GLOBAL DYNAMICS of THE LORENZ SYSTEM WITH INVARIANT ALGEBRAIC SURFACES

Abstract

In this paper by using the Poincare compactification of R(3) we describe the global dynamics of the Lorenz system(x) over dot = s(-x + y), (y) over dot = rx - y - xz, (z) over dot = -bz + xy,having some invariant algebraic surfaces. of course ( x, y, z) is an element of R(3) are the state variables and (s, r, b) is an element of R(3) are the parameters. For six sets of the parameter values, the Lorenz system has invariant algebraic surfaces. For these six sets, we provide the global phase portrait of the system in the Poincare ball (i.e. in the compactification of R(3) with the sphere S(2) of the infinity).