Finite-time scaling in local bifurcations
aa r X i v : . [ n li n . AO ] A p r Finite-time scaling in local bifurcations ´Alvaro Corral,
1, 2, 3, 4
Josep Sardany´es,
1, 3 and Llu´ıs Alsed`a
3, 2 Centre de Recerca Matem`atica, Edifici C,Campus Bellaterra, E-08193 Barcelona, Spain Departament de Matem`atiques, Facultat de Ci`encies,Universitat Aut`onoma de Barcelona, E-08193 Barcelona, Spain Barcelona Graduate School of Mathematics, Campus de Bellaterra,Edifici C, 08193 Bellaterra, Barcelona, Spain Complexity Science Hub Vienna, Josefst¨adter Stra β e 39, 1080 Vienna, Austria (Dated: April 12, 2018) Abstract
Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finitesystems. Here we use the analogous concept of finite-time scaling to describe the bifurcationdiagram at finite times in discrete dynamical systems. We analytically derive finite-time scalinglaws for two ubiquitous transitions given by the transcritical and the saddle-node bifurcation,obtaining exact expressions for the critical exponents and scaling functions. One of the scalinglaws, corresponding to the distance of the dynamical variable to the attractor, turns out to beuniversal. Our work establishes a new connection between thermodynamic phase transitions andbifurcations in low-dimensional dynamical systems, and opens new avenues to identify the natureof dynamical shifts in systems for which only short time series are available.
PACS numbers: ntroduction Bifurcations separate qualitatively different dynamics in dynamical systems as one ormore parameters are changed. Bifurcations have been mathematically characterized inelastic-plastic materials [1], electronic circuits [2], or in open quantum systems [3]. Also,bifurcations have been theoretically described in population dynamics [4–6], in socioecolog-ical systems [7, 8], as well as in fixation of alleles in population genetics and computer viruspropagation, to name a few examples [9, 10]. More important, bifurcations have been iden-tified experimentally in physical [11–14], chemical [15, 16], and biological systems [17, 18].The simplest cases of local bifurcations, such as the transcritical and the saddle-node bifur-cations, only involve changes in the stability and existence of fixed points.Although, strictly speaking, attractors (such as stable fixed points) are only reached inthe infinite-time limit, some studies near local bifurcations have focused on the dependenceof the characteristic time needed to approach the attractor as a function of the distance of thebifurcation parameter to the bifurcation point. For example, for the transcritical bifurcationit is known that the transient time, τ , diverges as a power law [19], as τ ∼ | µ − µ c | − , with µ and µ c being, respectively, the bifurcation parameter and the bifurcation point, while for thesaddle-node bifurcation this time goes as τ ∼ | µ − µ c | − / [20] (see [12] for an experimentalevidence of this power law in an electronic circuit).Thermodynamic phase transitions [21, 22], where an order parameter sudden changes itsbehavior as a response to small changes in one or several control parameters, can be con-sidered as bifurcations. Three important peculiarities of thermodynamic phase transitionswithin this picture are that the order parameter has to be equal to zero in one of the phasesor regimes, that the bifurcation does not arise (in principle) from a simple low-dimensionaldynamical system but from the cooperative effects of many-body interactions, and thatat thermodynamic equilibrium there is no (macroscopic) dynamics at all. Non-equilibriumphase transitions [23, 24] are also bifurcations and share these characteristics, except the lastone. Particular interest has been paid to second-order phase transitions, where the suddenchange of the order parameter is nevertheless continuous and associated to the existence ofa critical point.A key ingredient of second-order phase transitions is finite-size scaling [25, 26], which de-scribes how the sharpness of the transition emerges in the thermodynamic (infinite-system)2imit. For instance, if m is magnetization (order parameter), T temperature (control param-eter), and ℓ system size, for zero applied field and close to the critical