Transient dynamics of the Lorenz system with a parameter drift
aa r X i v : . [ n li n . C D ] S e p September 24, 2020 0:45 IJBC-D-20-00266
International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
Transient dynamics of the Lorenz system with a parameter drift
Julia Cantis´an
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de F´ısica, Universidad ReyJuan CarlosTulip´an s/n, 28933 M´ostoles, Madrid, Spain
Jes´us M. Seoane
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de F´ısica, Universidad ReyJuan CarlosTulip´an s/n, 28933 M´ostoles, Madrid, Spain
Miguel A.F. Sanju´an
Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de F´ısica, Universidad ReyJuan CarlosTulip´an s/n, 28933 M´ostoles, Madrid, SpainDepartment of Applied Informatics, Kaunas University of TechnologyStudentu 50-415, Kaunas LT-51368, Lithuania
Received (to be inserted by publisher)
Non-autonomous dynamical systems help us to understand the implications of real systems whichare in contact with their environment as it actually occurs in nature. Here, we focus on systemswhere a parameter changes with time at small but non-negligible rates before settling at a stablevalue, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explainedby the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover ascaling law relating the duration of the transient to the rate of change of the parameter for acase where a chaotic attractor is involved. Additionally, we analyze the viability of recoveringthe transient dynamics by reversing the parameter to its original value, as an alternative tothe control theory for systems with parameter drifts. We obtain the relationship between theparamater change rate and the number of trajectories that tip back to the initial attractorcorresponding to the transient state.
Keywords : Transient Dynamics, Non-autonomous System, Parameter Shift, Dynamic Bifurca-tion, Rate-Induced Tipping
1. Introduction
The evolution of a system with time is typically divided into two different regimes: the transient and thesteady states. The latter corresponds to the asymptotic dynamics: after some time the system settles in thisstate for an indefinite time unless it is perturbed. This final state might include not only fixed points, butalso limit cycles or chaotic attractors. The dynamics before the system settles in any of these attractors is eptember 24, 2020 0:45 IJBC-D-20-00266 J. Cantis´an et al. what we call transient dynamics. This type of dynamics is also very rich and includes, as a classic example,decaying oscillations before a fixed point, but also chaotic motion. When the system behaves in a chaoticway for a finite amount of time before reaching an attractor, it is said to present transient chaos.Traditionally, the study of dynamical systems has focused on the steady state as transients usually lastfor a short-time scale and it has been assumed that the system’s dynamics can be reflected by the asymp-totic behavior of models describing these systems. However, some systems present long-lasting transientscompared to the scale of the system. Additionally, the relevant time scales may correspond to the transientregime rather than to the asymptotic one. Furthermore, transients may provide an explanation for suddenregime shifts, even without any underlying change in external conditions.Transient phenomena in autonomous systems has been studied in the past few years in several scientificdisciplines. Much of the work has focused on analyzing the factors that make a system prone to long tran-sients, for example, time-delay [Morozov et al. , 2016]. Evidence of systems presenting relevant long-lastingtransients include ecological systems [Hastings, 2004; Hastings et al. , 2018; Morozov et al. , 2020], but alsodifferent models in neuroscience [Rabinovich et al. , 2006, 2008], power electronics [Warecki & Gajdzica,2014], earthquake activity in seismology [Picozzi et al. , 2019], and gravitational waves [Thrane et al. ,2015]. Furthermore, transient chaos is relevant in a wide variety of systems ranging from optomechan-ics [Wang et al. , 2016] to electronic engineering [Bo-Cheng et al. , 2010]. For a recent review on transientchaos see [Lai & T´el, 2011]Here, we focus on the study of transient phenomena in non-autonomous dynamical systems. In partic-ular, in systems where one of the parameters varies slowly with time. The system can be mathematicallywritten as d x dt = F ( x , p ( εt )) , (1)where ε is very small compared to the natural time scale of the system. This type of systems are fun-damental to understand the relation of the system with its environment. Real systems are affected byexternal conditions, which can be reflected as a gradual change in a parameter. Sometimes, the parameteris controllable. One example is a ferromagnet within a low frequency magnetic field. In other cases, it isnot controllable, as in the context of climate dynamics.Time series with long-lasting and physically relevant transients are often found in systems with pa-rameter drift. Because of the duration of transients one may think that the system is at its steady state.However, for a later time, they show a sudden transition to their real steady state.Due to the fact that the time dependence of the parameter represents a small perturbation, theassociated frozen-in system, that is, the system with fixed parameters, provides useful information about thenon-autonomous system [Berglund, 2000]. In