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Updated: Dec 16, 2025
13:51
Cross-Modal Multivariate Pattern Analysis
Published on: November 9, 2011
20.3K
Universal visibility patterns of unimodal maps.
Juan Carlos Nuño1, Francisco J Muñoz1
1Department of Applied Mathematics, Universidad Politécnica de Madrid, 28040 Madrid, Spain.
Chaos (Woodbury, N.Y.)
|July 3, 2020
Summary
Horizontal visibility analysis reveals unique patterns in unimodal maps, showing how visibility patterns emerge recursively in bifurcation cascades. These patterns converge at the onset of chaos, offering insights into non-linear dynamics.
Area of Science:
- Non-linear Dynamics
- Chaos Theory
- Complex Systems
Background:
- Bifurcation diagrams in unimodal maps characterize transitions to chaos.
- Periodic time series exhibit complex structures that are not fully understood.
- The horizontal visibility graph is a novel method for time series analysis.
Purpose of the Study:
- To investigate unimodal maps using horizontal visibility analysis.
- To identify and characterize visibility patterns in bifurcation diagrams.
- To explore the relationship between visibility patterns and the onset of chaos.
Main Methods:
- Application of the horizontal visibility graph to unimodal maps.
- Analysis of visibility patterns in periodic time series and their cascades.
- Recursive generation of visibility patterns from elementary blocks.
Main Results:
- Visibility patterns are observed in each cascade of the bifurcation diagram, converging at the onset of chaos.
- Elementary blocks of visibility are recursively embedded, forming patterns for each period.
- An infinite visibility pattern emerges at the onset of chaos, encompassing all periodic patterns.
Conclusions:
- Visibility patterns possess unique properties related to period and elementary blocks.
- Unique periodic windows (2-period and 3-period) yield unique visibility patterns.
- These findings provide a new perspective on the structure of chaos and its precursors in non-linear systems.
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