The second largest Lyapunov exponent and transition to chaos of natural convection in a rectangular cavity

In this study we have proposed an accurate and simple method to evaluate the Lyapunov spectrum. The method is suitable for any discretization method that finally expresses a governing equation system in the form of an ordinary differential equation system. The method was applied to evaluate up to the second largest Lyapunov exponents for natural convection in a rectangular cavity with heated and cooled side walls. The main results are as follows: (1) the largest and second largest Lyapunov exponents can be evaluated without any parameters that affects the exponents. (2) The second largest Lyapunov exponent makes it possible to classify quantitatively thermal convection fields into five regimes against the Rayleigh number and to clarify the transition route from steady state to chaos by identifying the first and second Hopf bifurcations. (3) The fluctuation in thermal convection fields just over the critical Rayleigh number at which Hopf bifurcation occurs can be quantitatively explained by using normalized Lyapunov vectors, associated with the computation of the Lyapunov exponents, just under the critical point.