fact, the origin of the previously mentioned sudden transitionmay be found in the crossing of a bifurcation in the associated frozen-in system. This type of bifurcationsare called dynamic bifurcations [Benoˆıt, 1991]. Recently, they have been also called bifurcation-inducedtipping points [Ashwin et al. , 2012]. Either way, they refer to the regime shift that is produced due to theslow passage through a bifurcation.Dynamic bifurcations were studied when only regular attractors are involved. The case of the super-critical Hopf bifurcation has been studied in [Neishtadt, 1987, 1988; Baer et al. , 1989]. As a result, it wasfound that the appearance of oscillations was delayed when the parameter is slowly drifting and that thedelay depends on the rate of the drift, what is called the delay effect. However, when chaotic attractors areinvolved, dynamic bifurcations have been only studied so far for maps. In particular, the case of a Lorenz-type map when the control parameter varies monotonically with time and when it induces a periodicforcing, has been analyzed in [Maslennikov & Nekorkin, 2013; Maslennikov et al. , 2018] respectively. Here,we aim to broaden the current knowledge on dynamic bifurcations to flows presenting strange attractors.Another interesting phenomenon occurs to systems with parameter drift when there is multistability. Inthis case, the parameter drift may cause the system to tip to another state. Sometimes, this happens becausethe current attractor disappears and it has to tip to any of the rest of the attractors. More surprisingly,this can happen for certain rates above a critical value, even if the first attractor is still stable. In thiscase, the system cannot track the attractor and it tips to another one with a certain probability. This hasbeen called rate-induced tipping [Ashwin et al. , 2012]. Besides, the study of the tipping probabilities foreptember 24, 2020 0:45 IJBC-D-20-00266 Transient dynamics of the Lorenz system with a parameter drift a monodimensional system including the passage through a chaotic attractor was shown in [Kasz´as et al. ,2019], but they do not consider the case when the chaotic attractor coexists with other attractors.Our goal here is to study the transient phenomena for a multistable system with parameter drift, inparticular the Lorenz system. The nature and duration of the transient dynamics and the transition to thesteady state can be explained by the delay effect previously mentioned. The bifurcation diagram of thefrozen-in system gives us information about the dynamics that has to be corrected by a scaling law thatrelates the magnitude of the delay with the parameter rate of change.The paper is organized as follows. We start in Sec. 2 with the description of the frozen-in system,including the basins of attraction for the starting and ending points of the parameter shift. In Sec. 3, welet the parameter evolve with time in such a way that the heteroclinic bifurcation is crossed. We uncoveran interesting scaling law that predicts the duration of the transient for the non-autonomous system. Also,we show that the presence of a chaotic attractor induces unpredictability, causing a random tipping.Next, in Sec. 4, we study the possibility of reversing the transient dynamics once the transition tothe steady state has taken place by reversing the parameter to its original value. For that purpose, higherparameter rates have to be considered and rate-induced tipping is the phenomenon responsible for thetransient state recovery. Finally, the main conclusions of this work are discussed.
2. The Lorenz system
We have chosen the Lorenz system [Lorenz, 1963] described by the Eqs. 2, since it can be consideredas a paradigmatic example of a multistable system that presents both chaos and transient chaos. Theseequations were proposed by Edward Lorenz as a simple model of convection dynamics in the atmosphere(Rayleigh-Bnard convection). These equations also arise in models of lasers [Li et al. , 1990] and chemicalreactions [Poland, 1993], among others. The equations of the system read as follows˙ x = − σx + σy, ˙ y = rx − y − xz, ˙ z = − βz + xy, (2)where σ , β and r are the system parameters. In the context of convection dynamics, σ is the Prandtlnumber and is characteristic of the fluid, β depends on the geometry of the container and has no specificname. Finally, r is the Rayleigh number and it accounts for the temperature gradient. In this context, x represents the rotation frequency of convection rolls, while y and z correspond to variables associated tothe temperature field. We fix the classical parameter values as σ = 10 and β = 8 /
3. And we explore thedynamics of the system in terms of the variation of r by using a bifurcation diagram.Depending on the temperature gradient, i.e., the value of r , convection rolls may exist or not. Thiscan be seen in detail in the bifurcation diagram shown in Fig. 1). For r < , , r = 1, the origin becomes a saddle through a supercriticalpitchfork bifurcation, and this instability is reflected in the bifurcation diagram by a discontinuous line for r >
1. Two symmetrical branches C ± = ( ± p b ( r − , ± p b ( r − , r −
1) (3)are created and are stable until r = 24 .
74. The unstable manifold of the origin, W s ( ), separates theirrespective basins of attraction. The fixed points attractors C ± correspond to convection rolls with the twopossible directions of rotation.On the other hand, at r = 24 .
