The invariant algebraic surfaces of the Lorenz system

Abstract

The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz systemx˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)

Liouvillian Integrability Versus Darboux Polynomials

“…In general Darboux integrability depends on the existence of Darboux polynomials and exponential factors. There are plenty of results studying the existence of Darboux polynomials, see for example [1,14,23,24,32,33,34,40] and the references mentioned there.…”

Darboux polynomials and global phase portraits for the D2 vector field

“…Therefore, a number of systems have been studied from this perspective. Let us focus here on articles [4,5]. In these articles the Darboux integrability of two famous systems was studied, namely those due to Lorenz and Rössler.…”

Takens–Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system

“…To do that, A C C E P T E D M A N U S C R I P T Table 1 Invariant algebraic surfaces for the Lorenz system [65][66][67].…”

“…To demonstrate this fact simply consider the six invariant algebraic surfaces for the Lorenz system [65][66][67] (see Table 1) and impose the conditions for the TakensBogdanov bifurcation given in Eq. (3).…”