Abstract:

The local integrability and linearizability of two dimensional Lotka-Volterra systems has been investigated by many authors. Our aim here is to generalize these problems first to three dimensional Lotka-Volterra systems with $(lambda,mu, u)$ resonance and second to more general three dimensional systems. More precisely, necessary and sufficient conditions for both integrability and linearizability to three dimensional Lotka-Volterra systems are obtained for $(1;-1; 1), (2;-1; 1)$ and $(1;-2; 1)$-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.
This is joint work with Colin Christopher.