Search results for "Biological applications of bifurcation theory"

showing 3 items of 3 documents

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

2012

In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form …

Hopf bifurcationApplied MathematicsMechanical EngineeringMathematical analysisAerospace EngineeringOcean EngineeringContext (language use)Parameter spaceBiological applications of bifurcation theoryNonlinear Sciences::Chaotic Dynamicssymbols.namesakePitchfork bifurcationControl and Systems EngineeringControl theoryQuasiperiodic functionAttractorsymbolsElectrical and Electronic EngineeringDouble-Hopf bifurcations – Normal forms – Modified post-bifurcation dynamicsSettore MAT/07 - Fisica MatematicaNonlinear Sciences::Pattern Formation and SolitonsBifurcationMathematicsNonlinear Dynamics
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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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Remarks on the economic interpretation of Hopf bifurcations

1999

Abstract The Hopf bifurcation theorem has become a frequently used tool in the study of nonlinear dynamical economic systems. In this paper, it is shown that phenomena like multiple limit cycles, hysteresis loops and catastrophic transitions may possibly accompany a Hopf bifurcation. The theoretical argument is illustrated in Foley's liquidity cost–business cycle model.

Period-doubling bifurcationHopf bifurcationEconomics and EconometricsPure mathematicsSaddle-node bifurcationBifurcation diagramBiological applications of bifurcation theoryNonlinear systemsymbols.namesakeHysteresis (economics)symbolsInfinite-period bifurcationMathematical economicsFinanceMathematicsEconomics Letters
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