An integrating factor matrix method to find first integrals

Abstract

Abstract. In this paper we developed an integrating factor matrix method to derive conditions for the existence of first integrals. We use this novel method to obtain first integrals, along with the conditions for their existence, for two and three dimensional Lotka-Volterra systems with constant terms. The results are compared to previous results obtained by other methods.

The integration of three-dimensional Lotka–Volterra systems

“…, x d ) a polynomial (usually though not always of degree 1 with constant term allowed). Results along this line include those of Cairó and co-workers [29][30][31][32], who in many cases used the classical technique of Darboux polynomials (DPs) [33]; and recent results based on an integrating factor technique [34]. In the case when d = 3 and a single first integral of a form similar to (1.2) is known to exist, there has been some work on the construction of a second, functionally independent first integral of a less algebraic form, involving a quadrature [35][36][37][38].…”

“…, x d ) a polynomial (usually though not always of degree 1 with constant term allowed). Results along this line include those of Cairó and collaborators (Cairó & Feix 1992;Cairó et al 1999;Cairó 2000;Cairó & Llibre 2000), who in many cases used the classical technique of Darboux polynomials (Goriely 2001); and results based on an integrating factor technique (Saputra et al 2010). When d = 3 and a single first integral of a form similar to (1.2) exists, there has been some work on the construction of a second, functionally independent first integral of a more complicated and less algebraic form (Grammaticos et al 1990;Goriely 1992;Gao 2000;Bustamante & Hojman 2003).…”

“…A d = 3 counterpart to their d = 2 analysis, which is lengthy, is not yet available. The many possible configurations of invariant planes have not been fully classified, though partial results have been obtained (Cairó 2000;Saputra et al 2010).…”

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

“…This type of dynamical systems occurs frequently in applications especially in the mathematical models for population biology [Jansen, 2001;Saputra et al, 2010a;De Witte et al, 2014] and for the spread of diseases [Chitnis et al, 2006;Llensa et al, 2014]. It is because in population models, if a species dies out it cannot be regenerated therefore it is always natural to have coordinate axes as the invariant manifold.…”

The inverse integrating factor for some classes of n-th order autonomous differential systems

“…Recently, there appear several papers about searching for inverse integrating factors of high order autonomous differential systems or using inverse integrating factors to study the systems. For example, in [14], the authors developed an integrating factor matrix method to derive conditions for the existence of first integrals, and used the method to study the integrability of two and three dimensional Lotka-Volterra systems with constant terms. We gave a method for deriving integrating factor by using invariant manifolds of system in [10].…”