06, a chaotic attractor is born through an heteroclinic bifurcation whichmakes the fluid become turbulent. For 24 . < r < .
74, the system is multistable ( C + , C − and thechaotic attractor coexist) and the attractor to which a given trajectory goes to depends on the initialconditions.Furthermore, at r = 24 .
74, the fixed points C ± lose the stability through a subcritical Hopf bifurcationand the chaotic attractor becomes the global attractor. Another important phenomenon occurs at r =13 . C ± , which is known as transienteptember 24, 2020 0:45 IJBC-D-20-00266 J. Cantis´an et al.
Fig. 1.
Bifurcation diagram for the Lorenz system.
For r = 1 a supercritical pitchfork bifurcation takes place and theorigin becomes a saddle point. For 1 < r < .
06, two attractors coexist: the upper branch corresponds to C + and the lowerbranch corresponds to C − . For this range of r values, past the homoclinic bifurcation (when r > . r > .
06, the chaotic attractor is born in a heteroclinicbifurcation and the three attractors coexist. Finally, for r > .
74 a subcritical Hopf bifurcation leaves the chaotic attractor asthe global attractor. In the insets, the attractors are represented in phase space for the regions where multistability is present(the stars mark the initial conditions). chaos or preturbulence. This is not reflected in the bifurcation diagram as it only shows the steady statedynamics. In phase space, the chaotic behavior is confined to the vicinity of the chaotic saddle [T´el et al. ,2006]. The lifetime of these chaotic transients increases with r until it reaches the value 24 .
06 when thelifetime becomes infinite and the chaotic attractor appears.Also, at r = 13 .
926 a pair of unstable limit cycles Γ ± (not represented in the diagram), called homoclinicorbits, are created and last until they are absorbed in the Hopf bifurcation at r = 24 .
74. For further detailsabout the homoclinic and heteroclinic bifurcations in terms of the organization of the respective two-dimensional manifolds of , C ± and Γ ± see [Doedel et al. , 2006].Now, we focus on the dynamics before and after r = 24 .
06 in order to explore the effects of a parameterdrift when a strange attractor appears/disappears. This is why we present the basins of attraction for r = 20, where transient chaos is present, and r = 24 .
5, where the chaotic attractor and the fixed pointattractors C ± coexist.For this purpose, we distribute N = 10 initial conditions uniformly, preserving approximately thesame density of points for any area on the sphere, that is, avoiding accumulation of points in the poles.The radius of the sphere is fixed to 30 for all the simulations in this paper, but similar results are foundfor other radii. The sphere is likewise centered at the halfway between C ± : (0 , , r − . C + /C − . For the basin at r = 24 .
5, we take a sufficiently long integrationtime (the maximum time for trajectories to arrive to C ± is around 90 and we take t f = 2000 as the finaleptember 24, 2020 0:45 IJBC-D-20-00266 Transient dynamics of the Lorenz system with a parameter drift integration time) and we consider that trajectories that do not converge to C + /C − , converge to the chaoticattractor. Fig. 2.
Basins of attraction for r = 20 . (a) The sphere of initial conditions is represented with the trajectories that endup in the C + attractor depicted in red and the ones that end up in C − depicted in blue. The attractors are also represented inthe inside as points with matching colors. (b) We present a view in the ( x, y ) plane. Fractality appears in the basin boundariesfor z > ( r −
1) = 19.
The basins of attraction for r = 20 are represented in Fig. 2: in red, the initial conditions that endup in C + , and in blue the ones that end up in C − ; the attractors are also represented in the inside aspoints with matching colors. The sphere is divided in two, following approximately the symmetry plane x + y = 0. More precisely, the basin boundary is defined by the stable manifold of the origin W s ( ). It isimportant to notice that the blue face is nearer the red attractor and vice versa. There is also a circularblue region that is immersed in the red region, and symmetrically there is a circular red region in thebackwards of Fig. 2(a). Finally, the basin boundary is smooth for z < ( r −
1) = 19 (downside) and fractalfor z > ( r −
1) = 19 (upside) and around the immersed circle. This fractality is a trace of chaos.
Fig. 3.
Time to reach the attractors for r = 20 . The numerically computed lifetime of the transients is representedshowing a wide range of lifetimes, denoted in the color bar, with higher values on the corresponding fractal regions from Fig. 2.This corresponds to the trajectories presenting transient chaos.
The time that the trajectory needs to reach the attractors is depicted in Fig. 3. The lifetime is increasedeptember 24, 2020 0:45 IJBC-D-20-00266 J. Cantis´an et al. for higher values of the temperature gradient, if r < .
