Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems

This work deals with planar polynomial differential systems , . We give a set of necessary conditions for a system to have an invariant algebraic curve. These conditions are determined from the value of the cofactor at the singular points of the system, once considered in a compact space. We apply these results to show the non-Liouvillian integrability of several families of quadratic systems with an algebraic limit cycle.     

Polynomial differential systems having a given Darbouxian first integral

The Darbouxian theory of integrability allows to determine when a polynomial differential system in has a first integral of the kind f₁^λ₁?f_p^λ_pexp(g/h) where f_i, g and h are polynomials in , and for i=1,…,p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.     

A stabilization criterion for matrices

Given a simple linear system which is unstable in the sense that A has eigenvalues in the closed right-half of the complex plane, it is shown how the system can be dilated to a stable system of larger size. The cases of real matrices and complex matrices are considered separately.     

The periodic problem for curvature-like equations with asymmetric perturbations

We discuss existence and multiplicity of solutions of the periodic problem for the curvature-like equation

     

Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A∈C^m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA^∗ and A^∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices and . More generally, we consider the matrix triple , where are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form , where X,Y are nonsingular.     

Riccati inequality and functional properties of differential operators on the half line

Given a piecewise continuous function and a projection P₁ onto a subspace X₁ of CN, we investigate the injectivity, surjectivity and, more generally, the Fredholm properties of the ordinary differential operator with boundary condition . This operator acts from the “natural” space into L²×X₁. A main novelty is that it is not assumed that A is bounded or that has any dichotomy, except to discuss the impact of the results on this special case. We show that all the functional properties of interest, including the characterization of the Fredholm index, can be related to the existence of a selfadjoint solution H of the Riccati differential inequality . Special attention is given to the simple case when H=A+A^∗ satisfies this inequality. When H is known, all the other hypotheses and criteria are easily verifiable in most concrete problems.     

A fractional Helly theorem for convex lattice sets

A set of the form , where is convex and denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2^d. We prove that the fractional Helly number is only d+1: For every d and every α∈(0,1] there exists β>0 such that whenever F₁,…,F_n are convex lattice sets in such that for at least index sets I⊆{1,2,…,n} of size d+1, then there exists a (lattice) point common to at least βn of the F_i. This implies a (p,d+1)-theorem for every p?d+1; that is, if is a finite family of convex lattice sets in such that among every p sets of , some d+1 intersect, then has a transversal of size bounded by a function of d and p.     

Almost periodic solutions of forced vectorial Liénard equations

We describe the set of bounded or almost periodic solutions of the following Liénard system: , where is almost periodic, is a symmetric and nonsingular linear operator, and ∇F denotes the gradient of the convex function F on R^N. Then, we state a result of existence and uniqueness of almost periodic solution.     

Oscillation susceptibility analysis along the path of longitudinal flight equilibriums

In this paper the oscillation susceptibility of an aircraft in a longitudinal flight with constant forward velocity is analyzed in different flight models. Conditions which ensure such a flight, and equations governing the flight are presented. The stability of the equilibriums appearing is analyzed and the existence of Hopf bifurcations and saddle-node bifurcations is researched. For two aircrafts in a simplified model it is shown that saddle-node bifurcations are present and there are no Hopf bifurcations. It is shown that for the elevator deflection there are two turning points , having the property that if , then the angle of attack α and the pitch rate q oscillate with the same period, while the pitch angle θ increases (decreases) tending to . The behavior of the aircraft is simulated in the simplified model when the elevator deflection δ_e varies in the range and when δ_e leaves this range. For one of the aircrafts the analysis is performed also in the not simplified model, showing the differences between the results obtained in different models.     

Stability of singular Hopf bifurcations

A stability formula is given for the singular Hopf bifurcation arising in singularly perturbed systems of the form , in this paper. The derivation of the formula is based on a reduction technique and on an existing stability formula for Hopf bifurcation.     

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation , after a small polynomial perturbation. We first show that, under small perturbations of the form , where is a polynomial of degree 2m−1 in which the power of z is odd and the power of is even, the only possible distribution of limit cycles is (u,u) for all values of u=0,1,2,…,m−3. Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of is m−3, for m≥4. Then we consider a perturbation of the form , where is a polynomial of degree m in which the power of z is odd and obtain the upper bound m−5, for m≥6. Moreover, we show that the distribution (u,v) of limit cycles is possible for 0≤u≤m−5, 0≤v≤m−5 with u+v≤m−2 and m≥9.     

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system , where D^γ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation , in is given.