Twin Vector Fields and Independence of Spectra for Quadratic Vector Fields

Abstract

The object of this paper is to address the following question: When is a polynomial vector field on C 2 completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector fields this is not the case: given a generic quadratic vector field there is, up to affine equivalence, exactly one other vector field which has the same spectra of singularities. Moreover, we will see that we can always assume that both vector fields have the same singular locus and at each singularity both vector fields have the same spectrum. Let us say that two vector fields are twin vector fields if they have the same singular locus and the same spectrum at each singularity.To formalize the above claim we shall prove the following: any two generic quadratic vector fields with the same spectra of singularities (yet possibly different singular locus) can be transformed by suitable affine maps to be either the same vector field or a pair of twin vector fields.We then analyze the case of quadratic Hamiltonian vector fields in more detail and find necessary and sufficient conditions for a collection of non-zero complex numbers to arise as the spectra of singularities of a quadratic Hamiltonian vector field.Lastly, we show that a generic quadratic vector field is completely determined (up to affine equivalence) by the spectra of its singularities together with the characteristic numbers of its singular points at infinity.

Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities

“…Nowadays isochronicity appears in many physical problems and it is closely related with the existence and uniqueness of solutions for certain bifurcation problems or boundary value problems (see for instance [8] and the references therein). In the last decade the study of the isochronicity has been grown specially in the case of polynomial differential systems due to the appearance of powerful methods of computational analysis, see for instance [1,3,4,14] to cite just few of them. A polynomial differential system of degree n is a differential system (1) ẋ = P (x, y), ẏ = Q(x, y), with P and Q polynomials such that the maximum of their degrees is n. We denote by χ = (P, Q) the polynomial vector field associated to system (1).…”

Spectra of quadratic vector fields on $\mathbb{C}^2$: The missing relation

“…The formulas for t k are linear in a j , and the formulas for d k have degree 2. As in [Ram17], Theorem 2.1 implies that the fourth zero of a vector field given by ( 10) is the point − d4 d2 , − d4 d3 . This fact adds two more linear equations to our system, and we need to solve a quadratic equation on the line in C 6 given by 5 linear equations.…”

“…Given a generic vector field v ∈ V 2 we can compute its twin v ′ by solving a simple system of algebraic equations (cf. [Ram17]). The coefficients defining v ′ are expressed as rational functions on the coefficients defining v. Thus, we obtain a rational map (which is an involution) τ : V 2 V 2 .…”

“…, d 3 ) is a dominant rational map whose generic fiber is two points corresponding to a pair of twins (cf. [Ram17]), thus once we pass to the quotient by the involution τ we obtain a birational isomorphism.…”

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

“…The spectral map Ψ. The first step in the analysis of the spectra of a quadratic self-map with invariant line is the following theorem, which is analogous to the main result of [Ram17] for quadratic vector fields on C 2 (see theorem 2.3 below).…”