Abstract
Finitesize scaling is a key tool in statistical physics, used to infer critical behavior in finite systems. Here we have made use of the analogous concept of finitetime scaling to describe the bifurcation diagram at finite times in discrete (deterministic) dynamical systems. We analytically derive finitetime scaling laws for two ubiquitous transitions given by the transcritical and the saddlenode bifurcation, obtaining exact expressions for the critical exponents and scaling functions. One of the scaling laws, corresponding to the distance of the dynamical variable to the attractor, turns out to be universal, in the sense that it holds for both bifurcations, yielding the same exponents and scaling function. Remarkably, the resulting scaling behavior in the transcritical bifurcation is precisely the same as the one in the (stochastic) GaltonWatson process. Our work establishes a new connection between thermodynamic phase transitions and bifurcations in lowdimensional dynamical systems, and opens new avenues to identify the nature of dynamical shifts in systems for which only short time series are available.
Introduction
Bifurcations separate qualitatively different dynamics in dynamical systems as one or more parameters are changed. Bifurcations have been mathematically characterized in elasticplastic materials^{1}, electronic circuits^{2}, or in open quantum systems^{3}. Also, bifurcations have been theoretically described in population dynamics^{4,5,6}, in socioecological systems^{7,8}, as well as in fixation of alleles in population genetics and computer virus propagation, to name a few examples^{9,10}. More importantly, bifurcations have been identified experimentally in physical^{11,12,13,14}, chemical^{15,16}, and biological systems^{17,18}. The simplest cases of local bifurcations, such as the transcritical and the saddlenode bifurcations, only involve changes in the stability and existence of fixed points.
Although, strictly speaking, attractors (such as stable fixed points) are only reached in the infinitetime limit, some studies near local bifurcations have focused on the dependence of the characteristic time needed to approach the attractor as a function of the distance of the bifurcation parameter to the bifurcation point. For example, for the transcritical bifurcation it is known that the transient time, τ, diverges as a power law^{19}, as τ ~ μ − μ_{c}^{−1}, with μ and μ_{c} being the bifurcation parameter and the bifurcation point, respectively, while for the saddlenode bifurcation^{20} this time goes as τ ~ μ − μ_{c}^{−1/2} (see^{12} for an experimental evidence of this power law in an electronic circuit).
Thermodynamic phase transitions^{21,22}, where an order parameter suddenly changes its behavior as a response to small changes in one or several control parameters, can be considered as bifurcations^{23}. Three important peculiarities of thermodynamic phase transitions within this picture are that the order parameter has to be equal to zero in one of the phases or regimes, that the bifurcation does not arise (in principle) from a simple lowdimensional dynamical system but from the cooperative effects of manybody interactions, and that at thermodynamic equilibrium there is no (macroscopic) dynamics at all. Nonequilibrium phase transitions^{24,25} are also bifurcations and share these characteristics, except the last one. Particular interest has been paid to secondorder phase transitions, where the sudden change of the order parameter is nevertheless continuous and associated to the existence of a critical point.
A key ingredient of secondorder phase transitions is finitesize scaling^{26,27}, which describes how the sharpness of the transition emerges in the thermodynamic (infinitesystem) limit. For instance, if m is magnetization (order parameter), T temperature (control parameter), and \(\ell \) system’s size, then for zero applied field and close to the critical point, the equation of state can be approximated as a finitesize scaling law,
with T_{c} the critical temperature, β and ν two critical exponents, and g[y] a scaling function fulfilling g[y] ∝ (−y)^{β} for y → −∞ and g[y] → 0 for y → ∞.
It has been recently shown that the GaltonWatson branching process (a fundamental stochastic model for the growth and extinction of populations, nuclear reactions, and avalanche phenomena) can be understood as displaying a secondorder phase transition^{28} with finitesize scaling^{29,30}. In a similar spirit, in this article we show how bifurcations in onedimensional discrete dynamical systems display “finitetime scaling”, analogous to finitesize scaling with time playing the role of system size. We analyze the transcritical and the saddlenode bifurcations for iterated maps and find analytically welldefined scaling functions that generalize the bifurcation diagrams for finite times. The sharpness of each bifurcation is naturally recovered in the infinitetime limit. The finitesize behavior of the GaltonWatson process becomes just one instance of our general finding for the transcritical bifurcation. And as a byproduct, we derive the powerlaw divergence of the characteristic time τ when μ is kept constant, off criticality^{19,20}.