06. Comparing this figure with Fig. 2, it can be seenthat the fractal areas correspond to initial conditions that take the longest time to reach the attractors.The explanation for this phenomenon is related to the stable and unstable manifolds of the chaotic saddle.Points exactly on the stable manifold necessarily reach the chaotic saddle and never leave it (these pointsare exceptional and are not shown), and points initially far from the stable manifold escape the chaoticsaddle quickly (not showing transient chaos), while points near the stable manifold have a longer lifetime.The closer to the stable manifold, the longer the lifetime of the transients is. Thus, the structure seen inlight blue in Fig. 3 is due the cut of the stable manifold of the chaotic saddle with the sphere of initialconditions.
Fig. 4.
Basins of attraction for r = 24 . . (a) The sphere of initial conditions is represented with the same color code asbefore: red accounts for C + and blue for C − . Yellow regions account for trajectories that head to the chaotic attractor. The(b) and (c) panels show the ( x, y ) plane for z >
19 (view from above) and z <
19 (view from the downside) respectively.
The basins of attraction for r = 24 . C + /C − . The initial conditions that end upin the chaotic attractor are depicted in yellow. For the 10 trajectories, 10348 correspond to the red basin,10130 to the blue basin and 79522 to the yellow basin. Thus, approximately the 20% of the trajectoriesgo to C + /C − . This time, the structure for the red and blue basins consists on two loops for z <
19 thatoverlap in the plane x + y = 0 and four circular regions for z >
19 (one of each color on top and anothertwo smaller ones on the sides).The lifetime of the trajectories that end up in either C + /C − were also computed. We can observethat there is a wide range of lifetimes in Fig. 5. This is not due to transient chaos but to the fact that theoscillations around C + /C − are slowly damped until they reach the attractor.
3. Dynamic Heteroclinic Bifurcation
In this section, we explore the dynamics of the system when the parameter that accounts for the temperaturegradient, r , slowly varies. The bifurcation diagram in Fig. 1 represents the dynamics for each value of r when the system evolves with the corresponding fixed value of r . We are interested in the case when theparameter varies during the evolution of the system. In other words, the parameter turns into a slowlyvarying function of time of the form: r = r + εt , where ε is a sufficiently small parameter compared to thenatural time scale of the system. Bifurcations that are crossed due to this parameter time-dependence arecalled dynamic bifurcations [Benoˆıt, 1991].As previously stated, this type of bifurcations have been studied when they imply the appear-ance/disappearance of regular attractors. For example, the dynamic pitchfork bifurcation for the Lorenzsystem has been deeply studied. It was found that when the temperature gradient is increased, convectionrolls appear suddenly at a r >
1, which is what is called the delay effect. When the temperature gradient isdecreased they slowly decelerate and finally disappear for a r <
1, showing hysteresis. Furthermore, theyalways follow the same equilibrium, i.e., roll in the same direction, to which one depends on the initialconditions. The area enclosed in the hysteresis diagram depends on the adiabatic parameter, ε , and thisrelation is defined by its corresponding scaling law [Berglund & Kunz, 1999].eptember 24, 2020 0:45 IJBC-D-20-00266 Transient dynamics of the Lorenz system with a parameter drift
Time to reach the C + /C − attractors for r = 24 . . (a) Red/blue points correspond to the attractors C + /C − .Only the initial conditions that end in C + /C − are depicted, the rest of the sphere corresponds to initial conditions that endup in the chaotic attractor. As we can see, the borders of the structure show higher lifetimes due to the slow oscillation decayto the C + /C − attractors.(b) This panel shows the ( x, y ) plane of the previous panel. As far as we know, the crossing of a bifurcation due to a slow parameter drift when chaotic attractors areimplied has been only studied so far for maps, for instance, in the Lorenz map in [Maslennikov & Nekorkin,2013] and [Maslennikov et al. , 2018]. Here, we study the dynamic heteroclinic bifurcation at r = 24 .
06 forthe Lorenz system. This implies a change in the number of the attractors and the appearance/disappearanceof a chaotic attractor.In analogy with the delay effect found for the pitchfork bifurcation where the origin becomes unstablebut the system tracks that path for a period of time, we start our analysis for a value of r > .
