The Center Problem in 3-Dimensional Systems and Applications

For high dimensional systems, it is a well-known method to restrict the system to its a center manifold, but the approximation of the center manifold brings a greater complexity in the computation of Lyapunov quantities and their dependence even if the original system is not of high degree. Furthermore, in order to determine if a center-focus equilibrium is a center, criteria for planar systems, such as time reversibility and integrability, are not available on the approximated center manifold. In spite of this, we have recently been able to obtain some good results for integrability in a 3-dimensional system. We do this by finding a global center manifold of the system and reducing the higher dimensional system to the global center manifold. This result gives a useful method for identifying a focus or center in high dimensional systems.