Integrability and global dynamics of quadratic vector fields defined on R3 with Quadrics as invariant surfaces

Abstract

With the present research project we propose the global analysis of polynomial differential systems defined on the space R3, which has a quadric as an invariant surface. The global analysis proposed consists basicaly in three steps: 1) determination of the quadratic vector fields which has a quadric as an invariant algebraic surfaces; 2) Poincaré compactification of the systems, which enables their extension to analytic systems defined on the closed ball of radius one (Poincaré ball), whose boundary, the sphere S2 (Poincaré sphere) is invariant under the flow of the extended system and corresponds to the points of R3 at infinity; 3) study of the solutions on the invariant algebraic surfaces and how this surfaces are contained in the Poincaré ball; study of the end of these surfaces at infinity (intersection with the Poincaré sphere) and consequently the description of the dynamics at infinity. The proposed analysis enable us to describe important global structures of the polynomial systems on the whole phase space R3. Furthermore, an analytical/numerical study shows that small perturbations of these global structures by varying the parameters of the systems may lead to the creation of chaotic dynamics. In this way, the understanding of such structures is an important starting point to the understanding of the complex dynamical behavior of the solutions of polynomial systems on R3. In the proposed analysis we will use the classical results of the qualitative theory and bifurcations of ordinary differential equations, combined with numerical simulations through the software MAPLE. (AU)