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Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems

A. Raouf Chouikha (2005)

Applicationes Mathematicae

We first examine conditions implying monotonicity of the period function for potential systems with a center at 0 (in the whole period annulus). We also present a short comparative survey of the different criteria. We apply these results to quadratic Loud systems ( L D , F ) for various values of the parameters D and F. In the case of noncritical periods we propose an algorithm to test the monotonicity of the period function for ( L D , F ) . Our results may be viewed as a contribution to proving (or disproving) a conjecture...

Monotonicity of the period function for some planar differential systems. Part II: Liénard and related systems

A. Raouf Chouikha (2005)

Applicationes Mathematicae

We are interested in conditions under which the two-dimensional autonomous system ẋ = y, ẏ = -g(x) - f(x)y, has a local center with monotonic period function. When f and g are (non-odd) analytic functions, Christopher and Devlin [C-D] gave a simple necessary and sufficient condition for the period to be constant. We propose a simple proof of their result. Moreover, in the case when f and g are of class C³, the Liénard systems can have a monotonic period function...

Monotonicity of the principal eigenvalue related to a non-isotropic vibrating string

Behrouz Emamizadeh, Amin Farjudian (2014)

Nonautonomous Dynamical Systems

In this paper we consider a parametric eigenvalue problem related to a vibrating string which is constructed out of two different materials. Using elementary analysis we show that the corresponding principal eigenvalue is increasing with respect to the parameter. Using a rearrangement technique we recapture a part of our main result, in case the difference between the densities of the two materials is sufficiently small. Finally, a simple numerical algorithm will be presented which will also provide...

Monotonicity properties of oscillatory solutions of differential equation ( a ( t ) | y ' | p - 1 y ' ) ' + f ( t , y , y ' ) = 0

Miroslav Bartušek, Chrysi G. Kokologiannaki (2013)

Archivum Mathematicum

We obtain monotonicity results concerning the oscillatory solutions of the differential equation ( a ( t ) | y ' | p - 1 y ' ) ' + f ( t , y , y ' ) = 0 . The obtained results generalize the results given by the first author in [1] (1976). We also give some results concerning a special case of the above differential equation.

Morales-Ramis Theorems via Malgrange pseudogroup

Guy Casale (2009)

Annales de l’institut Fourier

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

Motion of spirals by crystalline curvature

Hitoshi Imai, Naoyuki Ishimura, TaKeo Ushijima (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

Motion of spiral-shaped polygonal curves by nonlinear crystalline motion with a rotating tip motion

Tetsuya Ishiwata (2015)

Mathematica Bohemica

We consider a motion of spiral-shaped piecewise linear curves governed by a crystalline curvature flow with a driving force and a tip motion which is a simple model of a step motion of a crystal surface. We extend our previous result on global existence of a spiral-shaped solution to a linear crystalline motion for a power type nonlinear crystalline motion with a given rotating tip motion. We show that self-intersection of the solution curves never occurs and also show that facet extinction never...

Motion with friction of a heavy particle on a manifold - applications to optimization

Alexandre Cabot (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Let Φ : H → R be a C2 function on a real Hilbert space and ∑ ⊂ H x R the manifold defined by ∑ := Graph (Φ). We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force (characterized by g>0), the reaction force and the friction force ( γ > 0 is the friction parameter). For any initial conditions at time t=0, we prove the existence of a trajectory x(.) defined on R+. We are then interested in the asymptotic behaviour of...

Motion with friction of a heavy particle on a manifold. Applications to optimization

Alexandre Cabot (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Let Φ : H → ℝ be a 𝒞 2 function on a real Hilbert space and Σ ⊂ H × ℝ the manifold defined by Σ : = Graph ( Φ ) . We study the motion of a material point with unit mass, subjected to stay on Σ and which moves under the action of the gravity force (characterized by g > 0 ), the reaction force and the friction force ( γ > 0 is the friction parameter). For any initial conditions at time t = 0 , we prove the existence of a trajectory x ( . ) defined on ℝ + . We are then interested in the asymptotic behaviour of the trajectories when t → + ∞ . More precisely,...

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