systems exhibiting rate-inducedcritical transitions but which fall outside of the standardset of assumptions that encompass the theoretical devel-opments of the rate-induced tipping field. As such, onecannot technically call the transitions we observe in thesesystems “rate-induced tipping.” In light of these exam-ples, we propose a new definition of rate-induced tippingas the lost of forward asymptotic stability of a nonau-tonomous attractor. Encouragingly, this new definitionis equivalent to the current definition for asymptoticallyconstant scalar systems although it is not known if it gen-eralizes to asymptotically constant N -dimensional systemsor N -dimensional systems with other types of parameterchange. I. INTRODUCTION
Tipping points in the scientific literature are characterizedby a sudden, qualitative shift in the behavior or state of thesystem due to a relatively small change in inputs . In naturalsystems this behavior can have a variety of underlying causes,such as changing external conditions or self-excitation. Forexample, nutrient run-off from agricultural practices can causedownstream lakes to “tip” from a healthy state to a eutrophicstate resulting in excessive algae growth and death of animallife. In mathematical models, tipping may be caused by a bi-furcation due to a change in parameter values, noise in the sys-tem, or the rate at which parameters are changing . This lastcause, called rate-induced tipping (sometimes rate-dependent a) Electronic mail: [email protected] b) Electronic mail: [email protected] tipping , rate tipping , or simply R-tipping ), can tip a systemto a different state even when the parameter change does notpass through a bifurcation point of the system .The mathematical study of rate-induced tipping has almostexclusively been concerned with deterministic, continuoustime systems of the form˙ x = f ( x , λ ( rt )) , x ∈ R N , λ ( rt ) ∈ R M (1)especially where N = M =
1. Although recent researchhas been conducted on understanding rate-induced tipping innoisy systems or time-dependent maps , here we will fo-cus only on deterministic, continuous time systems. Findingrate-induced tipping in system (1) amounts to varying the pa-rameter r , referred to as the rate parameter , and assessing thesystem’s sensitivity to the change of the parameter λ ( rt ) intime.It is natural to compare the time-dependent system (1) withthe corresponding family of autonomous systems ˙ x = f ( x , λ ) where λ is constant. Although recent studies have consideredtipping in systems with periodic orbits , it is standard toassume that equilibria of the corresponding autonomous sys-tem are hyperbolic. In the hyperbolic context, one may con-sider the curve of equilibria x ∗ ( λ ( rs )) generated by the sta-tionary system ˙ x = f ( x , λ ( rs )) as s is varied, called a quasi-static equilibrium or QSE .The definition of rate-induced tipping has evolved over timeas mathematicians try to describe the phenomenon rigorously.Rate tipping was originally determined by how close a trajec-tory remained to a stable QSE: trajectories that left a predeter-mined radius around a stable QSE had tipped . This seemedlike a natural definition as several early examples were foundwhere the boundary of the basin of attraction stayed a con-stant distance away from the stable QSE. However, in othersystems, solutions may drift away from the stable QSE con-tinuously in the rate parameter r as r is varied without havinga critical transition, leaving the definition lacking. In SectionIII A 1, we will discuss a system where solutions exhibit thisdrifting behavior relative to the stable QSE.The current working definition of rate tipping also relieson behavior of trajectories of the system in relation to thesystem’s QSEs . In systems where the change in the pa-rameter λ ( rt ) is asymptotically constant—limiting to λ ± when t → ± ∞ —the definition of rate-induced tipping relies on thelimiting behavior of (local) pullback attractors of the sys-tem. A curve γ P ( t ) is a (local) pullback attractor if trajecto-ries x ( t ; x , t ) get closer to γ P ( t ) as the initial condition x is a r X i v : . [ m a t h . D S ] F e b ethinking the Definition of Rate-Induced Tipping 2started further back in time, i.e. for x in some ball around γ P ( t ) , lim s →− ∞ | x ( t ; x , s ) − γ P ( t ) | = t . (2)Ashwin, Perryman, and Wieczorek show that for each sta-ble hyperbolic equilibrium of the autonomous system ˙ x = f ( x , λ − ) , there is a corresponding solution x P ( t ) to (1) whichis local pullback attracting and converges to the stable QSElimit x ∗ ( λ − ) as t → − ∞ . If the local pullback attractor con-verges to the stable QSE limit x ∗ ( λ + ) as t → ∞ , the solutionis said to end-point track the QSE . Rate-induced tipping issaid to occur if the local pullback attractor x P ( t ) does not end-point track the stable QSE, i.e. does not limit to the forwardlimit of the corresponding equilibrium x ∗ ( λ + ) as t → ∞ as therate parameter r is varied.However, this definition of rate tipping as a pullback attrac-tor not end-point tracking a stable QSE may be too restric-tive, as it does not generalize to systems with locally boundedparameter change as illustrated below in Section III. Instead,it may be more appropriate to define rate-induced tipping interms of the loss of forward attraction of the (local) pull-back attractor because it is equivalent to end-point trackingfor scalar equations with asymptotically constant parameterchange and encompasses rate-induced tipping-like behaviorin systems where the current definition is not applicable.In nonautonomous systems, there are two types of attrac-tors: pullback attractors and forward attractors. In contrastwith pullback attractors, a curve γ F ( s ) is a (local) forward at-tractor if trajectories x ( t ; x , t ) get closer to γ F ( t ) as they flowforward in time, i.e. for x in some ball around γ F ( s ) ,lim s → ∞ | x ( s ; x , t ) − γ F ( s ) | = t . (3)In autonomous systems, the pullback and forward definitionsof attraction are equivalent; however, in nonautonomous sys-tems pullback attractors need not be forward attracting andforward attractors need not be pullback attracting .We propose a new definition of rate tipping as follows Definition 1.
System (1) undergoes rate-induced tipping whena forward attracting (local) pullback attractor γ ( t ) of the sys-tem goes through a nonautonomous bifurcation and loses itsforward stability at a critical value of the rate parameterr = r ∗ . Note that for the bifurcation at r = r ∗ in the proposed defini-tion, the pullback attractor may remain in the system and onlylose its forward stability or the pullback attractor may be an-nihilated in a collision with another globally defined solutionas in the examples in Sections III A 2 and III B.The remainder of this note is laid out as follows. In Sec-tion II we discuss asymptotically constant systems. In Sec-tion II A we briefly summarize a recent paper showing that inscalar systems the definition for rate-induced tipping proposedin Definition 1 above is equivalent to the current definition ofrate-induced tipping used in the literature . In Section II B,we prove that the conditions that guarantee that the asymptot-ically constant N -dimensional system end-point tracks for all r > r >
0. Although this doesn’t show that the defi-nitions are equivalent for N -dimensional systems with asymp-totically constant parameter change, it lends credence to thepossibility that they are equivalent. In Section III we dis-cuss systems with locally bounded parameter change. In Sec-tions III A 2, we discuss a system with a pullback attractor thatdrifts away from a stable QSE as the rate parameter is variedbut does not exhibit a critical transition. In Sections III A 2and III B, we provide examples of systems which undergorate-dependent critical transitions due to a pullback attractorlosing its forward stability as the rate parameter is varies butwhich would not be called “rate-induced tipping” under thecurrent definition. In Section III C, we prove a more generalstatement encompassing the examples from Sections III A 2and III B. II. EXAMPLES WHEN λ ( rt ) IS ASYMPTOTICALLYCONSTANTA. Scalar Equations