Universality of Convergence to Attractive Fixed Points
In this paper, we consider a onedimensional 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 some nonexplicit parameters) and n is discrete time. It is assumed that the map has an attractive (i.e., stable) fixed point at x = q, for which f(q) = q, with f ′(q) < 1, where the prime denotes the derivative^{20}. Moreover, the initial condition, x_{0}, is assumed to belong to the basin of attraction of the fixed point. Additional conditions on x_{0} will be discussed below.
We are interested in the behavior of x_{n} = f ^{n}(x_{0}) for large but finite n, where f ^{n}(x_{0}) denotes the iterated application of the map n times. Naturally, for sufficiently large n, f ^{n}(x_{0}) will be close to the attractive fixed point q and we will be able to expand f(f ^{n}(x_{0})) around q, resulting in
with
By rearranging and introducing the variable c_{n+1}, the inverse of the “distance” to the fixed point at iteration n + 1, we arrive at
(we may talk about a distance because we calculate the difference in such a way that it is always positive). Iterating this transformation \(\ell \) times leads to
where only the lowestorder terms have been considered^{29}. When the variable z, defined as \(z={\ell }^{\alpha }(M1)\), is kept finite (with \(\ell \to \infty \) and M → 1) a nontrivial limit of the previous expression exists if α = 1. It is found that the righthand side of the expression is dominated by the second term, which grows linearly with \(\ell \). Therefore, for large \(\ell \), we arrive at \({c}_{n+\ell }\simeq C\ell ({e}^{z}1){e}^{z}/z\), and taking the inverse, we obtain
with scaling function
Observe that the sequence \({\{{f}^{\ell }({x}_{0})\}}_{\ell =1}^{\infty }\) is convergent and thus, for \(\ell \) large enough with respect to n, \({f}^{\ell +n}({x}_{0})\simeq {f}^{\ell }({x}_{0})\). Consequently,
This is exactly the same result as the one derived in ref.^{29} for the GaltonWatson model, leading to the realization that this model is governed by a transcritical bifurcation (but restricted to a fixed initial condition x_{0} = 0).
The scaling law (4) means that any attractor of a onedimensional map is approached in the same universal way, as long as a Taylor expansion as the one in Eq. (2) holds, in particular if f ″(q) ≠ 0. In this sense one may talk about a “universality class”, as displayed in Fig. 1. The idea is that for each value of the number of iterations \(\ell \) one has to pick a value of M (which depends on the parameters of f(x)) for which \(z=\ell (M1)\) (the rescaled difference concerning the point M = 1) remains constant. Note that, in order to have a finite z, as \(\ell \) is large, M = f ′(q) will be close to 1, implying that the system will be close to its bifurcation point, corresponding to M = 1 (where the attractive fixed point will lose its stability). Therefore, in the scaling law, C can be replaced by its value at the bifurcation point \({C}_{\ast }\), so, we write \(C={C}_{\ast }\) in Eq. (4).
In principle, the value of the initial condition x_{0} is not of fundamental importance. The same results can be obtained, for example, by taking x_{1} = f(x_{0}) as the initial condition and then replacing \(\ell \) by \(\ell 1\) because, for very large \(\ell \), \(\ell \simeq \ell 1\). Therefore, as \(\ell \) grows, memory of the initial condition is erased, as \(\ell \) can be made as large as desired. However, x_{0} has to fulfill x_{0} < q if \({C}_{\ast } > 0\) and x_{0} > q if \({C}_{\ast } < 0\), in the same way that all the iterations x_{n} must also satisfy these inequalities (i.e., all the iterations have to be on the same “side” of the point q, see the caption of Fig. 1 for the concrete conditions). The scaling law implies that plotting \([q{f}^{\ell }({x}_{0})]{C}_{\ast }\ell \) as a function of \(\ell (M1)\) must yield a data collapse of the curves corresponding to different values of \(\ell \) onto the scaling function G.
