Finite-time scaling on low-dimensional map bifurcations

TL;DR

This paper extends finite-time scaling methods to analyze bifurcations in low-dimensional maps, introducing new observables and applying the approach to 1D and 2D systems, revealing fundamental dynamical insights.

Contribution

It generalizes finite-time scaling to 1D and 2D maps, introduces finite-time susceptibility and Lyapunov exponent, and applies the method to complex bifurcations including discontinuities.

Findings

Finite-time scaling effectively characterizes bifurcations in 1D maps.

New observables display consistent scaling near bifurcation points.

Method successfully applied to 2D Chialvo map with complex bifurcations.

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.