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  1. Home
  3. Homoclinic Snaking: Structure And Stability.
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  3. Homoclinic Snaking: Structure And Stability.

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Homoclinic snaking: structure and stability.

John Burke1, Edgar Knobloch

  • 1Department of Physics, University of California, Berkeley, California 94720, USA. burkej8@socrates.berkeley.edu

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|October 2, 2007

View abstract on PubMed

Summary
This summary is machine-generated.

The bistable Swift-Hohenberg equation reveals complex localized states organized in a snakes-and-ladders structure. Stability analysis in one and two dimensions clarifies their behavior in physical systems.

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Area of Science:

  • Nonlinear dynamics
  • Mathematical physics

Background:

  • The Swift-Hohenberg equation models pattern formation in various physical systems.
  • Understanding localized states is crucial for predicting system behavior near bifurcations.

Purpose of the Study:

  • To review the origin of the snakes-and-ladders structure in one spatial dimension.
  • To describe the stability of localized states in one and two dimensions.
  • To discuss the relevance to physical systems.

Main Methods:

  • Analysis of the bistable Swift-Hohenberg equation.
  • Investigation of spatially localized states near the Maxwell point.
  • Stability analysis with respect to perturbations.

Main Results:

  • Identified multiple stable and unstable spatially localized states of arbitrary length.
  • Characterized the snakes-and-ladders organization of these states.
  • Described stability properties in 1D and 2D.

Conclusions:

  • The snakes-and-ladders structure arises from the interplay of homogeneous and periodic states.
  • Stability analysis provides insights into the behavior of localized states.
  • Results are relevant to diverse physical phenomena.