Asymptotic chaos

We provide in table I a list of normal forms of ordinary differential equations describing the dynamics of physical systems in conditions near to the simultaneous onset of up three instabilities. The first (quadratic) terms in the Taylor series for the nonlinear terms in these amplitude equations (as they are called in fluid dynamics) are given in each case. We focus on a particular example involving three instabilities and derive an asymptotic version of the corresponding normal form as a limit of small dissipation is approached. The numerical investigation of this asymptotic normal form strongly suggests that chaotic behavior occurs as close as one wants to the onset of the triple instability. This chaotic behavior is also exhibited by a return map constructed by direct numerical experiments on the amplitude equation. We also derive by analytic methods a model return map that qualitatively reproduces much of the dynamics observed numerically in the solutions of the asymptotic normal form in nearly homoclinic conditions. In the limit of strong contraction, this model map of the plane reduces to a unidimensional map that is valuable in understanding the dynamics of the original system.