Instability and Bifurcation in a Trend Depending Price Formation Model

González, María del Mar; Gualdani, Maria Pia; Solà-Morales, J.

doi:10.1007/s10440-016-0043-8

Cited by 6 publications

(5 citation statements)

References 16 publications

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“…The Hopf bifurcation phenomenon engendered by the nonlocal nature of the boundary condition has also inspired the research presented in [10] on a market price formation model introduced by J.M. Lasry and P.L.…”

Section: Introductionmentioning

confidence: 92%

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On the Maximal Parameter Range of Global Stability for a Nonlocal Thermostat Model

Guidotti¹,

Merino²

2019

Preprint

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The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions. Following the terminology used in [17] we need to distinguish between the concept of the attractor Âβ and the B-attractor A β when formulating our main results in Theorem 5.1. We point out that, in general, the attractor Âβ is a proper subset of the B-attractor A β . We prove that the continuous semiflow Φ β induced on H 1 (0, π) by the system (1.1) has a global attractor Âβ = {0} for β ∈ (0, β 0 ) and that for β ∈ (0, 4 π ) the B-attractor and the attractor coincide, i.e. A β = Âβ = {0} . In [7], the authors prove that the global B-attractor A β exists for β ∈ (0, ∞) and that A β = {0} for β ∈ (0, 1 π ) . We thus extend previous results by determining larger parameter ranges where the B-attractor and the attractor are shown to be equal to {0} . The existence of the B-attractor A β shown in [7] for β ∈ (0, ∞) implies, in particular, that all orbits are bounded in the underlying Banach space. Since H 1 (0, π) → C [0, π] , the orbits satisfy Φ β (t, u 0 ) ∞ ≤ c(u 0 ) < ∞, t ≥ 0.

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Section: Introductionmentioning

confidence: 92%

“…The series representation of the Laplace transform is obtained from (2.3) by elementary integrations. Its explicit representation is obtained in [12] (formula (10)) in a different context and is derived again in Section 3.3. A more elementary direct computation using a partial fraction expansion is also given in e.g.…”

Section: It Is Known ( [2]mentioning

confidence: 99%

On the Maximal Parameter Range of Global Stability for a Nonlocal Thermostat Model

Guidotti¹,

Merino²

2019

Preprint

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“…The Hopf bifurcation phenomenon engendered by the nonlocal nature of the boundary condition has also inspired an application, presented in Ref. 12, to a market price formation model introduced by J.M. Lasry and P.L.…”

Section: Introductionmentioning

confidence: 99%

“…The resulting system of equations studied in Ref. 12 can be reduced to a single equation by introducing a moving boundary. This leads to an additional difficulty caused by the need to deal with a time‐dependent source

δ_{x false(t false)}

instead of a source at a fixed position.…”

Section: Introductionmentioning

confidence: 99%

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

Guidotti

Merino

2021

Stud Appl Math

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A parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

show abstract

“…The Hopf bifurcation phenomenon engendered by the nonlocal nature of the boundary condition has also inspired an application, presented in [13], to a market price formation model introduced by J.M. Lasry and P.L.…”

Section: Introductionmentioning

confidence: 99%

On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation

Guidotti¹,

Merino²

2020

Preprint

View full text Add to dashboard Cite

A parameter dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non-self-adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that e.g. can be interpreted as a thermostat device or in the context of an agent based price formation model for a market. The existence and the stability of periodic self-oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation is proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. The bifurcation and stability results follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only it can be extended naturally to arbitrary euclidean dimension and to manifolds.

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Instability and Bifurcation in a Trend Depending Price Formation Model

Cited by 6 publications

References 16 publications

On the Maximal Parameter Range of Global Stability for a Nonlocal Thermostat Model

On the Maximal Parameter Range of Global Stability for a Nonlocal Thermostat Model

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation

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