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

Abstract

In this paper we perform a global analysis of the dynamics of the Chen systeṁwhere (x, y, z) ∈ R 3 and (a, b, c) ∈ R 3 . We give the complete description of its dynamics on the sphere at infinity. For six sets of the parameter values the system has invariant algebraic surfaces. In these cases we provide the global phase portrait of the Chen system and give a complete description of the α-and ω-limit sets of its orbits in the Poincaré ball, including its boundary S 2 , i.e. in the compactification of R 3 with the sphere S 2 of the infinity. Moreover, combining the analytical results obtained with an accurate numerical analysis, we prove the existence of a family with infinitely many heteroclinic orbits contained on invariant cylinders when the Chen system has a line of singularities and a first integral, which indicates the complicated dynamical behavior of the Chen system solutions even in the absence of chaotic dynamics.

On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

“…The following result shows that the dynamics in a neighborhood of the infinity for the unified chaotic system is the same as that in the Lorenz and in the Chen system, see for more details [Llibre et al, 2010] and [Llibre et al, 2012], respectively. The study of the infinity for a polynomial differential system is made using the Poincaré compactification, see section 2 for a brief introduction to such compactification.…”

Bifurcations at infinity, invariant algebraic surfaces, homoclinic and heteroclinic orbits and centers of a new Lorenz-like chaotic system

“…We observe that there is a difference between the phase portraits of system (1) at infinity given in this paper from the same type of phase portraits given for instance in [5][6][7][8]12,13]. Here the phase portrait at infinity in the local charts U 1 , U 2 and U 3 is represented in the Poincaré disk.…”

“…which also has a line of singularities given by the z 2 -axis. Considering the invariance of z 1 z 2 -plane, we study the phase portraits of system (8), which corresponds to the phase portraits of system (1) at infinity on the chart U 3 . They are shown in Figs.…”

“…Furthermore, some quadratic systems similar to system (2) also appeared, arising from physical models or proposed from an abstract point of view, which are shown to be chaotic [4,[10][11][12][14][15][16][17]19,20]. More recently, a series of papers have been published which present global dynamical analysis of Lorenz and Lorenz-like systems, giving a contribution to the understanding of these complex kinds of differential systems in R 3 ; see, for instance [5][6][7][8]12,13].…”

“…Beyond to complement the results presented in [4], the analysis of system (1) presented here makes part of a program aiming to describe global properties of quadratic three-dimensional differential systems defined in R 3 , which is being developed by several authors; see, for instance [5][6][7][8]12,13]. It is important to observe that these types of systems appear in the literature as mathematical models for several natural problems, coming from Physics, Biology and Engineering mathematical modeling; see, for instance, [3,5,6,9,13,21] and references therein.…”