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Finite-time scaling in local bifurcations
Álvaro Corral1,2,3,4, Josep Sardanyés5,6, Lluís Alsedà7,8
1Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193, Bellaterra, Barcelona, Spain. acorral@crm.cat.
Scientific Reports
|August 8, 2018
Summary
Finite-time scaling reveals universal laws for dynamical systems, analogous to finite-size scaling in physics. This method precisely describes bifurcations, offering insights into system dynamics from short time series.
Area of Science:
- Statistical Physics
- Dynamical Systems Theory
- Nonlinear Dynamics
Background:
- Finite-size scaling is crucial for understanding critical phenomena in statistical physics.
- Deterministic dynamical systems exhibit bifurcations, which are critical transitions.
- Analyzing these transitions at finite times is essential for real-world applications.
Purpose of the Study:
- To adapt finite-size scaling concepts to finite-time scaling in discrete dynamical systems.
- To analytically derive finite-time scaling laws for transcritical and saddle-node bifurcations.
- To establish a connection between thermodynamic phase transitions and dynamical system bifurcations.
Main Methods:
- Analytical derivation of finite-time scaling laws.
- Investigation of transcritical and saddle-node bifurcations.
- Comparison of scaling behavior with stochastic processes like the Galton-Watson process.
Main Results:
- Exact expressions for critical exponents and scaling functions were obtained for both bifurcation types.
- A universal scaling law was identified for the distance of the dynamical variable to the attractor, applicable to both bifurcations.
- The scaling behavior in the transcritical bifurcation was found to be identical to that of the Galton-Watson process.
Conclusions:
- Finite-time scaling provides a powerful tool for analyzing bifurcations in discrete dynamical systems.
- The universality of certain scaling laws bridges concepts from statistical physics and dynamical systems.
- This approach enables the characterization of dynamical shifts even with limited time-series data.
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