The integrability problem consists in finding an explicit the class of functions that a first integral of a given planar polynomial differential system must belong to. In this paper we propose a ...method to find integrability via algebraic changes of variables in order to transform the planar polynomial differential into a simple system or equation to determine a first integral of it.
•A method to find integrability via algebraic changes of variables is proposed.•Looking for one transformation of planar polynomial differential system into a simple one.•The method is applied to several examples.
•The orbital normal form of the nondegenerate Hopf-zero singularity pro- vides necessary conditions for the existence offirst integrals for such singularity.•The relation between the existence ...offirst integrals and of inverse Jacobi multipliers is analized.•Some algorithmic procedures for determining the existence offirst inte- grals are presented.
In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields.
In this paper we study the analytic integrability of degenerate vector fields of the form (y3+2ax3y+⋯,−x5−3ax2y2+⋯) around the origin. For these vector fields it is proved that integrability does not ...imply formal orbital equivalence to the Hamiltonian leading part. Moreover, it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible.
We consider a nine-parameter familiy of 3D quadratic systems, x˙=x+P2(x,y,z), y˙=−y+Q2(x,y,z), z˙=−z+R2(x,y,z), where P2,Q2,R2 are quadratic polynomials, in terms of integrability. We find necessary ...and sufficient conditions for the existence of two independent first integrals of corresponding semi-persistent, weakly persistent, and persistent systems. Unlike some of the earlier works, which primarily focus on planar systems, our research covers three-dimensional spaces, offering new insights into the complex dynamics that are not typically apparent in lower dimensions.
We consider the complex differential system
where
f
is the analytic function
with
a
j
∈ ℂ. This system has a weak saddle at the origin and is a generalization of complex Liénard systems. In this work ...we study its local analytic integrability.
In this paper we study the analytic integrability around the origin inside a family of degenerate centers or perturbations of them. For this family analytic integrability does not imply formal ...orbital equivalence to a Hamiltonian system. It is shown how difficult is the integrability problem even inside this simple family of degenerate centers or perturbations of them.
We consider a complex differential system with a weak saddle at the origin and we characterize the existence of a local analytic first integral around the weak saddle. If the system does not have a ...fixed degree and instead the degree is arbitrarily large, the family can have a numerable infinite number of integrability cases.
PDF