06 and wedecrease this value past the heteroclinic bifurcation where the chaotic attractor is no longer stable.The first difference in this analysis with the one for regular attractors is that single trajectories areno longer representative and do not contain all the possible dynamics, thus we follow an ensemble oftrajectories starting on the same sphere of initial conditions used in the previous section.Moreover, we refer to Eqs. 2 as the frozen-in system when r is a fixed parameter and the non-autonomous system to the same set of equations when r is a function of time in the following form r = r for t < t r − ε · ( t − t ) for t < t < t r − ε · ( t − t ) for t > t r t t (4)where t is the time for which the parameter shift starts and t when it ends. This way, we let the systemevolve to its steady state, the chaotic attractor in our case, before the shift in r starts. Besides, we call t f > t the final observation time.In Fig. 6, the x -component of a trajectory starting at the sphere, with the following parameters: r = 24 . t = 800, t = 4000, t f = 4000 and ε = 10 − is shown. As we can see, the motion is chaoticup until a value (when the line turns blue) for which there is a sudden transition to the attractor C − .Equation 4 establishes a correspondence between time and the value of the temperature gradient, r , whichis also shown in the figure as a secondary x -axis. The transition from one regime to another appears for avalue of r < .
06 that we call the critical value and designate it by r cr .Thus, we conclude that the delay effect is also present when chaotic attractors are involved. Further-more, it is necessary a lower temperature gradient, exactly r cr = 23 . J. Cantis´an et al.
Fig. 6.
Time series for the non-autonomous Lorenz system with ε = 10 − . The secondary x -axis shows the timedependence in the r parameter. It can be seen that the transient lasts for a long period of time and that the transition to thesteady state starts at r < .
06, specifically, for r = 23 . a constant rotation frequency of the rolls. However, if we let the system evolve to its steady state for eachvalue of r , which translated to our analysis would mean to make ε →
0, the transition would take placeexactly at the value in the bifurcation diagram, r = 24 .
06. For 23 . < r < .
06, the system tracks thechaotic attractor although it is no longer stable. In other words, the chaotic attractor is a metastable state.As previously mentioned, when dealing with chaotic attractors a single trajectory is not informativeenough and we should consider an ensemble of trajectories. For that reason, we take 10 initial conditionson the sphere, and we exclude the ones that for r = 24 . C + /C − . These correspond to the red/bluebasins in Fig. 4.The attractor toward which the trajectory goes depends on the initial conditions. But for a singleinitial condition, it depends on the precise moment that the trajectory is caught wandering in the chaoticattractor, that is on t . Using the same terminology as in [Kasz´as et al. , 2019], we call the basins for thenon-autonomous system: scenario-dependent basins. After this clarification, we fix t = 800 , ε = 10 − , r = 24 . t = 4000 and t f = 4000, as before and we let the ensemble of trajectories evolve.The value of r cr for the dynamic heteroclinic bifurcation is different for each initial condition unlikefor the dynamic pitchfork bifurcation for which the transition occurred at the same value of r for everyinitial condition. In Fig. 7(a), we can see how the values of r cr follow a normal distribution with mean h r cr i = 23 .
071 and standard deviation σ = 0 . ε to study the effect of the parameter rateof change. We focus on small, but non-negligible rates compared to the natural time scale of the system.In our case, the period of revolution of convection rolls is of order 10 − . In Fig. 7(b), we calculated the r cr for more values of ε on the range 10 − − − which are marked as black stars. As we can see, forhigher rates the delay effect is more pronounced. For example, for ε = 10 − , the temperature gradient forwhich the system suffers a sudden transition from a chaotic state to a stable state is h r cr i = 22 .
32, which issignificantly a smaller value than the r cr for ε = 10 − . Another consequence of this, is that for rapid shiftsin the parameter, the metastable state lasts longer. Also, as ε →
0, the standard deviation is reduced. Inthe limit, that is, the frozen-in system, all the initial conditions suffer the transition at the same value of h r cr i .Finally, we derived the corresponding scaling law for the heteroclinic bifurcation. This equation relatesthe value of the parameter for which the system abandons the chaotic attractor with the parameter changerate. For that purpose, points in Fig. 7(b) are fitted to a power law of the form h r cr i = − a · ε / + r , (5)eptember 24, 2020 0:45 IJBC-D-20-00266 Transient dynamics of the Lorenz system with a parameter drift
Critical value of the parameter r and scaling law for the heteroclinic dynamic bifurcation. (a) Thenormal distribution of r cr for an ensemble of trajectories with a specific rate of change of ε = 10 − . (b) The mean value of r cr for different values of ε . The stars correspond to numerically calculated values of h r cr i , the power law fit is also shown in red. where a > r = 24 . R -square: R = 0 . r cr ∝ ε / . Unlikethe scaling law for the pitchfork bifurcation, our scaling law deals with the presence of chaotic attractors,which to the best of our knowledge is a novel finding. This implies that the value of r cr in the scaling lawis a mean value from the ensemble approach.We may ask ourselves about the spatial distribution of r cr in the sphere of initial conditions. However,due to the presence of the chaotic attractor, there is no pattern and the initial conditions that lead to longerand shorter metastable states are completely intermingled. We may say that the chaotic attractor acts asa memory-loss agent. In the same way, the scenario-dependent basins do not show a pattern and bothbasins ( C + /C − : red/blue basin) are intermingled. The final destination of the trajectories is related to theprecise moment that the trajectory is caught wandering in the chaotic attractor when the parameter shiftstarts rather than to the initial condition. In other words, predictability of individual trajectories is lostbecause the passage through the chaotic attractor induces fractal basins of attraction. Similar phenomenahas been addressed as a random tipping in [Kasz´as et al. , 2019]. Transient chaos interpretation
So far, we have defined the delay effect for the dynamic heteroclinic bifurcation, but this phenomenon canbe interpreted from a different point of view in the context of transient dynamics. The Lorenz equationswith r defined by Eq. 4 form a non-autonomous system, which can be considered to present transient chaosas the duration of the chaotic dynamics is finite (see Fig. 6). In this context, the scaling law predicts theend of the transient state.In the previous section, the scaling law predicted the value of the temperature gradient for which thedynamics changed as past r = 24 .
