Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 2, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 3, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 4, we consider the problem of local...

In this paper we deal with a model describing the evolution in time of the density of a neural population in a state space, where the state is given by Izhikevich’s two - dimensional single neuron model. The main goal is to mathematically describe the occurrence of a significant phenomenon observed in neurons populations, the synchronization. To this end, we are making the transition to phase density population, and use Malkin theorem to calculate...

Tuberculosis (TB) remains a major global health problem. A possible risk factor for TB is diabetes (DM), which is predicted to increase dramatically over the next two decades, particularly in low and middle income countries, where TB is widespread. This study aimed to assess the strength of the association between TB and DM. We present a deterministic model for TB in a community in order to determine the impact of DM in the spread of the disease....

The aim of the contribution is to study ODE predator-prey system with a prey population embodying the Allee effect. Particular stationary points are analyzed and the results are illustrated by graphs of numerical solutions for various values of model parameters.

This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing...

Associated to analytic Hamiltonian vector fields in ${ℂ}^4$ having an equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

A decomposition technique of the solution of an n-th order linear differential equation into a set of solutions of 2-nd order linear differential equations is presented.

This paper is concerned with a second-order functional differential equation of the form $x^{\text{'}\text{'}}\left(z\right)=x(az+b{x}^{\text{'}}\left(z\right))$ with the distinctive feature that the argument of the unknown function depends on the state derivative. An existence theorem is established for analytic solutions and systematic methods for deriving explicit solutions are also given.

We study existence of analytic solutions of a second-order iterative functional differential equation $x^{\text{'}\text{'}}\left(z\right)={\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(z\right){\left(x^{\left[j\right]}\left(z\right)\right)}^t+G\left(z\right)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $\alpha ²y^{\text{'}\text{'}}\left(\alpha z\right)y^{\text{'}}\left(z\right)=\alpha y^{\text{'}}\left(\alpha z\right)y^{\text{'}\text{'}}\left(z\right)+\left[y^{\text{'}}\left(z\right)\right]³[{\sum }_{j=0}^k{\sum }_{t=1}^{\infty }{C}_{t,j}\left(y\left(z\right)\right){\left(y\left({\alpha }^{j}z\right)\right)}^t+G\left(y\left(z\right)\right)]$ invertible analytic solutions of the form $y\left(\alpha y^{-1}\left(z\right)\right)$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the...