Hopf Bifurcation in a Scalar Reaction Diffusion Equation

Guidotti, Patrick; Merino, Sandro

doi:10.1006/jdeq.1997.3307

Cited by 23 publications

(47 citation statements)

References 10 publications

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“…The critical values of the parameter γ (γ = 1.819705 and γ = 17.798542) coincide with the ones obtained by Guidotti & Merino in [7]. In their paper the equivalent values of γ are 0.5792 and 5.6655 since their problem is defined over (0, π) rather then (0, 1).…”

Section: Determination Of the Polessupporting

confidence: 83%

“…Our motivation for studying this problem comes from a paper written by Guidotti and Merino [7], where a similar initial-boundary value problem was associated with an abstract Cauchy problem using the general results presented by Amann [1]: in this work the principle of linearized stability (see Drangeid [4]) was applied and the spectrum of the associated linear operator was investigated through the Hopf bifurcation theorem (see e.g. Guckenheimer & Holmes [6]).…”

Section: Introductionmentioning

confidence: 99%

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The Thermostat Problem

Kalna¹,

McKee²

2002

TEMA - Tendências Em Matemática Aplicada E Computacional

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Abstract. A paradigm model for an air-conditioning system is studied: heat flux to and from one end of a bar is a (nonlinear) function of the temperature at the other end. The behaviour of this model is studied through semi-discretisation in the spatial variable and local linearisation. This procedure produces an autonomous system of ordinary differential equations whose local stability may be studied through its spectrum of eigenvalues and related back to the original problem through the Hopf bifurcation theorem. It will be shown that the proportionality constant γ is a bifurcation parameter which gives rise to three qualitatively different solutions: one stable, where the temperature tends exponentially to zero; one stable that is bounded by an envelope which tends exponentially to zero; and an unstable solution that oscillates with ever increasing amplitude. An almost local problem is also studied with similar results: the three qualitative solutions arise as before with the bifurcation parameter decreasing as the problem becomes closer to the local problem.Integral equation characterisations of the nonlinear problem are developed and existence and uniqueness are demonstrated. For the linear problem the general analytic solution is provided and its numerical evaluation is discussed.

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Section: Determination Of the Polessupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

The Thermostat Problem

Kalna¹,

McKee²

2002

TEMA - Tendências Em Matemática Aplicada E Computacional

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“…And the contrary if p (t) < 0. This is a kind of non-local regulatory behavior of the type of that of [12].…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 97%

“…An inspiring example of destabilization of an equilibrium in a diffusion equation has been that of P. Guidotti and S. Merino [12]. In that example a kind of heat regulation mechanism produces oscillations in the temperature when a parameter becomes sufficiently large.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 98%

Instability and Bifurcation in a Trend Depending Price Formation Model

2016

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A well-known model due to J.-M. Lasry and P.L. Lions that presents the evolution of prices in a market as the evolution of a free boundary in a diffusion equation is modified in order to show instabilities for some values of the parameters. This loss of stability is associated to the appearance of new types of solutions, namely periodic solutions, due to a Hopf bifurcation and representing price oscillations; and traveling waves, that represent either inflationary or deflationary behavior.

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“…It was in fact shown in [13] that resolvent positivity does not follow from the validity of a maximum principle in the case non separated boundary conditions. In that case we would have to assume resolvent positivity the operator A associated to (A, B) directly.…”

Section: Remark 36mentioning

confidence: 96%

Elliptic and parabolic problems in unbounded domains

Guidotti

2004

Mathematische Nachrichten

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We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi-explicit representation formulas via operator-valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles.

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Hopf Bifurcation in a Scalar Reaction Diffusion Equation

Cited by 23 publications

References 10 publications

The Thermostat Problem

The Thermostat Problem

Instability and Bifurcation in a Trend Depending Price Formation Model

Elliptic and parabolic problems in unbounded domains

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