point the equation ofstate can be approximated as a finite-size scaling law, m ≃ ℓ β/ν g [ ℓ /ν ( T − T c )] , (1)with T c the critical temperature, β and ν two critical exponents, and g [ y ] a scaling functionfulfilling g [ y ] ∝ ( − y ) β for y → −∞ and g [ y ] → y → ∞ .It has been recently shown that the Galton-Watson branching process (a fundamen-tal stochastic model for the growth and extinction of populations, nuclear reactions, andavalanche phenomena) can be understood as displaying a second-order phase transition [27]with finite-size scaling [28, 29]. In a similar spirit, in this article we show how bifurcationsin one-dimensional discrete dynamical systems display “finite-time scaling”, analogous tofinite-size scaling with time playing the role of system size. We analyze the transcriticaland the saddle-node bifurcations for iterated maps and find analytically well-defined scalingfunctions that generalize the bifurcation diagrams for finite times. The sharpness characterof each bifurcation is naturally recovered in the infinite-time limit. As a by-product, wederive the power-law divergence of the characteristic time τ when µ is kept constant, off ofcriticality [19, 20]. UNIVERSAL CONVERGENCE TO ATTRACTIVE FIXED POINTS
Let us consider a one-dimensional discrete dynamical system, or iterated map, x n +1 = f ( x n ) , where x is a real variable, f ( x ) is a univariate function (which will depend on somenon-explicit parameters) and n being discrete time. Let us consider also that the map hasan attractive (i.e., stable) fixed point at x = q , for which f ( q ) = q, and that x belongs tothe domain of attraction of the fixed point (more conditions on x later). It is important toremember that attractiveness in discrete-time systems is characterized by | f ′ ( q ) | < x n = f n ( x ) for large but finite n , where f n ( x )denotes the iterated application of the map n times. Naturally, for sufficient large n , f n ( x )will be close to the attractive fixed point q and we will be able to expand f ( f n ( x )) around3 , so, f n +1 ( x ) = f ( f n ( x )) = q + M ( f n ( x ) − q )+ C ( f n ( x ) − q ) + O ( q − f n ( x )) , (2)with M = f ′ ( q ) and C = f ′′ ( q )2 . Rearranging and introducing the variable c n +1 , the inverse of the distance to the fixed pointat iteration n + 1, we arrive to c n +1 = 1 q − f n +1 ( x ) = c n M + CM + O ( q − f n ( x ))(we may talk about a distance because, in practice, we calculate the difference in such a waythat it is always positive). Iterating this transformation ℓ times we get c n + ℓ = c n M ℓ + C (1 − M ℓ ) M ℓ +1 (1 − M ) , to the lowest order [28]. Introducing the new variable z = ℓ ( M − ℓ large onerealizes that the second term in the sum grows linearly with ℓ and overcomes the first one,and so, c n + ℓ ≃ Cℓ ( e z − e − z /z. Next, considering ℓ much larger than n , so that n + ℓ ≃ ℓ ,we get a scaling law for the dependence of the distance to the attractor on M and ℓ , q − f ℓ ( x ) = 1 c ℓ ≃ Cℓ G ( ℓ ( M − , (3)with scaling function G ( z ) = ze z e z − . (4)This result has also been obtained in Ref. [28] for the Galton-Watson model, leading us torealize that this model is governed by a transcritical bifurcation (but restricted to x = 0).The scaling law (3) means that any attractor of a one-dimensional map is approachedin the same universal way, as long as a Taylor expansion as the one in Eq. (2) holds, inparticular if f ′′ ( q ) = 0. So we may talk about a “universality class”. The idea is thatfor different number of iterations ℓ one is able to find a value of M (which depends onthe parameters of f ( x )) for which z = ℓ ( M −
1) keeps constant and therefore the rescaleddifference with respect the point M = 1 is constant as well. Note that, in order to have afinite z , as ℓ is large, M = f ′ ( q ) will be close to 1, so we will be close to a bifurcation point,4orresponding to M = 1 (where the attractive fixed point will lose its stability). Due to thisfact, in the scaling law we can replace C by its value at the bifurcation point C ∗ , so, wewrite C = C ∗ in Eq. (3).In principle, the value of the initial value x is not of fundamental importance, we couldtake for instance x = f ( x ) as the initial condition instead, and we would get the sameresult just replacing ℓ by ℓ −