Among the results of a recent paper by Kuehn and Longo isa theorem showing that for scalar systems with asymptoticallyconstant parameter change, and under other standard assump-tions used in the tipping literature, a pullback attractor doesnot endpoint track its associated stable QSE when it loses itsforward asymptotic stability at the rate parameter is varied .The authors show that this occurs when the forward attractingpullback attractor collides with a pullback repeller at a criticalvalue of r = r ∗ defining the tipping point .More precisely, the study assumes that f in (1) has twicedifferentiable partial derivatives that are continuous and x and λ are both scalars , i.e. f ∈ C ( R × R , R ) . Additional as-sumptions include that the autonomous problem ˙ x = f ( x , λ ) has a stable hyperbolic equilibrium which depends continu-ously on λ for all λ ∈ [ λ − , λ + ] , that all other equilibria ofthe autonomous problem are hyperbolic for all λ ∈ [ λ − , λ + ] except for at most finitely many points λ ∈ ( λ − , λ + ) , andthat λ ( rt ) is strictly increasing and contained in the interval [ λ − , λ + ] . Under these conditions they show that denoting r ∗ : = sup { r > | x ρ − ( · ) is uniformlyasymptotically stable for all 0 < ρ ≤ r } , where x ρ − ( · ) is a pullback attractor limiting to a stable QSE inbackward time, then x r − end-point tracks the respective curveof quasi-static equilibria for 0 < r < r ∗ and r ∗ < ∞ if andonly if x r ∗ − is globally defined but not uniformly asymptoti-cally stable and it does not end-point track the QSE (Kuehnand Longo, Theorem 3.6 ). At this critical value of the rateparameter r , the forward attracting pullback attractor in thesystem collides with a forward repelling pullback repeller .This result by Kuehn and Longo demonstrates that Defini-tion 1 is equivalent to the current definition of rate-inducedtipping for scalar systems under these assumptions.ethinking the Definition of Rate-Induced Tipping 3 B. N -Dimensional Systems It is not yet known if the results showing the equivalenceof a nonautonomous bifurcation and end-point tracking forscalar equations generalize to N -dimensions. However, belowwe show that the same conditions that guarantee end-pointtracking in N -dimensional systems when λ ( rt ) is asymptoti-cally constant also guarantee that the pullback attractor is for-ward attracting.Kiers and Jones give a condition to guarantee that thepullback attractor of a system in N -dimensions with asymp-totically constant parameter change end-point tracks the sta-ble QSE. Here we show that the assumptions establishing thisresult are strong enough to also guarantee that the pullbackattractor is a forward attractor for all positive values of therate parameter. This shows that the same conditions that guar-antee there is no rate-induced tipping are the same conditionsthat guarantee the pullback attractor is also a forward attractor.In order to prove this result, we need to recall two definitionsfrom earlier studies. Definition 2 (Stable path ) . Given an asymptotically con-stant parameter shift Λ ( s ) satisfying λ − < λ ( s ) < λ + , lim s →± ∞ = λ ± , and lim s →± ∞ d λ ( s ) / ds = , if for all fixeds ∈ R , X ( s ) is an attracting equilibrium of the autonomoussystem ˙ x = f ( x , λ ( s )) , then we say that ( X ( s ) , λ ( s )) is a stable path of the augmentedautonomous system ˙ x = f ( x , λ ( s )) , ˙ s = r . (4) Furthermore we defineX ± = lim s →± ∞ X ( s ) . Definition 3 (Forward inflowing stable ) . The stable path ( X ( s ) , λ ( s )) from X − to X + is forward inflowing stable (FIS)if for each s ∈ R there is a compact set K ( s ) such that1. X ( s ) ∈ IntK ( s ) for all s ∈ R ;2. if s < s then K ( s ) ⊂ K ( s ) ;3. if x ∈ ∂ K ( x ) then there exists t > so that x ( t : x , s ) ∈ IntK ( s ) for all t ∈ ( s , s + t )
4. X ± ∈ IntK ± where K − = (cid:84) s ∈ R K ( s ) and K + = (cid:83) s ∈ R K ( s ) ; and5. K + ⊂ B ( X + , λ + ) is compact. Proposition 1.
