Investigation of dynamical systems using tools of the theory of invariants and projective geometry (Q1371012)

The authors study the nonlinear dynamical system of the type \[ \dot x=P(x,y,z), \dot y=Q(x,y,z), \dot z=R(x,y,z) \] by means of reduction to some ordinary differential equations of second order of the form \[ y''+a_1(x,y)y'{}^3+3a_2(x,y)y'{}^2+3a_3(x,y)y'+a_4(x,y)=0. \] They apply the theory of invariants developed by S. Lie, R. Liouville and A. Tresse and the projective geometry of E. Cartan to evaluating the invariants of the ordinary differential equations. They study two concrete dynamical systems: the Lorenz system and the Rößler system and establish the connection of values of the invariants with characteristics of the dynamical systems.