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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

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Homoclinic snaking in bounded domains.

S M Houghton¹, E Knobloch

¹School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom. smh@maths.leeds.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 2, 2009

Summary

This summary is machine-generated.

Homoclinic snaking, a phenomenon in spatially reversible systems, exhibits oscillations that terminate differently in finite domains based on boundary conditions. This study details snaking termination in the Swift-Hohenberg equation, explaining observed behaviors in fluid convection.

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Area of Science:

Nonlinear dynamics
Mathematical physics
Computational fluid dynamics

Background:

Homoclinic snaking describes oscillations in spatially localized states within bistable systems.
In finite domains, snaking termination depends on boundary conditions, unlike infinite domains where it continues indefinitely.
Previous observations of "snaking without bistability" in binary fluid convection simulations lacked a detailed mechanistic explanation.

Purpose of the Study:

To elucidate the termination mechanisms of homoclinic snaking in finite domains.
To explain the phenomenon of "snaking without bistability" observed in binary fluid convection.
To provide a detailed analysis using the Swift-Hohenberg equation as a model system.

Main Methods:

Analysis of homoclinic snaking in the Swift-Hohenberg equation.
Investigation of behavior under different boundary conditions (periodic vs. non-Neumann).
Detailed examination of how localized structures interact with domain boundaries.

Main Results:

In finite domains, snaking branches terminate differently based on boundary conditions.
Periodic boundary conditions lead to termination on spatially periodic states.
Non-Neumann boundary conditions result in continuous transition to large-amplitude filling states.
The detailed behavior in the Swift-Hohenberg equation explains "snaking without bistability" in fluid convection.

Conclusions:

Boundary conditions critically determine the termination of homoclinic snaking in finite systems.
The study provides a unifying explanation for observed "snaking without bistability" phenomena.
The Swift-Hohenberg equation serves as a valuable model for understanding complex dynamics in spatially extended systems.