1. For very large ℓ this difference plays no role ( ℓ ≃ ℓ − ℓ grows, the influence of the initial condition gets lost, as we can make ℓ aslarge as desired. But on the other hand, x has to fulfill x < q if C ∗ > x > q if C ∗ <
0, in the same way that all the iterations x n (i.e., all the iterations have to be on thesame “side” of q ). The scaling law implies that plotting [ q − f ℓ ( x )] C ∗ ℓ versus ℓ ( M −
1) hasto yield a data collapse of the curves corresponding to different values of ℓ onto the scalingfunction G .For example, for the logistic (lo) map [20], f ( x ) = f lo ( x ) = µx (1 − x ), a transcriticalbifurcation takes place at µ = 1 and the attractor is at q = 0 for µ ≤ q = 1 − /µ for µ ≥
1, which leads to M lo = f ′ lo ( q ) = µ for µ ≤ M lo = 2 − µ for µ ≥
1, and alsoto C lo ∗ = −
1. Therefore, z = ℓ ( M −
1) = − ℓ | µ − | and f ℓlo ( x ) − q ≃ ℓ − G ( − ℓ | µ − | ) , for x > q . Thus, in order to verify the collapse of the curves onto the function G , one needs torepresent [ f ℓlo ( x ) − q ] ℓ versus − ℓ | µ − | , or, if one wants to see separately the two regimes, µ ≷
1, versus y = ℓ ( µ − G ( −| y | ).Figure 1(b) shows precisely this; the nearly perfect data collapse for large ℓ is the indicationof the fulfillment of the finite-time scaling law. For comparison, Fig. 1(a) shows the samedata with no rescaling (i.e., just the distance to the attractor as a function of the bifurcationparameter µ ).If one prefers the normal form of the transcritical (tc) bifurcation (in the discrete case), f tc ( x ) = (1 + µ ) x − x , then the bifurcation takes place at µ = 0 (with q = 0 for µ ≤ q = µ for µ ≥ z = − ℓ | µ | (or for y = ℓµ inorder to separate the two regimes, as shown overimposed in Fig. 1(b), again with very goodagreement).For the saddle-node (sn) bifurcation (also called fold or tangent bifurcation [30]), in itsnormal form (discrete system), f sn ( x ) = µ + x − x , the attractor is at q = √ µ (only for µ > µ = 0, which leads to M sn = 1 − √ µ and C sn ∗ = −
1. The5caling law can be written as f ℓsn ( x ) − √ µ ≃ ℓ G ( − ℓ √ µ ) . (5)To see the data collapse onto the function G one must represent [ f ℓsn ( x ) − √ µ ] ℓ versus z = − ℓ √ µ (or versus y = − z for clarity sake, as shown also in Fig. 1(b)). If one prefers ahorizontal axis linear in µ , one may define z = −√ u , and then f ℓsn ( x ) − √ µ ≃ F (4 ℓ µ ) /ℓ ,with a transformed scaling function F ( u ) = G ( −√ u ) = √ u/ ( e √ u − , and then use u = − z = 4 ℓ µ for the horizontal axis of the rescaled plot.Although the key idea of the finite-time scaling law, Eq. (3), is to compare the solutionof the system at “corresponding” values of ℓ and µ (such that z is constant, in a sort of lawof corresponding states [21]), the law can be used as well at fixed µ . At the bifurcation point( µ = µ c , so z = 0), we find that the distance to the attractor decays hyperbolically, i.e., | f ℓ ( x ) − q | = | C ∗ ℓ | − , as it is well known, see for instance Ref. [19]. Out of the bifurcationpoint, for non-vanishing µ − µ c we have z → −∞ (as ℓ → ∞ ) and then G ( z ) → e − z ,which leads to f ℓ ( x ) − q ≃ ℓ − e − z ≃ e − ℓ/τ , where, from the expression for z , we findthat the characteristic time τ diverges as τ = 1 / | µ − µ c | for the transcritical bifurcation(both in normal form and in the logistic form) and as τ = 1 / (2 √ µ − µ c ) for the saddle-nodebifurcation (with µ c = 0 in the normal form) [12]. These laws, mentioned in the introduction,have been reported in the literature as scaling laws [20], but in order to avoid confusion wesuggest to call them power-law divergence laws. Note that this sort of law arises because G ( z ) is asymptotically exponential; in contrast, the equivalent of G ( z ) in the equation ofstate of a magnetic system in the thermodynamic limit is a power law, which leads to theCurie-Weiss law [31]. SCALING LAW FOR THE DISTANCE TO THE FIXED POINT AT BIFURCA-TION FOR THE ITERATED VALUE x n IN THE TRANSCRITICAL BIFURCA-TION
