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Updated: Aug 6, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Localized dynamic patterns in the complex cubic-quintic Swift-Hohenberg equation in one and two dimensions
Hannes Uecker1, Nicolás Verschueren2,3,4,5, Edgar Knobloch2
1Institute for Mathematics, Carl von Ossietzky University of Oldenburg, Oldenburg, Germany.
Chaos (Woodbury, N.Y.)
|July 17, 2026
Summary
This study investigates dynamic patterns in the Swift-Hohenberg equation using bifurcation analysis. It reveals complex localized structures and provides a roadmap for understanding wall-attached states in 2D systems.
Area of Science:
- Nonlinear Dynamics
- Pattern Formation
- Mathematical Physics
Background:
- The Swift-Hohenberg equation models pattern formation in various physical systems.
- Understanding dynamic patterns and bifurcations is crucial for predicting system behavior.
- Previous studies have explored simpler cases, but complex symmetries and localized structures require further investigation.
Purpose of the Study:
- To analyze dynamic patterns in a complex Swift-Hohenberg equation on 1D and 2D domains.
- To investigate bifurcations, including Hopf and secondary bifurcations, leading to localized structures.
- To explore the organization of wall-attached states on 2D disks and their potential localization.
Main Methods:
- Bifurcation analysis to identify critical points and transitions in system behavior.
- Numerical methods, including numerical continuation, to compute and track dynamic patterns.
- Exploitation of gauge symmetry to efficiently compute relative equilibria and locate tertiary bifurcations.
Main Results:
- Identified Hopf bifurcations leading to traveling and standing waves in 1D.
- Observed secondary bifurcations to spatially localized structures like modulated traveling waves and localized standing waves.
- Demonstrated homoclinic snaking in localized structures and efficient computation of tertiary bifurcations to complex dynamic patterns.
- Established a framework for organizing wall-attached states on 2D disks, including rotating and breathing spots.
Conclusions:
- The study provides a comprehensive analysis of complex dynamic patterns and bifurcations in the Swift-Hohenberg equation.
- The findings offer a detailed 'road map' for understanding the emergence and organization of localized structures in both 1D and 2D systems.
- The methods developed are applicable to a broader range of nonlinear systems exhibiting complex spatio-temporal dynamics.
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