06 the chaotic attractor was considered to be a metastable state. Inthe time framework, the scaling law predicts when the system suffers a transition to its steady state. Thetransient dynamics may last for long periods of time; therefore, the transient is not a negligible part of thedynamics as it has been considered before, and it is fundamental to uncover a law that predicts the end ofthis state.To characterize a nonattracting chaotic set, as the chaotic saddle responsible for the transientchaos, we may analyze the decay in the number of trajectories that still present a chaotic behavior[Maslennikov & Nekorkin, 2013]. In Fig. 8 we represent, for different decay rates, N ( t ) as the normal-ized number of trajectories in the chaotic attractor for a time t . As we did earlier, we exclude the onesthat at r = 24 . C + /C − basins. Note that the decay in r starts at t = 800. When thecurves decrease to zero, the transient chaos phase ends and every trajectory reaches its steady state, i.e.,the C + /C − fixed point attractors.eptember 24, 2020 0:45 IJBC-D-20-00266 J. Cantis´an et al.
Fig. 8.
Normalized number of trajectories that remain in the chaotic attractor for a time t . At t = 800, theparameter starts decreasing, but the chaotic attractor constitutes a metastable state for some time later than before reachingthe steady state. Therefore, the transient behavior is shown. For a certain parameter change rate, the trajectories do not reachthe steady state at once; instead they follow a normal distribution. For slow parameter change rates the transient dynamicslasts for a longer period of time. This may create the false impression that the transient regime is the steady state. As we can see, in all cases, the decay with time follows a sigmoid which is related to the normaldistribution for r cr in Fig. 7(a). Note that r cr and time are equivalent through Eq. 4. In fact, we arerepresenting nothing more than the complementary cumulative distribution function of Fig. 7(a) in termsof time.The S -shape of Fig. 8 reflects that the majority of trajectories decay more or less at the same time(same r cr ), while some of them decay earlier or later. The curve for ε = 10 − is further from the rest ofthem as we showed in Fig. 7(b) that h r cr i ∝ − ε / , thus as ε →
0, the h r cr i decreases and the time for thetransition increases non-monotonically.The scaling law on Eq. 5 predicted that for high parameter change rates the temperature gradientcould be decreased to a low value before turbulence disappeared. In this new interpretation, we add thatin terms of time, the transient dynamics is shorter in that case. Finally, the scaling law can be written interms of time using Eq. 4: h τ i = a · ε − / + t , (6)where τ refers to the lifetime of the transient dynamics, t in our case is 800 and a is a positive constant.We conclude that in non-autonomous systems, the transient dynamics may present an unexpected long-term behavior. After a period of apparent equilibrium, in our case chaotic, the system suffers a transition toits truly steady state. The transient dynamics duration depends on the parameter rate of change throughthe scaling law. For slow rates, the transients last for a long period. This may be an undesired effect for aexperimentalist as a parameter may be changing too slowly to notice and the dynamics may seem stable.Additionally, due to the presence of the chaotic attractor, the final destination of the trajectory after thetransient dynamics becomes unpredictable.
4. Reversibility
Sometimes, the long-lasting transients that appear in systems with parameter drifts may be followed byan undesirable state. This problem has been addressed from a control theory approach, where a smallperturbation may keep the system in the desired transient state [Aguirre et al. , 2004]. For instance, thisapproach has been proposed to prevent species extinction maintaining the system in the transient chaosregime [Shulenburger et al. , 1999]. Here, we tackle the problem from a different perspective.eptember 24, 2020 0:45 IJBC-D-20-00266
Transient dynamics of the Lorenz system with a parameter drift For systems with parameter drifts, the parameter may be controllable and we may ask ourselves whetherit is possible to reverse the dynamics by reversing the parameter to its original value. For example, we cansee that in the non-autonomous Lorenz system, a chaotic transient precedes a sudden transition to a fixedpoint as shown in Fig. 6. An intuitive way to avoid the latter, is to increase the temperature gradient againin order to recover the chaotic dynamics that was lost. However, if we analyze the frozen-in bifurcationdiagram, the fixed point attractors C + /C − are stable before and after the heteroclinic bifurcation at r = 24 .
