Laurent Cairó

On the singularity analysis of ordinary differential equations invariant under time translation and rescaling

The PT is applied to the general second- and third-order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability. The effect of the two symmetries on the compatibility conditions is determined and we show that, generally, these conditions are automatically a consequence of the resonance condition. Use is made of truncated Laurent series both in ascending and descending powers. As an example, the case of the generalized Chazy equation is presented.

Darboux integrability for 3D Lotka-Volterra systems

We describe the improved Darboux theory of integrability for polynomial ordinary differential equations in three dimensions. Using this theory and computer algebra, we study the existence of first integrals for the three-dimensional Lotka-Volterra systems. Only working up to degree two with the invariant algebraic surfaces and the exponential factors, we find the major part of the known first integrals for such systems, and in addition we find three new classes of integrability. The method used is of general interest and can be applied to any polynomial ordinary differential equations in arbitrary dimension.

Liouvillian first integrals for the planar Lotka-Volterra system

We complete the classication of all Lotka-Volterra systemsx=x(ax+by+c),y=y(Ax+By+C), having a Liouvillian first integral. In our classification we take into account the first integrals coming from the existence of exponential factors.

A Class of Reversible Cubic Systems with an Isochronous Center

Abstract We study cubic polynomial differential systems having an isochronous center and an inverse integrating factor formed by two different parallel invariant straight lines. Such systems are time-reversible. We find nine subclasses of such cubic systems, see Theorem 8. We also prove that time-reversible polynomial differential systems with a nondegenerate center have half of the isochronous constants equal to zero, see Theorem 3. We present two open problems.

Connection between the Existence of First Integrals and the Painleve Property in Two-Dimensional Lotka-Volterra and Quadratic Systems

Taking advantage of the considerable amount of work done in the search for first integrals (invariants) for the two-dimensional Lotka-Volterra system and the quadratic system (lvs and qs), we compare the relations needed to exhibit invariants (one for the lvs, at least three for the qs) to the two conditions of the Painlevé test (index and compatibility). We find that, eventually restricting the invariants to those which are analytic (all exponents integers) and thereby adding new constraints, these constraints always coalesce with the two Painlevé conditions. We conclude that straightforward application of the Painlevé test picks up only these simple analytic invariants and that possession of the Painlevé property is too strong a condition for the existence of the invariants.

Darboux method and search of invariants for the Lotka–Volterra and complex quadratic systems

The Darboux method introduces algebraic solutions quite useful to obtain invariant and first integrals of polynomial differential systems. Here we study the 2D Lotka–Volterra (LVS) and the complex quadratic system (QS) using straight lines for both and conics for the LVS. The conditions needed to obtain these invariants are given and a study of the phase space portrait is done.

Generalized Hamiltonian structures for systems in three dimensions with a rescalable constant of motion

The generalized Hamiltonian structures of several three-dimensional dynamical systems of interest in physical applications are considered. In general, Hamiltonians exist only for systems that possess at least one time-independent constant of motion. Systems with only time-dependent constants of motion may sometimes be rescaled and their constant of motion made time-independent. When this is possible, the transformed system may be cast in a generalized Hamiltonian formalism with non-canonical structure functions.

Integrability of the 2D Lotka-Volterra system via polynomial first integrals and polynomial inverse integrating factors

We present new first integrals of the two-dimensional Lotka-Volterra systems which have a polynomial inverse integrating factor. Moreover, we characterize all the polynomial first integrals of the two-dimensional Lotka-Volterra systems.

Phase portraits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2

We classify all the global phase portraits of the cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2. For such vector fields there are exactly 28 different global phase portraits in the Poincare disc up to a reversal of sense of all orbits.

Darbouxian first integrals and invariants for real quadratic systems having an invariant conic

We apply the Darboux theory to study the integrability of real quadratic differential systems having an invariant conic. The fact that two intersecting straight lines or two parallel straight lines are particular cases of conics allows us to study simultaneously the integrability of quadratic systems having at least two invariant straight lines real or complex.