Analytical Integrability of Perturbations of Quadratic Systems

Analytic partial-integrability of a symmetric Hopf-zero degeneracy

“…We study the polynomial integrability of (necessary condition of formal integrability). We distinguish the following cases separately: Assume (or In such a case, (or ) is a polynomial first integral of . Assume If the origin of is an isolated singular point and has multiple factors, thus is not polynomially integrable, by [6, theorem 3.1]. Otherwise, and the vector field is of the form which is not polynomially integrable. Assume In this case, the result follows by changing by . Assume and From [3, proposition 1.7], as the factors of are and if it exists a polynomial first integral of , then it has the expression with natural numbers. By imposing we have that Thus, and are different from zero.…”

“…Assume and From [3, proposition 1.7], as the factors of are and if it exists a polynomial first integral of , then it has the expression with natural numbers. By imposing we have that Thus, and are different from zero.…”

“…Performing the scaled-change system (3.4) turns into The formal integrability of system (4.1) has been studied in [3]. By [3, theorem 2.11], if system (4.1) is formally integrable at the origin then it is linearizable (orbitally equivalent to its leading homogeneous term).…”

“…) real and J 2k+1 is unique module Ker( ˜ (3) 2k+1 ). We finish the proof, obtaining the expression of Ker( ˜ (3) 2k+1 ) and Cor( ˜ (3) 2k+1 ) for each case:…”

“…2 )} and we can choose Cor( ˜ (3) 2k+1 ) = Span{(x 2 + y 2 ) k z, z 2k+1 }. For the systems (1.2) with b 2 (b 1 − a 1 ), a 1 (a 2 − b 2 ) and a 1 b 2 − a 2 b 1 are rational numbers different from zero with the same sign, it has that ˜ (3) 2k ((…”