Inestabilidades cuasi reversibles genéricas: la universalidad de las ecuaciones de Lorenz y Maxwell Bloch y un caso predictible de caos à la Shilnikov

Generic instabilities are fundamental for our understanding of the behavior of physical systems. We shall study here a new generic instabilty arising in quasi-reversible systems, i.e. reversible systems in which the time inversion invariance is weakly broken, and we can consider the symmetry breaking terms as part of the unfolding. The generic instabilities of these systems , which occur very frequently in physics, are determined by the reversible part of the evolution equations and a stable equilibrium corresponds to a spectrum of the linearized system which lies completely in the imaginary axis. We have then two generic instabilities: a) collision of two frequencies at the origin (stationary instabilility) and b) collision of two frequencies at a finite frequency, called “confusion de fréquences” par Y. Rocard who find this instability in the wings of airplanes . We have shown with Marcel Clerc and Pierre Coullet that case a) with reflection symmetry and injection of energy through a neutral mode-a conserved quantity in the reversible system- has as normal form the wellknown Lorenz equations which then turn to be universal equations (PRL 2002), and case b) has as normal form, in the same conditions, the Maxwell-Bloch equations for the laser effect (Optics Communications, 2003). I will present now case a) without reflection symmetry. The normal form in this case is a special case of the normal form in Arnold’s notation and it leads to Shilnikov chaos which can be analytically predicted in the space of parameters. Moreover we construct a very simple mechanical example , the Shilnikov particle, which makes the instability and exhibits Shilnikov chaos.