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Finite-time scaling on low-dimensional map bifurcations

Daniel A. Martin1,2,*, Qian-Yuan Tang3,†, and Dante R. Chialvo1,2,3

  • *Contact author: dmartin@unsam.edu.ar
  • †Contact author: tangqy@hkbu.edu.hk
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Phys. Rev. Research 7, 043241 – Published 2 December, 2025

DOI: https://doi.org/10.1103/73cc-j6mx

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 one-dimensional 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 two-dimensional (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 avenues for studying complex dynamical behavior.

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Phys. Rev. Research 7, 043241– Published 2 December, 2025
  • Received 5 July 2025
  • Accepted 29 October 2025
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