For example, for the logistic (lo) map^{20}, f(x) = f_{lo}(x) = μx(1 − x), a transcritical bifurcation takes place at μ = 1 and the attractor is at q = 0 for μ ≤ 1 and at q = 1 − 1/μ for μ ≥ 1, which leads to \({M}_{lo}={f}_{lo}^{^{\prime} }(q)=\mu \) for μ ≤ 1 and M_{lo} = 2 − μ for μ ≥ 1, and also to \({C}_{lo\ast }=\,1\). Therefore, \(z=\ell (M1)=\,\ell \mu 1\) and \({f}_{lo}^{\ell }({x}_{0})q\simeq {\ell }^{1}\)\(G(\ell \mu 1)\) for x_{0} > q. Thus, in order to verify the collapse of the curves onto the function G, the quantity \([\,{f}_{lo}^{\ell }({x}_{0})q]\ell \) must be displayed as a function of −\(\ell \mu 1\); if the resulting plot does not change with the value of \(\ell \) the scaling law can be considered to hold. Alternatively, the two regimes \(\mu \,\gtrless \,1\), can be observed by writing \([\,{f}_{lo}^{\ell }({x}_{0})q]\ell \) as a function of \(y=\ell (\mu 1)\). In the latter case the scaling function turns out to be G(−y). Figure 1(b) shows precisely this; the nearly perfect data collapse for large \(\ell \) is the indication of the fulfillment of the finitetime scaling law. For comparison, Fig. 1(a) shows the same data with no rescaling (i.e., just the distance to the attractor as a function of the bifurcation parameter μ). In the case of the normal form of the transcritical (tc) bifurcation (in the discrete case), f_{tc}(x) = (1 + μ)x − x^{2}, the bifurcation takes place at μ = 0 (with q = 0 for μ ≤ 0 and q = μ for μ ≥ 0). This leads to exactly the same behavior for \(z=\,\ell \mu \) (or for \(y=\ell \mu \) in order to separate the two regimes, as shown overimposed in Fig. 1(b), again with very good agreement).
For the saddlenode (sn) bifurcation (also called fold or tangent bifurcation^{31}), in its normal form (discrete system), f_{sn}(x) = μ + x − x^{2}, the attractor is at \(q=\sqrt{\mu }\) (only for μ > 0), so the bifurcation is at μ = 0, which leads to \({M}_{sn}=12\sqrt{\mu }\) and \({C}_{sn\ast }=\,1\). The scaling law can be written as
To see the data collapse onto the function G one must represent \([{f}_{sn}^{\ell }({x}_{0})\sqrt{\mu }]\ell \) as a function of \(z=\,2\ell \sqrt{\mu }\) (or as a function of y = −z for clarity sake, as shown also in Fig. 1(b)). In order to create a horizontal axis that is linear in μ, we first define \(z=\,\sqrt{u}\), in which case \({f}_{sn}^{\ell }({x}_{0})\sqrt{\mu }\simeq F(4{\ell }^{2}\mu )/\ell \), with a transformed scaling function \(F(u)=G(\,\,\sqrt{u})=\sqrt{u}/({e}^{\sqrt{u}}1),\) and then use \(u=\,{z}^{2}=4{\ell }^{2}\mu \) for the horizontal axis of the rescaled plot.
Although the key idea of the finitetime scaling law, Eq. (4), is to compare the solution of the system at “corresponding” values of \(\ell \) and μ (such that z is constant, in a sort of law of corresponding states^{21}), the law can also be used 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., \(\,{f}^{\ell }({x}_{0})q={C}_{\ast }\ell {}^{1}\), as it is well known, see for instance ref.^{19}. Out of the bifurcation point, for nonvanishing μ − μ_{c} we have z → −∞ (as \(\ell \to \infty \)) and then G(z) → e^{−z}, which leads to \({f}^{\ell }({x}_{0})q\simeq {\ell }^{1}\)\({e}^{z}\simeq {e}^{\ell /\tau }\), where, from the expression for z, we find that 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 \(\tau =1/(2\sqrt{\mu {\mu }_{c}})\) for the saddlenode bifurcation (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 we propose calling them powerlaw divergence laws, and keep the term scaling law for behaviors such as those in Eqs (1), (4) and (5). Note that this sort of law arises because G(z) is asymptotically exponential for z → −∞; in contrast, the equivalent of G(z) in the equation of state of a magnetic system in the thermodynamic limit is a power law, which leads to the CurieWeiss law^{32}.