If the stable path ( X ( s ) , λ ( s )) from X − to X + is forward inflowing stable, then the pullback attractor to X − is end-point tracking and forward attracting for all r > . Proof. The result regarding end-point tracking is proved byKiers and Jones (see Proposition 1 in that study). The fol-lowing proof showing that the pullback attractor is forwardattracting uses much of the machinery developed in that proofand other results in the same paper.Fix r > ( X ( s ) , λ ( s )) from X − to X + is forward inflowing stable. Then there exists sets K ( s ) satisfying Definition 3. Let K = (cid:83) s ∈ R K ( s ) × { s } and notethat K is forward invariant under the flow of the autonomousaugmented system (4) as shown by Kiers and Jones .Under these assumptions there exists a pullback attractor γ r ( t ) limiting to X − as t → − ∞ (see Theorem 2.2 ). By thesame arguments in the proof of Proposition 1 , γ r ( t ) is in K ( rt ) and K + for all t ∈ R and converges to X + as t → ∞ .Given x ∈ ∂ K ( rt ) , let x ( t ; x , rt ) be the solution to (1)through x at time rt . By the third property of forward inflow-ing stable, we know that there is a s > x ( t ; x , rt ) ∈ Int K ( rt ) for all t ∈ ( rt , rt + s ) . Because K is forward in-variant, this implies that x ( t ; x , rt ) ∈ Int K ( t ) and, thus, K + for all t > rt . By Lemma 5 this implies that x ( t ; x , rt ) converges to X + as t → ∞ .Then we see thatlim t → ∞ (cid:107) x ( t ; x , rt ) − γ r ( t ) (cid:107)≤ lim t → ∞ ( (cid:107) x ( t ; x , rt ) − X + (cid:107) + (cid:107) X + − γ r ( t ) (cid:107) ) = . Since x and t were arbitrary, we can conclude that γ r ( t ) isforward attracting.Note that this result does not rule out that there could bean end-point tracking pullback attractor which is not forwardattracting nor does it rule out that there could be a forwardattracting pullback attractor which is not end-point tracking.More work is needed to determine if the ideas of not end-point tracking and loss of forward stability are equivalent in N − dimensional systems with asymptotically constant param-eter change. III. EXAMPLES WHEN λ ( rt ) IS LOCALLY BOUNDED
The theoretical framework for rate tipping is underdevel-oped for locally bounded parameter change. Below we givesome illuminating examples where the local pullback attractordoes not end-point track the stable QSE of the system. In thefirst case we show that there is never a rate-dependent criti-cal transition and in the second case we show there is a rate-dependent critical transition. These examples are not isolatedbehavior for systems with locally bounded parameter changes,showing that rate-dependent transitions in these type of sys-tems need not rely on the behavior of the pullback attractorin relation to the QSEs. This independence of rate-dependentcritical transitions from end-point tracking is the main reasonwe are advocating for removing the idea of the QSE from thedefinition of rate-induced tipping.ethinking the Definition of Rate-Induced Tipping 4 ���� �� - - x QSE x [ t ] x [ t ] attractor x [ t ] x [ t ] A t x ���� �� - - x QSE x [ t ] x [ t ] attractor x [ t ] x [ t ] B t x ���� �� - - x QSE x [ t ] x [ t ] attractor x [ t ] x [ t ] C t x FIG. 1. Time Series for ˙ x = − ( x − rt ) showing the system’s forward attracting pullback attractor (solid yellow), stable QSE (dashed yellow),and selected solutions (blue curves; dotted, solid, dashed, dot-dashed). A: r = / B: r = / C: r = A. Scalar Equations
1. Drift away from a stable QSE without a critical transition
Consider system (1) with N = f ( x , λ ( rt )) = − x + λ ( rt ) with λ ( rt ) = rt . Using integrating factors, we see thatthe family of solutions is x ( t ) = r ( t − ) + Ce − t where C = C ( x , t ) = ( x − r ( t − )) e t depends on the ini-tial condition. Trajectories, the pullback attractor and the QSEfor this system are plotted in Figure 1 for three values of therate parameter.For every choice of r ∈ R the solution γ r ( t ) = r ( t − ) is aforward attracting pullback attractor. The forward attractionis clear to see from the form of the solutions. To see that γ r ( t ) is a pullback attractor for the system, notice thatlim s →− ∞ | x ( t ; x , s ) − γ r ( t ) | = lim s →− ∞ | ( x − r ( s − )) e s − t | = . In fact, γ r ( t ) is the unique pullback attractor of the system andis the solid yellow line in Figure 1.This equation has one quasi-static equilibrium given by Q ( t ) = λ ( rt ) which is stable. However, since as t → ∞ , x ( t ) approaches the forward attracting pullback attractor γ r ( t ) forany initial condition , we see that for any 0 < a < | r | , there is atime where every solution x ( t ; x , t ) leaves the a -radius of thestable QSE.The pullback attractor does not limit to the stable QSE inforward time, so attempting to extend the current definitionof rate-induced tipping to this case would require us to callthis rate-induced tipping. However, the pullback attractor isglobally defined and forward attracting for all r In particular,there are no critical transitions as a result of varying the rateparameter r in this system and so it would be unfortunate tosay there is rate-induced tipping in this case.
