The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century
The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.
The principle of stationary action in the calculus of variations
We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Our main intention in this regard is to expose precise mathematical conditions for critical paths to be minimum solutions in a variety of situations of interest in Physics. Our claim is that, with a few elementary techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models...
The -product approach for linear ODEs: A numerical study of the scalar case
Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the...
The property (A) for a certain class of the third order ODE
The rate of satiation in a cooperative system
The ratio of eigenvalues of the Dirichlet eigenvalue problem for equations with one-dimensional -Laplacian.
The Rayleigh and van der Pol wave equations, some generalizations
The Region of Attraction of the Zero Solution of y' (t) = - (y(t-1)) 2m+1.
The relationship between the infinite eigenvalue assignment for singular systems and the solvability of polynomial matrix equations
Two related problems, namely the problem of the infinite eigenvalue assignment and that of the solvability of polynomial matrix equations are considered. Necessary and sufficient conditions for the existence of solutions to both the problems are established. The relationships between the problems are discussed and some applications from the field of the perfect observer design for singular linear systems are presented.
The return map for a planar vector field with nilpotent linear part: a direct and explicit derivation.
The return of the quartic oscillator. The complex WKB method
The Riccati differential equation with complex-valued coefficients
The Riccati differential equation with complex-valued coefficients and application to the equation
The Riccati equation: pinching of forcing and solutions.
The role of an integral inequality in the study of certain differential equations.
The role of center manifolds in ordinary differential equations
The role of delay in digestion of plankton by fish population: A fishery model.
The Schauder-Tychonoff Fixed Point Theorem and Applications
The Schröder equation and asymptotic properties of linear delay differential equations.
The Second Half-With a Quarter of a Century Delay
We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutions of delay equations apply directly to a resource-consumer model with age-structured consumer population.