Scaling Law for the Distance to the Fixed Point at Bifurcation in the Transcritical Bifurcation
In some cases, the distance between \({f}^{\ell }({x}_{0})\) and some constant value of reference will be of more interest than the distance to the attractive fixed point q, as the value of q may change with the bifurcation parameter. For the transcritical bifurcation we have two fixed points, q_{0} and q_{1}, and they collide and interchange their character (attractive to repulsive, and vice versa) at the bifurcation point. It will be assumed that q_{0} is constant independent of the bifurcation parameter (naturally, q_{1} will not be constant), and that “below” the bifurcation point q_{0} is attractive and q_{1} is repulsive, and vice versa “above” the bifurcation. We will be interested in the distance between q_{0} and \({f}^{\ell }({x}_{0})\), i.e., \({q}_{0}{f}^{\ell }({x}_{0})\), which, below the bifurcation point corresponds to the quantity calculated previously in Eq. (4), but not above. The reason is that, there, q was an attractor, but now q_{0} can be attractive or repulsive. Note that, without loss of generality, we can refer \({q}_{0}{f}^{\ell }({x}_{0})\) as the distance of \({f}^{\ell }({x}_{0})\) to the “origin”.
Following ref.^{29}, we seek a relationship between both fixed points when the system is close to the bifurcation point. As, in that case, \({q}_{1}\simeq {q}_{0}\), we can expand f(q_{1}) around q_{0} to obtain
which leads directly to
to the lowest order in (q_{1} − q_{0}). Naturally, M_{0} = f ′(q_{0}) and C_{0} = f ″(q_{0})/2. We also seek a relationship between M_{1} = f ′(q_{1}) and M_{0}. Expanding f ′(q_{1}) around q_{0}, \(f^{\prime} ({q}_{1})={M}_{1}={M}_{0}+2{C}_{0}({q}_{1}{q}_{0})+{\mathscr{O}}{({q}_{1}{q}_{0})}^{2},\) which, using Eq. (6), leads to
to the first order in (q_{1} − q_{0}).
We now write \({q}_{0}{f}^{\ell }({x}_{0})={q}_{0}{q}_{1}+{q}_{1}{f}^{\ell }({x}_{0})\). For q_{0} − q_{1} we will apply Eq. (6), and for \({q}_{1}{f}^{\ell }({x}_{0})\) we can apply Eq. (4), as q_{1} is of attractive nature “above” the bifurcation point; then
(with C_{1} = f ″(q_{1})/2), and defining \(y=\ell ({M}_{0}1)\) we obtain (with the form of the scaling function, Eq. (3)),
Using Eq. (7) it can be shown that
(so, the y introduced here is the same y introduced in the previous section), and therefore,
where we have also used that \({C}_{1}={C}_{0}={C}_{\ast }\), to the lowest order, with \({C}_{\ast }\) being the value at the bifurcation point. Therefore, we obtain the same scaling law as in the previous section:
with the same scaling function G(y) as in Eq. (3), although the rescaled variable y is different here (y ≠ z, in general). This is possible thanks to the property y + G(−y) = G(y) that the scaling function satisfies. Note that the scaling law (1) has the same form as the finitetime scaling (9) with y given by Eq. (8), and therefore we can identify β = ν = 1. Note also that we can identify M_{0} = f ′(q_{0}) with a bifurcation parameter, as it is M_{0} < 1 “below” the bifurcation point (M_{0} = 1) and M_{0} > 1 “above”. In fact, M_{0} can be considered as a natural bifurcation parameter, as the scaling law (4) expressed in terms of M_{0} becomes universal. M defined in the previous section cannot be a bifurcation parameter as it is never above one because it is defined with respect to the attractive fixed point.
For the transcritical bifurcation of the logistic map we identify q_{0} = 0 and M_{0} = μ, so \(y=\ell (\mu 1)\). For the normal form of the transcritical bifurcation, q_{0} = 0 but M_{0} = μ + 1, so \(y=\ell \mu \). Consequently, Fig. 2(a) shows \({f}^{\ell }({x}_{0})\) (the distance to q_{0} = 0) as a function of μ, for the logistic map and different \(\ell \), whereas Fig. 2(b) shows the same results under the corresponding rescaling, together with analogous results for the normal form of the transcritical bifurcation. The data collapse supports the validity of the scaling law (9) with scaling function given by Eq. (3).
