Abstract
Finite-size 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 finite-time scaling to describe the bifurcation diagram at finite times in discrete (deterministic) dynamical systems. We analytically derive finite-time scaling laws 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 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) Galton-Watson process. Our work establishes a new connection between thermodynamic phase transitions and bifurcations in low-dimensional dynamical systems, and opens new avenues to identify the nature of dynamical shifts in systems for which only short time series are available.
Highlights
Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finite systems
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 from a simple low-dimensional dynamical system but from the cooperative effects of many-body interactions, and that at thermodynamic equilibrium there is no dynamics at all
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 finite-time solution f (x0), and, on the other hand, the control parameter, the critical point, and the finite-size order parameter
Summary
(a) Distance between the –th iteration of the logistic map (lo) and its attractor, as a function of the bifurcation parameter μ, for different values of . In order to verify the collapse of the curves onto the function G, the quantity [ flo (x0) − q] must be displayed as a function of − |μ − 1|; if the resulting plot does not change with the value of the scaling law can be considered to hold. Scaling Law for the Distance to the Fixed Point at Bifurcation in the Transcritical. M0 can be considered as a natural bifurcation parameter, as the scaling law (4) expressed in terms of M0 becomes universal. Distance to q0 = 0) as a function of μ, for the logistic map and different , whereas Fig. 2(b) shows the same results under the corresponding rescaling, together with analogous results for the normal form of the transcritical bifurcation.
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