ALGEBRAIC STRUCTURES AND INVARIANT-MANIFOLDS OF DIFFERENTIAL-SYSTEMS

Algebraic tools are applied to find integrability properties of ODEs. Bilinear nonassociative algebras are associated to a large class of po lynomial and nonpolynomial systems of differential equations, since al l equations in this class are related to a canonical quadratic differe ntial system: the Lotka-Volterra system. These algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given. Their subalgebras are associated to nonlinear invariant manifolds in the ph ase space. These manifolds are calculated explicitly. More general alg ebraic invariant surfaces are also obtained by combining a theorem of Walcher and the Lotka-Volterra canonical form. Applications are given for Lorenz model, Lotka, May-Leonard, and Rikitake systems. (C) 1998 A merican Institute of Physics.