Dynamics at infinity and other global dynamical aspects of Shimizu–Morioka equations

Abstract

We present some global dynamical aspects of Shimizu–Morioka equations given by $$\dot{x}=y, \qquad\dot{y}=x-\lambda y-xz, \qquad\dot{z}=-\alpha z+x^2,$$ where (x,y,z)∈ℝ3 are the state variables and λ,α are real parameters. This system is a simplified model proposed for studying the dynamics of the well-known Lorenz system for large Rayleigh numbers. Using the Poincare compactification of a polynomial vector field in ℝ3, we give a complete description of the dynamics of Shimizu–Morioka equations at infinity. Then using analytical and numerical tools, we investigate for the case α=0 the existence of infinitely many singularly degenerate heteroclinic cycles, each one consisting of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of these equilibria. The dynamical consequences of the existence of these cycles are also investigated. The present study is part of an effort aiming to describe global properties of quadratic three-dimensional vector fields with chaotic dynamical behavior, as made for instance in (Dias et al. in Nonlinear Anal. Real World Appl. 11(5):3491–3500, 2010; Kokubu and Roussarie in J. Dyn. Differ. Equ. 16(2):513–557, 2004; Llibre and Messias in Physica D 238(3):241–252, 2009; Llibre et al. in J. Phys. A, Math. Theor. 41:275210, 2008; Llibre et al. in Int. J. Bifurc. Chaos Appl. Sci. Eng. 20(10):3137–3155, 2010; Lorenz in J. Atmos. Sci. 20:130–141, 1963; Lu et al. in Int. J. Bifurc. Chaos Appl. Sci. Eng. 14(5):1507–1537, 2004; Mello et al. in Chaos Solitons Fractals 37:1244–1255, 2008; Messias in J. Phys. A, Math. Theor. 42:115101, 2009; Messias et al. in TEMA Tend. Mat. Apl. Comput. 9(2):275–285, 2008).