The Chen System Having an Invariant Algebraic Surface

Abstract

In this paper, we characterize the dynamics of the Chen system ẋ = a(y - x), ẏ = (c - a)x - xz + cy, ż = xy - bz which has an invariant algebraic surface.

Global Dynamics in the Poincaré Ball of the Chen System Having Invariant Algebraic Surfaces

“…However this analysis does not correspond completely to the analysis of the solutions on the real invariant algebraic surfaces when they are considered in the whole space R 3 . Also some dynamical objects as heteroclinic orbits, connecting two singular points on a line of singularities, which were numerically detected by us and are described more precisely ahead in this note, were not described in [Cao et al, 2008]. The global study of how the invariant algebraic surfaces of the Chen system and the solutions on them reach the infinity, which completes the analysis presented in [Cao et al, 2008], is done here by using the 3-dimensional compactification technique of Poincaré.…”

“…Also some dynamical objects as heteroclinic orbits, connecting two singular points on a line of singularities, which were numerically detected by us and are described more precisely ahead in this note, were not described in [Cao et al, 2008]. The global study of how the invariant algebraic surfaces of the Chen system and the solutions on them reach the infinity, which completes the analysis presented in [Cao et al, 2008], is done here by using the 3-dimensional compactification technique of Poincaré. Similar global analysis for other quadratic polynomial differential systems in the compactification of R 3 were presented in [Buzzi et al, 2007;Llibre and Messias, 2009;Llibre et al, 2008Llibre et al, , 2010Messias, 2009].…”

“…The dynamics of Chen system on the invariant algebraic surfaces was studied in [Cao et al, 2008], where all the cases of Proposition 1.1, except the case (f), were considered. In such a work the authors gave a dynamical description of the solutions of system (1) in the projections of the invariant algebraic surfaces (a), .…”

“…In this way some dynamical aspects are missed. For instance, the dynamics on invariant straight lines and planes, which project orthogonally on the (x, y)-plane as points and straight lines respectively, do not appear in the analysis made in [Cao et al, 2008]. Moreover, the authors analyze the dynamics on the invariant surfaces extended to infinity on the Poincaré disc, after the projection of these surfaces on the xy-plane.…”

“…For a > 0 we have an analogous result. A complete analysis of the dynamics of the Chen system on the parabolic cylinders z = x 2 /2a projected onto the (x, y)-plane can be found in [Cao et al, 2008]. In Table 2 we have the classification of the singular points.…”