Scaling Law for the Iterated Value x _{n} in the SaddleNode Bifurcation
In the case of a saddlenode bifurcation, the \(\ell \)–th iterate can be isolated from Eq. (5) to obtain
with \(y=\,z=2\ell \sqrt{\mu }\) and H(y) = y(e^{y} + 1)(e^{y} − 1)^{−1}/2. Therefore, the representation of \(\ell {f}^{\ell }({x}_{0})\) as a function of \(2\ell \sqrt{\mu }\) unveils the shape of the scaling function H. In terms of \(u={y}^{2}=4{\ell }^{2}\mu \),
and, therefore, plotting \(\ell {f}^{\ell }({x}_{0})\) as a function of \(4{\ell }^{2}\mu \) must lead to the collapse of the data onto the scaling function I(u), as shown in Fig. 3. Comparison with the finitesize scaling law (1) allows one to establish β = ν = 1/2 for this bifurcation (and bifurcation parameter μ, not \(\sqrt{\mu }\)).
Conclusions
By means of scaling laws, we have made a clear analogy between bifurcations and phase transitions^{23}, with a direct correspondence between, on the one hand, the bifurcation parameter, the bifurcation point, and the finitetime solution \({f}^{\ell }({x}_{0})\), and, on the other hand, the control parameter, the critical point, and the finitesize order parameter. However, in phase transitions, the sharp change of the order parameter at the critical point arises in the limit of infinite system size; in contrast, in bifurcations, the sharpness at the bifurcation point shows up in the infinitetime limit, \(\ell \to \infty \). So, finitesize scaling in one case corresponds to finitetime scaling in the other. Specifically, we conclude that the finitesize scaling behavior derived in ref.^{29} can be directly understood from the transcritical bifurcation underlying the GaltonWatson branching process. It is remarkable that the critical behavior of such a stochastic process is governed by a bifurcation of a deterministic dynamical system.
Moreover, by using numerical simulations we have tested that the finitetime scaling laws also hold for dynamical systems continuous in time, as well as for the pitchfork bifurcation in discrete time, although with different exponents and scaling function in this case (this is due to the fact that the condition f ″(q) ≠ 0 does not hold). The use of the finitetime scaling concept by other authors does not correspond with ours. For instance, although ref.^{33} presents a scaling law for finite times, the corresponding exponent ν there turns out to be negative, which is not in agreement with the genuine finitesize scaling around a critical point. In addition, we have also been able to derive the powerlaw divergence of the transient time to reach the attractor out of criticality^{12,19,20}.
Our results could be useful for interpreting different types of fixed points found in renormalization group theory^{23}. Also, they might allow to idenfity the type of bifurcations in systems for which information is limited to short transients, such as in ecological systems. In this way, the scaling relations established in this article could be used as warning signals^{34} to anticipate the nature of collapses or changes in ecosystems^{5,6,34,35,36} (due to, e.g., transcritical or saddlenode bifurcations) and in other dynamical systems suffering shifts.
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Acknowledgements
We are very indebted to Matt Hennessy for his indepth revision of the manuscript and to Jordi GarcaOjalvo for clever suggestions. The research leading to these results has received funding from ‘la Caixa’ Foundation and from a MINECO grant awarded to the Barcelona Graduate School of Mathematics (BGSMath) under the ‘María de Maeztu’ Program (grant MDM20140445). This work has also been funded by projects FIS201571851P, MTM201452209C21P and MTM201786795C31P from the Spanish MINECO, from 2014SGR1307 (AGAUR), and from the CERCA Programme of the Generalitat de Catalunya. Josep Sardanyés has been also funded by a Ramón y Cajal Fellowship (RYC201722243).
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A.C. and L.A. performed the mathematical calculations. A.C., L.A. and J.S. carried out the numerical computations. All authors analysed and discussed the results. All authors reviewed the manuscript.
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Corral, Á., Sardanyés, J. & Alsedà, L. Finitetime scaling in local bifurcations. Sci Rep 8, 11783 (2018). https://doi.org/10.1038/s4159801830136y
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DOI: https://doi.org/10.1038/s4159801830136y
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