06. This implies that it may not be so easy to avoid these states and a complete study of themwould be needed.For that purpose, in this section, we analyze what happens to the system if the temperature gradi-ent increases with time and the heteroclinic bifurcation is crossed ‘from the left’. The equations for theparameter r read: r = r for t < t r + ε · ( t − t ) for t < t < t r + ε · ( t − t ) for t > t r t t (7)Once again, we take a sphere of 10 initial conditions and we let these trajectories evolve with time.Regarding the parameters in Eq. 7, there are some restrictions in order to reverse the dynamics. First of all,we take the initial temperature gradient to be in the range 13 . < r < .
06. In particular, we consider r = 20 since for this parameter we have already calculated the corresponding basins of attraction in thefrozen-in system (see Fig. 2). We aim to increase the temperature gradient in such a way that those redand blue basins corresponding to C + /C − (the undesirable state), map to the yellow basin correspondingto the chaotic attractor (the desirable state).The ideal situation would be to fix t to a higher value than the maximum lifetime calculated in Fig. 3,that is, t > C + /C − . However, it canbe checked that once a trajectory is close enough to C + /C − it cannot escape, no matter the rest of theparameter values in Eq. 7. By close enough, we refer to the same criterion used for the basins of attraction,that is, that the trajectories enter into a sphere of radius 0 . C + /C − . Higher values of theradii, around 5, mark the no return limit. Taking that argument into account, we set t = 0, so that theparameter starts increasing from the beginning of the trajectory, when it is still on the sphere, far enoughfrom the C + /C − attractors.For the same reason as before, if we take slow parameter rates, the trajectories will enter the no returnlimit. Thus, the third restriction is about ε . For the range of ε considered in the previous section, thesystem does not tip to the chaotic attractor. In fact, we found that a minimum rate close to 5 · − isneeded. This type of tipping, where the system fails to track a continuously changing quasi-static attractorfor a certain critical rate is called rate-induced tipping or R -tipping [Ashwin et al. , 2012]. In our case, as weare not letting the system to reach C + /C − , no traditional rate-induced tipping is possible [Kasz´as et al. ,2019].On the other hand, the last parameter, t , is fixed in such a way that for every ε , the temperaturegradient stops at r = 24 .
5. For example, for ε = 10 − , the value of t is 45, since 20 + 10 − · (45 −
0) = 24 . ε = 5 · − , − , · − and 5 · − . Additionally, we have included the basins forthe frozen-in equations at the starting and ending values of the parameter shift, that is, r = 20 and 24 . r = 24 .
5. For slower parameterrates, as ε = 10 − , we obtain that the scenario-dependent basin is the same as the basin for r = 20. In othereptember 24, 2020 0:45 IJBC-D-20-00266 J. Cantis´an et al.
Fig. 9.
Rate-induced tipping phenomenon.
The scenario-dependent basins of attraction for different values of ε showingthat for high parameter change rates a significant number of trajectories tip to the chaotic attractor. We also show the basinsof attraction of the frozen-in system for r = 20 and r = 24 . t = 0 , r = 20 and r f = 24 . words, no tipping from C + /C − to the chaotic attractor is possible. For ε = 5 · − , a few trajectoriesstarting near the stable manifold of the chaotic saddle at r = 20 tip to the chaotic attractor. This isbecause points in that region have longer lifetimes and show transient chaos in the static case, see Fig. 3.For ε = 7 · − and 10 − , this effect is stressed. And for ε = 5 · − the number of trajectories that tipto the chaotic attractor are significantly increased and the basin boundaries are smoothed.Even for the initial conditions that do not tip, these trajectories are affected by an increase in the timeto reach the C + /C − attractors. This long-lasting transient is explained by the fact that the attractor iscontinuously changing and the difficulty to track its path is more severe for higher parameter change ratesas shown in Fig. 10 for ε = 5 · − . The transient lasts for a longer time compared to the frozen-in case,even if it does not tip to the chaotic attractor.Finally, we have calculated the tipping probability as defined in [Kasz´as et al. , 2019]. In our case, itcorresponds to the proportion of the part of the basin of attraction of C + /C − at r = 20 that is mappedto the chaotic attractor, A , at r = 24 . t . The evolution of the tipping probability with therate of change of the parameter r is shown in Fig. 11. As we can see, the probability increases with ε , sincemore initial conditions tip to the chaotic attractors for higher parameter change rates. At a value around10 − the curvature changes sign and around 5 · − the tipping probability saturates and approximatelyhalf of the trajectories tip to the chaotic attractor.The shape of the numerically calculated points describe a sigmoid, specifically, a Gompertz typefunction of the form P = a · e − b · e − c · ε , (8)where a is the saturation value, i.e., a = 0 . b = 6 .
