Periodic solutions for a Lotka–Volterra mutualism system with several delays

In the present paper, a Lotka–Volterra type mutualism system with several delays is studied. Some new and interesting sufficient conditions are obtained for the global existence of positive periodic solutions of the mutualism system. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions. Our results are different from the existing ones such as those in of Yang et al. [F. Yang, D. Jiang, A. Ying, Existence of positive solution of multidelays facultative mutualism system, J. Eng. Math. 3 (2002) 64–68] and Chen et al. [F. Chen, J. Shi, X. Chen, Periodicity in a Lotka–Volterra facultative mutualism system with several delays, J. Eng. Math. 21 (3) (2004) 403–409].     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems

We study systems that are monotone in a generalized sense with respect to cones of rank 2. The main result of the paper is the existence of a Poincaré-Bendixson property for some solutions of those systems.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

A new approach to center conditions for simple analytic monodromic singularities

In this paper we present an alternative algorithm for computing Poincaré-Lyapunov constants of simple monodromic singularities of planar analytic vector fields based on the concept of inverse integrating factor. Simple monodromic singular points are those for which after performing the first (generalized) polar blow-up, there appear no singular points. In other words, the associated Poincaré return map is analytic. An improvement of the method determines a priori the minimum number of Poincaré-Lyapunov constants which must cancel to ensure that the monodromic singularity is in fact a center when the explicit Laurent series of an inverse integrating factor is known in (generalized) polar coordinates. Several examples show the usefulness of the method.     

A multiplicity result for periodic solutions of second order differential equations with a singularity

By the use of the Poincaré–Birkhoff fixed point theorem, we prove a multiplicity result for periodic solutions of a second order differential equation, where the nonlinearity exhibits a singularity of repulsive type at the origin and has linear growth at infinity. Our main theorem is related to previous results by Rebelo (1996, 1997)  and  and Rebelo and Zanolin (1996)  and , in connection with a problem raised by del Pino et al. (1992) [1].     

On the remarkable values of the rational first integrals of polynomial vector fields

The remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by Poincaré. He was mainly interested in their algebraic aspects. Here we are interested in their dynamic aspects; i.e. how they contribute to the phase portrait of the system, to its separatrices, to its singular points, etc. The relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor.     

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2

We classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 2. In other words we characterize all the global phase portraits of the quadratic polynomial vector fields having all their orbits contained in conics. For such a vector field there are exactly 25 different global phase portraits in the Poincaré disc, up to a reversal of sense.     

Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Part III

We study the asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Thanks to an additional (fixed) parameter, we show that two different critical exponents play a crucial role in the asymptotic analysis, giving an explanation of the phenomena discovered in Gazzola et al. (Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear) and Gazzola and Serrin (Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 477).     

Singular perturbations and vanishing passage through a turning point

The paper deals with planar slow-fast cycles containing a unique generic turning point. We address the question on how to study canard cycles when the slow dynamics can be singular at the turning point. We more precisely accept a generic saddle-node bifurcation to pass through the turning point. It reveals that in this case the slow divergence integral is no longer the good tool to use, but its derivative with respect to the layer variable still is. We provide general results as well as a number of applications. We show how to treat the open problems presented in Artés et al. (2009) [1] and Dumortier and Rousseau (2009) [13], dealing respectively with the graphics DI2a and DF1a from Dumortier et al. (1994) [14].     

Multiple periodic solutions of scalar second order differential equations

We prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity, thus generalizing previous results by Lazer and McKenna (1987) [1] and Del Pino, Manasevich and Murua (1992) [2]. The main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity. The proof is based on the use of the Poincaré-Birkhoff Fixed Point Theorem.     

Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields.     

Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields.     

Structural stability of piecewise-linear vector fields

Stability and genericity properties established for polynomial vector fields in the plane, extended to the Poincaré sphere, are proved for a class of piecewise-linear vector fields.     

Analytic normalization of analytically integrable differential systems near a periodic orbit

For an analytic differential system in Rⁿ${\mathbb{R}}^n$ with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has n−1$n-1$ functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincaré–Dulac type normal form. This result is an extension of analytically integrable differential systems around a singularity to the ones around a periodic orbit.     

Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré-Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17-30] the Poincaré-Liapunov constants are calculated for polynomial systems of the form , , where P_n and Q_n are homogeneous polynomials of degree n. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré-Liapunov constants, through the study of the highest fine focus order for n=4 and n=5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré-Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870-1877]. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and the determination of the number of functionally independent ones are also discussed.     

Jumps of the fundamental solution matrix in discontinuous systems and applications

This paper deals with differential equations with discontinuous right-hand side. The concept of a solution for a discontinuous system is defined on the basis of differential inclusions using Filippov’s method. We study in particular the behaviour of solutions crossing a discontinuity surface transversally. A formula characterizing jumps of the fundamental solution matrix is derived. As an application of it, the concept of Poincaré mapping is defined for such systems.     

Homoclinic bifurcation with nonhyperbolic equilibria

The problem of homoclinic bifurcation is studied for a high dimensional system with nonhyperbolic equilibria. By constructing local coordinate systems near the unperturbed homoclinic orbit, Poincaré maps for the new system are established. Then the persistence of the homoclinic orbit and the bifurcation of the periodic orbit for the system accompanied with pitchfork bifurcation are obtained. Some known results are extended.     

On a closed orbit of some constrained system

We study on what one calls a constrained system of ODEs on It consists of two ordinary differential equations and an algebraic equation with respect to three unknown functions. We seek closed orbits of such a system. A necessary and sufficient condition for the system to have non-trivial closed orbits is given. Elementary tools such as Lyapunov functions and Poincaré’s index theory are used in the proof of the result. As an application we consider a constrained system associated with a non-constraint system of ODEs called the modified Bonhöffer-van der Pol system.     

A corollary of the Poincaré-Bendixson theorem and periodic canards

The role of topological methods in the analysis of canard-type periodic trajectories is discussed. A special corollary of the Poincaré-Bendixson theorem is used to prove the existence of periodic planar canards.