Recent developments in applications of dynamical system theory

摘要

Throughout history, challenging practical problems act as a driving force that push forward theoretical development and even foster the establishment of new research areas. This is exactly the case related to the dynamical system theory. This theory appears as a outgrowth of the remarkable work of Poincare in trying to find out whether our Solar System is stable or not. To make this problem more mathematically tractable, he considered a special case in which two massive particles, the primaries, move in circular orbits around their center of mass, while another particle with a smaller mass moves in the plane of the primaries so that it suffers their gravitational influence but its presence does not disturb the motion of the primaries. This is the so called circular restricted three-body problem whose analysis, despite its apparently simplicity, repre sents a tremendous mathematical challenge. In his undertaken [1 ], Poincare developed and used innovative qualitative tech niques based on geometry to understand the global behavior of the solutions. As so, he introduced the concept of (Poincare) surface of section which he used to analyze solutions in the neighborhood of periodic orbits; defined the concepts of stable and unstable manifolds and described the complicated dynamical behavior that happens when these invariant sets have a transverse intersection. Actually, by identifying this behavior and understand its consequences, Poincare is considered to be first one who discover the nowadays named chaotic behavior [2].