2. Drift sway from a Stable QSE with a critical transition
Consider system (1) with N = f ( x , λ ( rt )) = − ( x − λ ( rt ))( x − λ ( rt ) − δ ) with λ ( rt ) = rt , pictured for three val- ues of r in Figure 2 with δ = /
2. The equation is straight-forward to solve once converted into co-moving coordinates.Namely, let y ( t ) = x ( t ) − λ ( rt ) − δ /
2. Then the co-movingequation is the autonomous equation˙ y = − ( y − δ / )( y + δ / ) − r = − y + (cid:18) δ − r (cid:19) with family of solutions for r ≤ δ / y ( t ) = (cid:40) ρ tanh ( ρ ( t + C )) , | y | < ρρ coth ( ρ ( t + C )) , | y | > ρ where ρ = (cid:114) δ − r , C ( y , t ) = ρ log (cid:18) y + ρρ − y (cid:19) − t , and C ( y , t ) = ρ log (cid:18) y + ρρ − y (cid:19) − t . The constant solutions y ( t ) = ± ρ are equilibria of the equa-tion. Notice that if the initial condition satisfies | y | > ρ , thenthere is a finite time singularity at t = − C . If y > ρ thissingularity happens in backward time and if y < − ρ this sin-gularity happens in forward time.Transforming back to the stationary coordinates, we seethat solutions take the form x ( t ) = (cid:40) rt + ( δ / ) + ρ tanh ( ρ ( t + C )) , | x − rt − δ / | < ρ rt + ( δ / ) + ρ coth ( ρ ( t + C )) , | x − rt − δ / | > ρ . when r < δ /
4. For this range of r there is a forward attractingpullback attractor given by γ r ( t ) = rt + ( δ / ) + ρ , corresponding to the stable equilibrium in the co-moving sys-tem. For the same values of r there is a forward repellingpullback repeller given by ζ r ( t ) = rt + ( δ / ) − ρ ethinking the Definition of Rate-Induced Tipping 5 ���� �� x t x [ t ] attractorrepellor x [ t ] x [ t ] Stable QSEUnstable QSE A t x ���� �� x t x [ t ] node x [ t ] Stable QSEUnstable QSE B t x ���� �� x x [ t ] x [ t ] x [ t ] x [ t ] Stable QSEUnstable QSE C t x FIG. 2. Time Series for ˙ x = − ( x − rt )( x − rt − δ ) showing the system’s forward attracting pullback attractor (solid yellow), stable QSE (dashedyellow), forward repelling pullback repeller (solid green), unstable QSE (dashed green), and selected solutions (blue curves; dotted, solid,dashed, dot-dashed). A: r = / B: r = r ∗ = / C: r = / corresponding to the unstable equilibrium in the co-movingsystem. The stable and unstable QSEs of the equation aregiven by Q S ( t ) = rt + δ , Q U ( t ) = rt , respectively. As with the example in Section III A 1, the pull-back attractor drifts away from the stable QSE as r increases.In Figure 2b, the system has a nonautonomous saddle nodebifurcation where the pullback attractor collides with the pull-back repeller when r = δ /
4. For this value of r , the pull-back attractor γ r loses forward stability (see Figure 2b and 2c),which can be seen in both the stationary coordinates and theco-moving coordinates. In particular, we see that for this r the value of ρ = (cid:112) δ / − r is zero. Then solutions with x > rt + δ / γ r ( t ) = rt + δ / x < rt + δ / γ r ( t ) = rt + δ / t = − C ( x , t ) .When r > δ /
4, one may show, in the same way as above,that the comoving system has gone through a saddle node bi-furcation and there are no equilibria, stable or otherwise, inthe system. In the nonautonomous system, the family of solu-tions is given by x ( t ) = rt + ( δ / ) − ˜ ρ tan ( ˜ ρ ( t + C )) where ˜ ρ = (cid:113) r − δ / C ( x , t ) = − ρ arctan (cid:18) x − rt − δ / ρ (cid:19) − t . This demonstrates that all solutions blow up in finite time and,thus, that the pullback attractor has been annihilated by thecollision with the pullback repeller at r = r ∗ = δ / r = r ∗ = δ /
4. Specifically, for r < r ∗ there are an uncountable number of globally defined solutions(solutions contained between the pullback attractor γ r ( t ) andthe pullback repeller ζ r ( t ) ), at r = r ∗ there is exactly one glob-ally defined solution (the curve rt + δ / r > r ∗ thereare no globally defined solutions. B. N -Dimensional Systems Similar behavior can occur in higher dimensional systems.Consider the system˙ x = − y ˙ y = ( x − λ ( t ))( x − λ ( t ) − δ ) − y (5)where λ ( t ) = rt . As with the previous example, we may trans-form to the co-moving frame with the change of coordinates z ( t ) = x ( t ) − λ ( t ) − δ /
2, yielding the autonomous system˙ z = − y − r ˙ y = (cid:18) z − δ (cid:19) − y . (6)The nullclines and sample trajectories the the co-moving sys-tem are plotted for δ = r in Figure 3.For values of r satisfying r < δ /