139 and c is the growth rate. In our case, c = 21 . R -square of 0 . ε and it reaches a saturation value for which no matter how fast the parameter is shifted,approximately half of the trajectories never tip to the chaotic attractor. These trajectories correspond tothe ones with lower lifetimes in the frozen-in case.The results presented above show that it is possible to change the fate of trajectories that would reachthe C + /C − attractors by increasing the parameter r . At the beginning of this section, we considered aeptember 24, 2020 0:45 IJBC-D-20-00266 Transient dynamics of the Lorenz system with a parameter drift
Transient lifetime comparison for the frozen-in case and the non-autonomous system with ε = 5 · − . Even for the trajectories that do not tip to the chaotic attractor, the transient regime duration is increased when the parametershifts as it is more difficult to track the attractor.Fig. 11.
Tipping probability dependence with ε . This probability accounts for the number of trajectories that tip from C + /C − to the chaotic attractor, A , by the end of the parameter shift. For higher parameter change rates more trajectoriesthat belonged to the red/blue basins at r = 20, tip to the chaotic attractor when the parameter shifts. The numericallycalculated points are fitted to a sigmoid, specifically, the Gompertz function. trajectory with a chaotic transient preceding a sudden transition to an undesirable state, like the one inFig. 6, and we asked ourselves whether it is possible to reverse this dynamics by reversing the parameterto its original value, that is r = 24 .
5. In the light of the results shown, this is not always possible.Once a trajectory has decayed to either one of the fixed points, that is, time has passed so that r hasdecreased below r cr , reversing the parameter to its original value is not enough to reverse the dynamics.However, if we observe this behavior for a single trajectory, we may still be able to reverse the dynamicsfor other trajectories. As we showed in Fig. 7(a), each initial condition decays to the fixed point at adifferent time, or likewise at a different parameter value r . This means that for r > ( h r cr i − σ ), there arestill trajectories in the chaotic region which after some time would decay to C + /C − , and an increase in thetemperature gradient may keep these in the desirable state. For higher parameter change rates, a highereptember 24, 2020 0:45 IJBC-D-20-00266 REFERENCES number of trajectories will tip to the chaotic attractor as shown in Fig. 11.Another consequence is that if we want to make sure that the chaotic behavior is not recovered oncethe temperature gradient is increased for any trajectory, we have to decrease this parameter to a value of r < h r cr i − σ .
5. Conclusions
In this paper we have studied the phenomenon of transient dynamics in a non-autonomous system. Inparticular, we have analyzed the Lorenz system subjected to a small parameter drift. First of all, we havecharacterized the associated frozen-in system. We have observed that when the parameter crossed theheteroclinic bifurcation value, the system continued tracking the chaotic attractor for further parametervalues until it reached a critical value at which it jumped to its truly steady state. Thus, we have concludedthat the so-called delay effect is also present in systems with strange attractors. Furthermore, it constitutesthe origin for the long-term transient dynamics before the system settles at its steady state.We derived a scaling law to relate the duration of the transient in the non-autonomous system to theparameter rate of change. We have shown that for higher parameter change rates, the transient dynamicsis shorter, while the deviation from the bifurcation value of the parameter that causes the transition islarger. This relation is governed by a power law.Finally, we have analyzed the possibility of recovering the transient dynamics by reversing the param-eter value to its original state, as an alternative to the control theory for non-autonomous systems. For thispurpose, higher parameter rates were considered. Above a critical rate, rate-induced tipping takes placeand some trajectories initially far from the fixed point attractors tip back to the chaotic attractor. We alsohave showed the sigmoid relation between the number of trajectories that change their fate and end up inthe chaotic attractor and the parameter change rate. For this purpose, the reverse on the parameter muststart before r > h r cr i − σ . Even for the trajectories for which the system does not tip, we have showed thatthe transient dynamics duration is enlarged. Acknowledgments
This work has been supported by the Spanish State Research Agency (AEI) and the European RegionalDevelopment Fund (ERDF, EU) under Projects No. FIS2016-76883-P and No. PID2019-105554GB-I00.
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