Stability analysis of periodic orbits in nonlinear dynamical systems using Chebyshev polynomials

We propose an algorithm to identify numerically periodic solutions of high-dimensional dynamical systems and their local stability properties. One of the most popular approaches is the Harmonic Balance Method (HBM), which expresses the cycle as a sum of Fourier modes and analyses its stability using the Hill's method. A drawback of Hill's method is that the relevant Floquet exponents have to be chosen from all the computed exponents. To overcome this problem the current work discusses the application of Chebyshev polynomials to the description of the time dependence of the periodic dynamics. The stability characteristics of the periodic orbit are directly extracted from the linearisation around the periodic orbit. The method is compared with the HBM with examples from Lorenz and Langford systems. The main advantage of the present method is that, unlike HBM, it allows for an unambiguous determination of the Floquet exponents. The method is applied to natural convection in a differentially heated cavity which demonstrates its potential for large scale problems arising from the discretisation of the incompressible Navier-Stokes equations.