4, the autonomous systemhas a stable equilibrium at x ∗ s = ( (cid:112) δ / − r , − r ) and a saddleequilibrium at x ∗ u = ( − (cid:112) δ / − r , − r ) . When r = δ /
4, thisautonomous system has a saddle node bifurcation where thesink and saddle collide and annihilate each other.The equilibria of the co-moving system correspond to glob-ally defined solutions for the nonautonomous system. For r < δ /
4, the stable equilibrium in the co-moving system cor-responds to a forward attracting pullback attractor γ r ( t ) = (cid:20) (cid:112) δ / − r + rt + δ − r (cid:21) in the nonautonomous system. Some solutions for the nonau-tonomous system are plotted in Figure 4.In Figure 4 we plot the x and y components of solutions tothe nonautonomous equation and see a critical transition in-duced by varying the rate parameter r . We plot trajectoriesof the system (5) with δ = r , one valuebelow the tipping threshold ( r = / r that causes tipping ( r ∗ = /
4) and one value above the tippingthreshold ( r = / x and y components ofthe pullback attractor (solid yellow), the globally defined so-lution associated with the saddle (solid green), and the stableethinking the Definition of Rate-Induced Tipping 6 ���� �� - - - - y A z y ���� �� - - - - y B z y ���� �� - - - - y C z y FIG. 3. Trajectories (dashed blue curves) and nullclines (red solid curves) for the co-moving system (6) with δ =
1. The system has a saddlenode bifurcation at r = r ∗ = / A: r = / B: r = r ∗ = / C: r = / ���� �� x attractorrepellersolutionsStable QSEUnstable QSE ���� �� - y attractorsolutionsStable QSE A t x y ���� �� x nodesolutionsStable QSEUnstable QSE ���� �� - y nodesolutionsStable QSE B t x y ���� �� x x [ t ] x [ t ] x [ t ] Stable QSEUnstable QSE ���� �� - y y [ t ] y [ t ] y [ t ] Stable QSE C t x y ���� �� x attractorrepellersolutionsStable QSEUnstable QSE ���� �� - y attractorsolutionsStable QSE A t x y ���� �� x nodesolutionsStable QSEUnstable QSE ���� �� - y nodesolutionsStable QSE B t x y ���� �� x x [ t ] x [ t ] x [ t ] Stable QSEUnstable QSE ���� �� - y y [ t ] y [ t ] y [ t ] Stable QSE C t x y ���� �� x attractorrepellersolutionsStable QSEUnstable QSE ���� �� - y attractorsolutionsStable QSE A t x y ���� �� x nodesolutionsStable QSEUnstable QSE ���� �� - y nodesolutionsStable QSE B t x y ���� �� x x [ t ] x [ t ] x [ t ] Stable QSEUnstable QSE ���� �� - y y [ t ] y [ t ] y [ t ] Stable QSE C t x y FIG. 4. Time Series for (5) showing the x and y components of the system’s forward attracting pullback attractor (solid yellow), stable QSE(dashed yellow), the globally defined solution associated with the saddle from (6) (solid green), the stable and saddle QSEs (dashed yellow andgreen), and selected solutions (dotted blue). The system has a nonautonomous bifurcation at r = r ∗ = / A: r = / B: r = r ∗ = / C: r = / and saddle QSEs (dashed yellow and green). As with the one-dimensional example, tipping corresponds to the collision be-tween the two globally defined solutions of the system at thecritical value r = r ∗ . For larger r the pullback attractor hasbeen annihilated and no attractors remain in the system. C. Tipping when a Transformation to a Co-moving Systemis Possible
It is possible to generalize these ideas to N -dimensionalsystems with locally bounded parameter changes, providedthey can be transformed through a co-moving coordinatechange to an autonomous system. Here we consider trans-lated systems like the ones from our examples, namely wherethe parameter changing time, λ ( rt ) , only appears in the sys-tem with a difference to state variables. More precisely, con- sidering the fully general case from the introduction˙ x = f ( x , λ ( rt )) , x ∈ R N , λ ( rt ) ∈ R M we instead take˙ x = f ( x − Λ ( rt ) , t , µ ) , x , Λ ( rt ) ∈ R N , µ ∈ R M , Λ i ( rt ) = a i λ ( rt ) , a i , λ ( rt ) ∈ R (7)where µ denotes the constant parameters of the system. Al-though this choice is certainly restricting, similar restrictionsare found in other rate-induced tipping studies and are called parameter shift systems . Of course we also take f smoothenough to ensure existence and uniqueness of solutions. Withthese types of systems in mind, we have the following result. Proposition 2.
