Local Integrability and Linearizability of a $$(1:-1:-1)$$ ( 1 : - 1 : - 1 ) Resonant Quadratic System

Integrability and linearizability of a family of three-dimensional quadratic systems

“…There is a big challenge to understand the mechanisms of integrability of non Lotka-Volterra systems. Some authors have been studied some non Lotka-Volterra systems: Dukarić et al [16] gave necessary and sufficient conditions for integrability and linearizability for a family of three dimensional quadratic systems. They used mainly Darboux theory of integrability and other methods to prove the sufficiency of their conditions.…”

“…So we consider φ 1 and φ 2 in the form (10) and we write them as power series up to degree 15. In order to find the necessary conditions we compute the obstructions to form first integrals which are known as resonant focus quantity, for more details see [16]. Then, a factorized Gröbner basis was found using the Computer Algebra system Reduce.…”

Three-Dimensional Lotka–Volterra Systems with 3:−1:2-Resonance

“…The integrability of some Lotka-Voltera families was considered by several authors, see for example [15,16,18,31,34,36]. Similar technics applied to some more general families like the ones in [5,23]. Some general properties of these systems are studied in the works [12,13].…”

A note on local integrability of differential systems

“…x ∈ R n , with f (x) = O * ( x 2 ) ∈ C ω (R n , 0), the study of the theory of local integrability or of the existence of first integrals at the origin can be traced back to Poincaré [16]. Since then, the theory of local integrability has been greatly developed, see for example [1,2,4,5,6,8,15,17,18,19,20,21,22]. Hereafter, O * ( x 2 ) denotes a function (or a vector-valued function) without constant and linear terms in its Taylor expansion, and C ω (R n , 0) denotes the set of analytic functions defined in a neighborhood of the origin.…”

“…M λ = ∅, there are certain known results which provide necessary conditions ensuring the existence and number of functionally independent local analytic or formal first integrals of system (1). For more details, see [2,4,6,15,18]. About the equivalent characterization of analytic integrability via normal form, there are also some known results on the existence of analytic normalization of analytically integrable differential systems to their Poincaré-Dulac normal forms.…”