In some cases, the distance between f ℓ ( x ) and some constant value of reference will beof more interest than the distance to the attractive fixed point q , as the value of q maychange with the bifurcation parameter. For the transcritical bifurcation we have two fixedpoints, q and q , and they collide and interchange their character (attractive to repulsive,6nd vice versa) at the bifurcation point. Let us consider that q is constant independentlyof the bifurcation parameter (naturally, q will not be constant), and that “below” thebifurcation point q is attractive and q is repulsive, and vice versa “above” the bifurcation.We will be interested in the distance between q and f ℓ ( x ), i.e., q − f ℓ ( x ), which, belowthe bifurcation point corresponds to the quantity calculated previously in Eq. (3), but notabove. The reason is that, in there, q was an attractor, but now q can be attractive orrepulsive. Note that, without loss of generality, we can refer q − f ℓ ( x ) as the distance of f ℓ ( x ) to the “origin”.Following Ref. [28], we need a relation between both fixed points when we are close tothe bifurcation point. As, in that case, q ≃ q , we can expand f ( q ) around q , to get f ( q ) = q = q + M ( q − q ) + C ( q − q ) + O ( q − q ) , which leads directly to M − C ( q − q ) , (6)to the lowest order in ( q − q ). Naturally, M = f ′ ( q ) and C = f ′′ ( q ) /
2. We will alsoneed a relation between M = f ′ ( q ) and M . Expanding f ′ ( q ) around q , f ′ ( q ) = M = M + 2 C ( q − q ) + O ( q − q ) , which, using Eq. (6), leads to M − − M , (7)to the lowest order.Now let us write q − f ℓ ( x ) = q − q + q − f ℓ ( x ). For q − q we will apply Eq. (6), andfor q − f ℓ ( x ) we can apply Eq. (3), as q is of attractive nature “above” the bifurcationpoint; then q − f ℓ ( x ) ≃ M − C + 1 C ℓ G ( ℓ ( M − C = f ′′ ( q ) / y = ℓ ( M −
1) we get (with the form of the scalingfunction, Eq. (4)), q − f ℓ ( x ) ≃ yC ℓ + 1 C ℓ (cid:18) ze z e z − (cid:19) . Using Eq. (7) one realizes that z = ℓ ( M −
1) = − ℓ ( M −
1) = − y (so, the y introducedhere is the same y introduced above), and therefore, q − f ℓ ( x ) ≃ C ∗ ℓ (cid:18) y + − ye − y e − y − (cid:19) = 1 C ∗ ℓ ye y e y − , C = C = C ∗ , to the lowest order, with C ∗ the value at thebifurcation point. Therefore, we obtain the same scaling law as in the previous section: q − f ℓ ( x ) ≃ C ∗ ℓ G ( y ) , (8)with the same scaling function G ( y ) as in Eq. (4), although the rescaled variable y is differenthere ( y = z , in general). This is possible thanks to the property y + G ( − y ) = G ( y ) that thescaling function verifies. Note that the scaling law (1) has the same form as the finite-timescaling (8) and we can identify β = ν = 1.Note also that we can identify M = f ′ ( q ) with a bifurcation parameter, as it is M < M = 1) and M > M defined in the previoussection cannot be a bifurcation parameter as it is never above 1, due to the fact that it isdefined with respect the attractive fixed point).For the transcritical bifurcation of the logistic map we identify q = 0 and M = µ , so y = ℓ ( µ − q = 0 but M = µ + 1,so y = ℓµ . Consequently, Fig. 2(a) shows f ℓ ( x ) (the distance to q = 0) as a functionof µ , for the logistic map and different ℓ , whereas Fig. 2(b) shows the same results underthe corresponding rescaling, together with analogous results for the normal form of thetranscritical bifurcation. The data collapse supports the validity of the scaling law (8) withscaling function given by Eq. (4). SCALING LAW FOR THE ITERATED VALUE x n IN THE SADDLE-NODE BI-FURCATION
Coming back to the saddle-node bifurcation, from Eq. (5) we can isolate the ℓ − th iterateto get, f ℓ ( x ) ≃ ℓ (cid:20) ℓ √ µ G ( − ℓ √ µ ) (cid:21) = 1 ℓ H ( y )with y = − z = 2 ℓ √ µ and H ( y ) = y ( e y + 1)( e y − − / . Therefore, the representation of ℓf ℓ ( x ) versus 2 ℓ √ µ unveils the shape of the scaling function H . In terms of u = y = 4 ℓ µ , f ℓ ( x ) ≃ ℓ I ( u ) , with I ( u ) = H ( √ u ) = √ u e √ u + 1)( e √ u − , (9)and so, ℓf ℓ ( x ) against 4 ℓ µ leads to the collapse of the data onto the scaling function I ( u ),as shown in Fig. 3. Comparison with the finite-size scaling law (1) allows one to establish β = ν = 1 / µ , not √ µ ).8 ONCLUSIONS
