Existence of limit cycles in a predator-prey system with a functional response.
Existence of positive solutions of -dimensional system of nonlinear differential equations entering into a singular point
First phases of the associated equation with parameters for the differential equation
Five limit cycles for a simple cubic system.
We resolve the centre-focus problem for a specific class of cubic systems and determine the number of limit cycles which can bifurcate from a fine focus. We also describe the methods which we have developed to investigate these questions in general. These involve extensive use of Computer Algebra; we have chosen to use REDUCE.
Generalized Hopf bifurcations and applications to planar quadratic systems
Global qualitative investigation, limit cycle bifurcations and Applications of Polynomial Dynamical Systems
Homoclinic orbits and Lie rotated vector fields.
Hopf-like bifurcations in planar piecewise linear systems.
Continuous planar piecewise linear systems with two linear zones are considered. Due to their low differentiability specific techniques of analysis must be developed. Several bifurcations giving rise to limit cycles are pointed out.
Implicit Differential Equations From the Singularity Theory Viewpoint
Integrability and limit cycles for Abel equations
Abel equations are among the most natural ordinary differential equations which have a Godbillon-Vey sequence of length 4. We show that the associated Poincaré mapping can be expressed by iterated integrals with three functions which are solutions of a system of partial differential equations.
Integrability of a linear center perturbed by a fifth degree homogeneous polynomial.
In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.
Integrability of a linear center perturbed by a fourth degree homogeneous polynomial.
In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.
Integrable systems in the plane with center type linear part
We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for these cases. Finally, two large integrable system classes are determined in the most general nonhomogeneous cases.
Integrable systems via inverse integrating factor.
Integraltransformationen zu singulären S-Hermiteschen Rand-Eigenwertproblemen.
Involutions and iterates of central dispersions
Limit behavior of solutions of ordinary linear differential equations.
Limit cycle for the Brusselator by He's variational method.
Limit cycles for vector fields with homogeneous components
We study planar polynomial differential equations with homogeneous components. This kind of equations present a simple and well known dynamics when the degrees (n and m) of both components coincide. Here we consider the case and we show that the dynamics is more complicated. In fact, we prove that such systems can exhibit periodic orbits only when nm is odd. Furthermore, for nm odd we give examples of such differential equations with at least (n+m)/2 limit cycles.
Limit cycles in a Kolmogorov-type model.