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On the Lagrange-Souriau form in classical field theory

D. R. Grigore, Octavian T. Popp (1998)

Mathematica Bohemica

The Euler-Lagrange equations are given in a geometrized framework using a differential form related to the Poincare-Cartan form. This new differential form is intrinsically characterized; the present approach does not suppose a distinction between the field and the space-time variables (i.e. a fibration). In connection with this problem we give another proof describing the most general Lagrangian leading to identically vanishing Euler-Lagrange equations. This gives the possibility to have a geometric...

On the local convergence of Kung-Traub's two-point method and its dynamics

Parandoosh Ataei Delshad, Taher Lotfi (2020)

Applications of Mathematics

In this paper, the local convergence analysis of the family of Kung-Traub's two-point method and the convergence ball for this family are obtained and the dynamical behavior on quadratic and cubic polynomials of the resulting family is studied. We use complex dynamic tools to analyze their stability and show that the region of stable members of this family is vast. Numerical examples are also presented in this study. This method is compared with several widely used solution methods by solving test...

On the long time behavior of KdV type equations

Nikolay Tzvetkov (2003/2004)

Séminaire Bourbaki

In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg- de Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.

On the long-time behaviour of a class of parabolic SPDE’s : monotonicity methods and exchange of stability

Benjamin Bergé, Bruno Saussereau (2005)

ESAIM: Probability and Statistics

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

On the long-time behaviour of a class of parabolic SPDE's: monotonicity methods and exchange of stability

Benjamin Bergé, Bruno Saussereau (2010)

ESAIM: Probability and Statistics

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional Brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

On the Lyapunov numbers

Sergiĭ Kolyada, Oleksandr Rybak (2013)

Colloquium Mathematicae

We introduce and study the Lyapunov numbers-quantitative measures of the sensitivity of a dynamical system (X,f) given by a compact metric space X and a continuous map f: X → X. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.

On the Mathematical Modelling of Microbial Growth: Some Computational Aspects

Markov, Svetoslav (2011)

Serdica Journal of Computing

We propose a new approach to the mathematical modelling of microbial growth. Our approach differs from familiar Monod type models by considering two phases in the physiological states of the microorganisms and makes use of basic relations from enzyme kinetics. Such an approach may be useful in the modelling and control of biotechnological processes, where microorganisms are used for various biodegradation purposes and are often put under extreme inhibitory conditions. Some computational experiments are...

On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus

Erwan Lanneau, Jean-Luc Thiffeault (2011)

Annales de l’institut Fourier

We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham’s proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus g = 2 to 5 , the minimum dilatation is the smallest Salem number for polynomials of degree 2 g .

On the mobility and efficiency of mechanical systems

Gershon Wolansky (2007)

ESAIM: Control, Optimisation and Calculus of Variations

It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.


On the multiplicity of brake orbits and homoclinics in Riemannian manifolds

Roberto Giambò, Fabio Giannoni, Paolo Piccione (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let M , g be a complete Riemannian manifold, Ω ⊂ M an open subset whose closure is diffeomorphic to an annulus. If ∂ ⁡ Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω ¯ = Ω ⁢ ⋃ ∂ ⁡ Ω starting orthogonally to one connected component of ∂ ⁡ Ω and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating...

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