Finite-time scaling on low-dimensional map bifurcations

TL;DR

Finite-time scaling is extended from 1D maps to period-doubling and discontinuous bifurcations and generalized to 2D maps, including the 2D Chialvo neuron map. The authors introduce the finite-time susceptibility $\chi_l$ and finite-time Lyapunov exponent $\lambda_l$ and derive scaling laws governed by the local nonlinearity exponent $k$, enabling data collapses with scaling variable $z$ that depends on iteration length $l$ and proximity to bifurcation. Across 1D and 2D systems, temporal observables capture local instability and display critical-like structure, linking transient dynamics to effective order parameters and universal scaling forms. The results offer a unified perspective on dynamical criticality and suggest practical early-warning diagnostics for transitions in complex systems through reduced, low-dimensional descriptions.

Abstract

Recent work has introduced the concept of finite-time scaling to characterize bifurcation diagrams at finite times in deterministic discrete dynamical systems, drawing an analogy with finite-size scaling used to study critical behavior in finite systems. In this work, we extend the finite-time scaling approach in several key directions. First, we present numerical results for 1D maps exhibiting period-doubling bifurcations and discontinuous transitions, analyzing selected paradigmatic examples. We then define two observables, the finite-time susceptibility and the finite-time Lyapunov exponent, that also display consistent scaling near bifurcation points. The method is further generalized to special cases of 2D maps including the 2D Chialvo map, capturing its bifurcation between a fixed point and a periodic orbit, while accounting for discontinuities and asymmetric periodic orbits. These results underscore fundamental connections between temporal and spatial observables in complex systems, suggesting new avenues for studying complex dynamical behavior.

Figures (9)

• Figure 1: Results for the period-doubling bifurcation of the logistic map (at $\mu=3$). Panel a: solutions $x_l$ for different values of $l$, as a function of $\mu$. A dashed line shows the average of the solutions, $p_m=\frac{p_1+p_2}{2}=\frac{\mu+1}{2\mu}$. Panel b: Distance to the closest fixed point $p_{1/2}=\frac{1}{2\mu} \left(\mu+1\pm \sqrt{(\mu-3) (\mu+1)}\right)$ Panel c: collapse of the distance to the midpoint, $p_m$, as a function of $z\doteq (\mu-3) l$. Panel d: Collapse of the distance to the closest function. In panels (c) and (d), analytical expressions are plotted in black continuous line.
• Figure 2: Results for 1D Chialvo Map near the period-doubling transition at $\mu^*=3-\log(3)$, for which $p^*=3$. The left panel corresponds to the original data, while the right panel corresponds to the collapsed one.
• Figure 3: Finite-time susceptibility and Lyapunov exponent for Logistic and 1D Chialvo Maps. Panel a: Finite-time susceptibility $\chi_l$ as a function of $\mu$ for the Logistic map about $\mu=3$, for different values of the number of iterations, $l=2^{10},2^{11},.., 2^{18}$. A red dashed curve joins the maximum $\chi_l$ values and the corresponding $\mu^*$ values. Panel b: Absolute value of the Finite-time Lyapunov exponent for the Logistic map, for the same values of $l$ as in panel a. A blue dashed line joins the maximum values of $|\lambda|$. Notice that the $y$-scale is inverted. Panel c: Maximum values of $\chi_l$ and $|\lambda_l|^{-1}$ as a function of $l$. Black continuous lines show the expected behavior: $\max(\chi_l) \propto l^{\gamma/\nu}$ with $\gamma/\nu=2$ and $max(\lambda_l)=\min(|\lambda_l|) \propto \log(l)/l$. Panels d-f: same results for the Chialvo 1D map. Initial conditions: $x_0=0.03$ for the Logistic map and $x_0=2.5$ for 1D Chialvo map.
• Figure 4: Stationary solutions and collapse on a unique function after rescaling in the discrete form of the extended supercritical pitchfork map. Panel a: stationary solutions with $x\geq 0$ as a function of $\mu$. Dashed lines stand for unstable solutions. Red arrows indicate the flow of the solutions. green circle and blue square stand for the limit points of lower and upper stable solutions. Panel b: $x_l$ as a function of $\mu$ for different initial conditions about the upper limit point (blue square in panel a), after $l=20$ and $l=100$ steps. Notice that the range of $\mu$ and $x$ values is smaller than in panel a. Panel c: function collapse for initial conditions about the upper limit point (blue square in panel a). In all panels, the initial conditions are: $x_0=0.8-0.95$ (collapse from above) and $x_0=1/\sqrt{2}$.
• Figure 5: Stationary solutions and collapse on a unique function after rescaling in Hopf bifurcation. Panel a: solutions $r_l$ for different values of $l$, as a function of $\mu$. Panel b: Distance to the closest fixed point. Panel c: collapse after rescaling of the distance to the midpoint $p_m$, plotted as a function of $z$. Panel d: Collapse after rescaling of the distance to the closest function.