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Scaling regions—intervals on a graph where the dependent variable depends linearly on the independent variable—abound in dynamical systems, notably in calculations of invariants like the correlation dimension or a Lyapunov exponent. In these applications, scaling regions are generally selected by hand, a process that is subjective and often challenging due to noise, algorithmic effects, and confirmation bias. In this paper, we propose an automated technique for extracting and characterizing such regions. Starting with a two-dimensional plot—e.g., the values of the correlation integral, calculated using the Grassberger–Procaccia algorithm over a range of scales—we create an ensemble of intervals by considering all possible combinations of end points, generating a distribution of slopes from least squares fits weighted by the length of the fitting line and the inverse square of the fit error. The mode of this distribution gives an estimate of the slope of the scaling region (if it exists). The end points of the intervals that correspond to the mode provide an estimate for the extent of that region. When there is no scaling region, the distributions will be wide and the resulting error estimates for the slope will be large. We demonstrate this method for computations of dimension and Lyapunov exponent for several dynamical systems and show that it can be useful in selecting values for the parameters in time-delay reconstructions.
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This content will become publicly available on April 1, 2026
Hierarchy of Chaotic Dynamics in Random Modular Networks
We introduce a model of randomly connected neural populations and study its dynamics by means of the dynamical mean-field theory and simulations. Our analysis uncovers a rich phase diagram, featuring high- and low-dimensional chaotic phases, separated by a crossover region characterized by low values of the maximal Lyapunov exponent and participation ratio dimension, but with high values of the Lyapunov dimension that change significantly across the region. Counterintuitively, chaos can be attenuated by either adding noise to strongly modular connectivity or by introducing modularity into random connectivity. Extending the model to include a multilevel, hierarchical connectivity reveals that a loose balance between activities across levels drives the system towards the edge of chaos.
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- Award ID(s):
- 2223725
- PAR ID:
- 10632889
- Publisher / Repository:
- Physical Review Journals
- Date Published:
- Journal Name:
- Physical Review Letters
- Volume:
- 134
- Issue:
- 14
- ISSN:
- 0031-9007
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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