Effective Stability in Reversible Systems

It is di cult in the theory of dynamical systems to draw a boundary line between con servation laws and symmetries because often their e ects on the dynamics can be almost the same An important example can be the case of motions in Hamiltonian and reversible systems For instance it is well known that the Kolmogorov Arnol d Moser theory applies in both type of systems see In the theory of Hamiltonian systems we also have the Nekhoroshev theorem that leads to the concept of e ective stability In this paper we present a procedure to put in normal form a nearly integrable re versible system not necessarily a Hamiltonian system Furthermore non resonant stabil ity estimates are obtained As an application we discuss the case of n harmonic oscillators with frequencies satisfying Diophantine conditions We will consider only linear reversibil ity because it appears often in Hydrodynamics Thermodynamics Fluid Mechanics and Electromagnetism Given a linear involution R of R n we will say that a dynamical system is R reversible or simply reversible if it is invariant under the transformationX RX t t where X R n In other words we can say that an autonomous system X F X is reversible if it veri es F t uF where the functional operator t u is de ned as t uF R F R Note that an important di erence between reversible systems and Hamiltonian systems is that we have no energy conservation in the rst ones