Suppose(i) λ ( rt ) is locally bounded for all r ethinking the Definition of Rate-Induced Tipping 7 (ii) for x = ( x , x , . . . , x N ) ∈ R N there is a set N ⊆{ , . . . , N } so that the time dependent vector v ( t ) hascomponents v i ( t ) = a i λ ( rt ) + b i for some constantsa i , b i ∈ R and i ∈ N and v j ( t ) = for j (cid:54)∈ N ;(iii) and the transformation y = x − v ( t ) results in an au-tonomous system ˙ y = g ( y , µ , r ) = f ( y , t , µ ) + ˙ v ( t ) , y ∈ R N . (8) Then any hyperbolic equilibria y ∗ r of the co-moving system (8) correspond to globally defined solutions of the originalnonautonomous system (7) . Furthermore,(a) y ∗ r is a sink if and only if the corresponding globally de-fined solution of (7) is a forward attracting pullback at-tractor; and(b) y ∗ r is a source if and only if the corresponding globallydefined solution of (7) is a forward repelling pullback re-peller.Proof. Let v ( t ) ∈ R N be defined as in the statement of theLemma and let y ( t ) be a solution to (8). Then x ( t ) = y ( t )+ v ( t ) is a solution to (7) and for each hyperbolic equilibrium y ∗ r of (8), x ∗ ( t ) = y ∗ r + v ( t ) is a globally defined solution to theoriginal system. By a similar argument, given any solution x ( t ; x , t ) to the nonautonomous system, y ( t ) = x ( t ; x , t ) − v ( t ) is a solution to the autonomous co-moving system.Suppose y ∗ r is a sink and take a ball B ( y ∗ r ) so that all so-lutions with initial conditions in B ( y ∗ r ) converge to y ∗ r . Let B ( t ) = B ( y ∗ r ) + v ( t ) so that B ( t ) is the ball B ( y ∗ r ) in themoving system. Fix t ∈ R . Given a solution x ( τ ; x , t ) tothe nonautonomous system with t < t and initial condition x ( t ; x , t ) = x ∈ B ( t ) , there is a corresponding solution y ( τ ) = x ( τ ; x , t ) − v ( τ ) to the autonomous co-moving sys-tem. Furthermore, since x ( t ; x , t ) = x ∈ B ( t ) , we have that y ( t ) = y ∈ B ( y ∗ r ) .This motivates us to write x = y + v ( t ) so that we may“pull back” the initial condition x to x s = y + v ( s ) with s < t .Then the solution to the nonautonomous system through thepoint x s at time s , x ( τ ; x s , s ) corresponds to the solution to theco-moving autonomous system which has initial condition y at time s . Due to time invariance of autonomous systems, thedistance between y ( t ) and y ∗ r is determined by the differencebetween the current time t and the initial time s , thus we seethat 0 = lim s →− ∞ | y ( t ) − y ∗ r | = lim s →− ∞ | x ( t ; x s , s ) − x ∗ ( t ) | . The forward attraction of x ∗ ( t ) follows from the asymptoticstability property of y ∗ r .To show the other direction we will show that if y ∗ r is not asink, then the corresponding globally defined solution x ∗ ( t ) ofthe nonautonoumous system is not forward attracting.Suppose y ∗ r is not a sink. Since we are assuming y ∗ r is ahyperbolixc equilibrium, it must be that y ∗ r has an unstablemanifold of dimension at least 1. Let φ ( x , t ) denote the flowassociated with the autonomous system (8) and W s ( y ∗ r ) (re-spectively W u ( y ∗ r ) ) denote the stable (unstable) manifold of y ∗ r . Let B ( y ∗ r ) be a ball containing y ∗ r satisfying the condi-tions that (1) its closure contains no other invariant sets, (2) φ ( B ( y ∗ r ) , t ) limits to W u ( y ∗ r ) in forward time, and (3) W s ( y ∗ r ) in backward time (or