Mathematical Theory of Bifurcation

Crandall, Michael G.; Rabinowitz, Paul H.

doi:10.1007/978-94-009-9004-3_1

Cited by 16 publications

(8 citation statements)

References 64 publications

(28 reference statements)

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“…A solution of the latter problem is understood to be extended to an even function on −π ≤ ξ ≤ π and then periodically over all of R (it is easy to see that if u(ξ) is periodic, then (g * u)(ξ) is also periodic with the same period). The problem is thus reduced to one of a simple eigenvalue on a compact domain, to which the theory of Crandall and Rabinowitz [10] applies. The bifurcating solution can be constructed using perturbation methods and its sub/supercriticality calculated.…”

Section: Bifurcations and Periodic Traveling Wavesmentioning

confidence: 99%

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

Gourley

2004

Journal of Mathematical Sciences

180

132

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We present a short survey on the biological modeling, dynamics analysis, and numerical simulation of nonlocal spatial effects, induced by time delays, in diffusion models for a single species confined to either a finite or an infinite domain. The nonlocality, a weighted average in space, arises when account is taken of the fact that individuals have been at different points in space at previous times. We discuss and compare two existing approaches to correctly derive the spatial averaging kernels, and we summarize some of the recent developments in both qualitative and numerical analysis of the nonlinear dynamics, including the existence, uniqueness (up to a translation), and stability of traveling wave fronts and periodic spatiotemporal patterns of the model equations in unbounded domains and the linear stability, boundedness, global convergence of solutions and bifurcations of the model equations in finite domains.

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Section: Bifurcations and Periodic Traveling Wavesmentioning

confidence: 99%

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

Gourley

2004

Journal of Mathematical Sciences

180

132

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“…According to Andronov et al [2], a necessary condition for structural stability is that no separatrix can connect two saddles, or connect back to the same saddle. 9. Global behavior of the system.…”

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confidence: 99%

Global Hopf Bifurcation in a Simple Climate Model

Ghil¹,

Tavantzis²

1983

SIAM J. Appl. Math.

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The mathematical structure of a simple climate model is investigated. The model is governed by a system of two nonlinear, autonomous differential equations for the evolution in time of global temperature T and meridional ice-sheet extent L. The system's solutions are studied by a combination of qualitative reasoning with explicit calculations, both analytical and numerical. For plausible values of the physical parameters, a branch of periodic solutions obtains, which is both orbitally and structurally stable.The amplitude of the stable periodic solutions in T and L correspond roughly to that obtained from proxy records of Quaternary glaciation cycles. The period of these solutions increases along the branch, until it becomes infinite, while the amplitude of the limiting solution is finite. The limiting solution is a homoclinic orbit formed by the reconnecting separatrix of a saddle. The exchange of stability between the branch of periodic solutions and the steady solution from which it arises is studied by a slight simplification of known methods [20], [21 ]. the orbital variations, on the other [13, Chap. 15]. The next step would be to verify the existence of causal relations between the orbital forcing and the climatic system's behavior.Equilibrium models of the climatic system based on radiation balance, so-called energy-balance models ([5, pp. 461-480], [22]) are now well understood. Insolation intensity appears in them as a bifurcation parameter, and response to its changes can be assumed to be quasi-static. Changes in insolation of the magnitude given by orbital variations, when imposed upon energy-balance models, fail to produce temperature changes larger than fractions of a degree Kelvin [22,.

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“…In section 4, we generalize Elbau-Felder's results using Crandall-Rabinowitz bifurcation theory from simple eigenvalues (see e.g. [17]) to construct a parameterization that allows us to let t 2 → 1/2 maintaining the parameter r away from 0. Remark 3.…”

Section: Remarkmentioning

confidence: 96%

“…and apply the constructive bifurcation theory from a simple eigenvalue developed by Crandall and Rabinowitz (see e.g. [17]). The theory applied to equation (31) uses the implicit function theorem with the role of ρ and λ exchanged.…”

Section: Bifurcating Curvesmentioning

confidence: 99%

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Conformal deformation of equilibrium measures in normal random ensembles

Veneziani

Pereira

Marchetti

2011

J. Phys. A: Math. Theor.

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Abstract. We investigate the eigenvalues statistics of ensembles of normal random matrices when their order N tends to infinite. In the model, the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve. We study the conformal deformations of equilibrium measures of normal random ensembles to the real line and give sufficient conditions for it to weakly converge to be a Wigner measure.

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Mathematical Theory of Bifurcation

Cited by 16 publications

References 64 publications

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics

Global Hopf Bifurcation in a Simple Climate Model

Conformal deformation of equilibrium measures in normal random ensembles

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