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

Abstract

We present a global dynamical analysis of the following quadratic differential system $$\dot{x}{=}a(y{-}x), \dot{y}\!=\!dy-xz, \dot{z}\!=\!-bz+fx^2+gxy$$ , where $$(x,y,z)\in {\mathbb {R}}^3$$ are the state variables and a, b, d, f, g are real parameters. This system has been proposed as a new type of chaotic system, having additional complex dynamical properties to the well-known chaotic systems defined in $${\mathbb {R}}^3$$ , alike Lorenz, Rossler, Chen and other. By using the Poincare compactification for a polynomial vector field in $${\mathbb {R}}^3$$ , we study the dynamics of this system on the Poincare ball, showing that it undergoes interesting types of bifurcations at infinity. We also investigate the existence of first integrals and study the dynamical behavior of the system on the invariant algebraic surfaces defined by these first integrals, showing the existence of families of homoclinic and heteroclinic orbits and centers contained on these invariant surfaces.