1 Routes to Chaos in Resonant Extrasolar Planetary Systems John D. Hadjidemetriou 1 and George Voyatzis 2 1 Department of Physics, University of Thessaloniki, Thessaloniki, Greece hadjidem@auth.gr 2 Department of Physics, University of Thessaloniki, Thessaloniki, Greece voyatzis@auth.gr Summary. We study the factors that affect the stability and the long term evolu- tion of a resonant planetary system. For the same resonance, the long term evolution of a resonant planetary system depends on the the relative orientation of the plan- etary orbits, on the phase of the planets on their orbits and on the proximity to a periodic resonant planetary system, either stable or unstable. Chaotic property is not always associated with a disruption of the system and the system may remain bounded for a long time. Key words: Periodic orbits, resonance, extrasolar systems, routes to chaos. 1.1 Introduction During the past 15 years many planetary systems were observed, called extrasolar planetary systems. Some of these systems are very different from our own Solar System, because they have massive planets close to the central star, which we will call the sun, of the order of the mass of Jupiter or larger, and large planetary eccentricities. In some of these planetary systems two or more planets were observed. In the case of two planets close to each other, the two planets are in mean motion resonance. Examples are HD 82943 and GLIESE 876, at the 2:1 resonance and 55Cnc at the 3:1 resonance. There are several problems associated with the study of the extrasolar systems, as their formation, their evolution and their dynamical stability. In this paper we will study some aspects of the dynamical stability of extrasolar planetary systems. There are different approaches to the study of the dynamical evolution of a plan- etary system and on the mechanisms that stabilize the system, or generate chaotic motion and instability: Beaug´ e and Michtchenko 2003, Beaug´ e et al. 2003, Gozd- jiewski et al. 2003, Michtchenko et al. 2006, Malhotra 2002, Lee 2004, Dvorak et al. 2005. In these papers different methods have been applied, as the averaging method, direct numerical integrations of orbits, or various numerical methods which provide indicators for the exponential separation of nearby orbits. In this way the regions