Solutions in closed form and as power series to the real Lorenz equations

Abstract

Using the method of the Lie theory of extended groups and for the parameter values σ = ½, b = 1 and r = 0 we construct explicitly the general exact solution to the real Lorenz equations in terms of Jacobian elliptic functions. In the context of our approach further possible completely integrable cases of the Lorenz system are discussed by considering the result of the Painleve analysis for σ = 1, b = 2 and r = 1/9 and negative values of r, the latter case, r 0 and σ > 0 not following from the Painleve test. For other positive parameter values and in the form of appropriate power series we find some particular exact solutions which do not possess the Painleve property.