CHAOTIC DYNAMICAL SYSTEMS AND APPLICATIONS TO THE SOLAR SYSTEM DYNAMICS
Dynamical systems are an essential tool to model physical phenomena in applied sciences whose state changes over time according to either differential or discrete difference equations. In this context two concepts are in opposition: “order”, or “integrability”, versus “chaos”. Integrable systems, for which all the solutions can be explicitly analytically determined, are special and represent only a crude approximation of the real dynamics. On the other hand, more accurate models are usually represented by non-integrable differential equations, whose solutions exhibit a highly sensitive dependence on initial conditions, termed as chaotic.
In this talk we discuss some of the main geometric and topological properties of deterministic chaos in connection with orbital stability of small objects in our solar system. After a short recap of the theory of non-linear dynamical systems, we present a modern approach of detecting and quantifying chaotic behaviors using finite time chaos indicators, a numerical strategy capable to capture the dynamical structure of the phase space. In the second part of the seminar, we introduce the restricted N-body problem in Hamiltonian mechanics and implement the above technique to discriminate between the realms of regular and chaotic motions of asteroids.