if y ∗ r is a source, then φ ( B ( y ∗ r ) , t ) lim-its to y ∗ r in backward time). These criteria ensure that B ( y ∗ r ) does not intersect with some other invariant set of the system,e.g. a periodic orbit. Such a ball is possible to find becausehyperbolic equilibria are isolated invariant sets.As before, let B ( t ) = B ( y ∗ r ) + v ( t ) and now also define W s ( t ) = W s ( y ∗ r ) + v ( t ) and W u ( t ) = W u ( y ∗ r ) + v ( t ) . Givena solution x ( τ ; x , t ) (cid:54) = x ∗ ( t ) to the nonautonomous systemwith initial condition x ( t ; x , t ) = x ∈ B ( t ) ∩ ( W s ( t )) c ,there is a corresponding solution y ( τ ) = x ( τ ; x , t ) − v ( τ ) tothe autonomous co-moving system with y ( t ) = y ∈ B ( y ∗ r ) ∩ ( W s ( y ∗ r )) c . Then there is some time T so that for t > T , y ( t ) is no longer in the ball B ( y ∗ r ) and, thus, x ( t ; x , t ) is no longerin B ( t ) . Since the globally defined solution x ∗ ( t ) is in B ( t ) forall time, x ( t ; x , t ) does not limit to x ∗ ( t ) in forward time.Showing part (b) follows in a similar manner.From the above proof, we see that y ∗ r has an unstable man-ifold of dimension at least 1 if and only if the correspondingglobally defined solution of (7) is not forward attracting. Thena bifurcation in the autonomous system caused by varying therate parameter which causes at least one of the dimensions ofthe stable manifold of a sink y ∗ r to become unstable is real-ized in the nonautonomous system as the pullback attractorloosing its forward stability. This is a rate-dependent criticaltransition that we see in the system as solutions in the sys-tem will converge to a different attractor or grow unbounded.Since this transition corresponds to a bifurcation in the co-moving autonomous system, all the regular bifurcation theoryfrom autonomous systems for hyperbolic equilibria apply. IV. FINAL REMARKS
Here we have proposed a new definition of rate-induced tip-ping which we find to be more inclusive of rate-dependentcritical transitions for deterministic, continuous time systems.We have restricted our discussion to nonautonomous systemswhose corresponding autonomous systems have only hyper-bolic fixed points. More work is needed to understand if theideas presented here could be generalized to other types ofsystems, for example where the corresponding autonomoussystem has a periodic orbit or more complicated behavior suchas a strange attractor.Although the examples discussed in Section III are far fromexhaustive, we nonetheless find them to be illustrative of be-haviors that systems with unbounded parameter change mayexhibit as their parameters are varied in time. In particular, apullback attractor not endpoint tracking a stable QSE is not in-dicative of a rate-dependent critical transition in systems withunbounded parameter change. This is the main motivator forour call for a new definition of rate-induced tipping.ethinking the Definition of Rate-Induced Tipping 8
ACKNOWLEDGMENTS
A portion of this work was completed during A.H-L.’ssabbatical which was partially supported by the HutchcroftFund and the Mathematics and Statistics Department at MountHolyoke College. The research of A.N.N. was supported byan NSF Mathematical Sciences Postdoctoral Research Fel-lowship, Award Number DMS-1902887.
DATA AVAILABILITY
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