  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.