By means of scaling laws, we have made clear an analogy between bifurcations andphase transitions, with a direct correspondence between, on the one hand, the bifurcationparameter, the bifurcation point, and the finite-time solution f ℓ ( x ), and, on the other hand,the control parameter, the critical point, and the finite-size order parameter. However, inphase transitions, the sharp change of the order parameter at the critical point arises in thelimit of infinite system size; in contrast, in bifurcations, the sharpness at the bifurcation pointshows up in the infinite-time limit, ℓ → ∞ . So, finite-size scaling in one case corresponds tofinite-time scaling in the other.In addition, we have also been able to derive the power-law divergence of the transienttime to reach the attractor off of criticality [12, 19, 20], and also conclude that the results ofRef. [28] can be directly understood from the transcritical bifurcation underlying the Galton-Watson branching process. Moreover, by using numerical simulations we have tested thatthe finite-time scaling laws also hold for dynamical systems continuous in time, as well asfor the pitchfork bifurcation in discrete time (although with different exponents and scalingfunction in this case). Let us mention that the use of the finite-time scaling concept by otherauthors does not correspond with ours. For instance, although Ref. [32] presents a scalinglaw for finite times, the corresponding exponent ν there turns to be negative, which is notin agreement with the genuine finite-size scaling around a critical point.Our results may also allow to identify the nature of bifurcations in systems for whichinformation is limited to short transients, such as in ecological systems. In this way, thescaling relations established in this article could be used as warning signals [33] to anticipatethe nature of collapses or changes in ecosystems [5, 6, 33–35] (due to, e.g., transcritical orsaddle-node bifurcations) and in other dynamical suffering dynamical shifts. [1] M. K. Nielsen and H. L. Schreyer. Bifurcations in elastic-plastic materials. Int. J. of SolidsStructures , 30:521–544, 1993.[2] S. Kahan and A. C. Sicardi-Schifino. Homoclinic bifurcations in Chua’s circuit.
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Acknowledgements
We have received funding from “La Caixa” Foundation and through the “Mar´ıa deMaeztu” Programme for Units of Excellence in R&D (MDM-2014-0445), as well asfrom projects FIS2015-71851-P and MTM2014-52209-C2-1-P from the Spanish MINECO,from 2014SGR-1307 (AGAUR), and from the CERCA Programme of the Generalitat deCatalunya.
Author contributions statement
All authors analysed and discussed the results. All authors reviewed the manuscript.
Additional information
Competing financial interests . The authors declare no competing interests.12 b FIG. 1: (a) Distance between the ℓ − th iteration of the logistic map (lo) and its attractor,as a function of the bifurcation parameter µ , for different values of ℓ . (b) The same data underrescaling (decreasing the density of points, for clarity sake), together with data from the transcriticalbifurcation in normal form (tc) and the saddle-node bifurcation (sn). The collapse of the curves intoa single one validates the scaling law, Eq. (3), and its universal character. The scaling function isin agreement with G ( −| y | ). Note that the initial condition x is taken uniformly randomly between0.25 and 0.75, which is inside the range necessary for all the iterations to be above the fixed point.This range is, below the bifurcation point, 0 < x < < x < µ (tc), and, above,1 − µ − < x < µ − (lo), µ < x < √ µ < x < − √ µ (sn). b FIG. 2: (a) ℓ − th iteration of the logistic map as a function of the bifurcation parameter µ , fordifferent values of ℓ . Same initial conditions as in previous figure. (b) Same data under rescaling(decreasing density of points), plus analogous data coming from the transcritical bifurcation innormal form. The data collapse shows the validity of the scaling law, Eq. (8), with scalingfunction G ( y ) from Eq. (4). b FIG. 3: (a) Same as Fig. 2(a) but for the saddle-node bifurcation in normal form. (b) Rescalingof the same data (with decreased density of points). The data collapse supports the scaling lawand the scaling function I ( u ) given by Eq. (9).) given by Eq. (9).