Search results for "Subcritical bifurcation"

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Pattern formation driven by cross–diffusion in a 2D domain

2012

Abstract In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.

Pattern formationFOS: Physical sciencesSaddle-node bifurcationPattern Formation and Solitons (nlin.PS)Dynamical Systems (math.DS)Bifurcation diagramDomain (mathematical analysis)Reaction–diffusion systemFOS: MathematicsMathematics - Dynamical SystemsBifurcationMathematical PhysicsMathematicsApplied MathematicsNonlinear diffusionTuring instabilityDegenerate energy levelsMathematical analysisGeneral EngineeringGeneral MedicineMathematical Physics (math-ph)Nonlinear Sciences - Pattern Formation and SolitonsBiological applications of bifurcation theoryComputational MathematicsAmplitude equationGeneral Economics Econometrics and FinanceSubcritical bifurcationAnalysis
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Pattern formation driven by cross-diffusion

2009

In this work we are interested in describing the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms (which take into account the self and the cross-diffusion effects). The reaction terms are chosen of the Lotka-Volterra type in the competitive interaction case. The cross-diffusion is proved to be the key mechanism of pattern formation via a linear stability analysis. A weakly nonlinear multiple scales analysis is carried out to predict the amplitude and the form of the pattern close to the bifurcation threshold. In particular, the Stuart-Landau equation which rules the evolution of the amplitude of the most unstable mode is found. In the subcritica…

Subcritical bifurcation hysteresis 2-d weakly nonlinear analysis
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