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

We present a global dynamical analysis of the following quadratic differential system (Formula presented.) , where (Formula presented.) 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 (Formula presented.) , alike Lorenz, Rössler, Chen and other. By using the Poincaré compactification for a polynomial vector field in (Formula presented.) , we study the dynamics of this system on the Poincaré 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.

Keywords

Centers on R3, Dynamics at infinity, First integral, Heteroclinic orbits, Homoclinic orbits, Invariant algebraic surfaces, Poincaré compactification, Quadratic system