Computation and stability analysis of periodic orbits using finite differences, Fourier or Chebyshev spectral expansions in time

TL;DR

This paper compares finite differences, Fourier, and Chebyshev spectral methods for computing and analyzing the stability of periodic orbits in high-dimensional dynamical systems, highlighting advantages of Chebyshev expansions.

Contribution

It introduces a novel approach using Chebyshev polynomials for stability analysis that directly yields Floquet exponents, improving over traditional Fourier methods.

Findings

Chebyshev and finite difference methods directly provide Floquet exponents.

Chebyshev expansion is effective for large-scale Navier-Stokes problems.

All methods show good convergence on benchmark systems.

Abstract

We analyse and compare several algorithms to compute numerically periodic solutions of high-dimensional dynamical systems and investigate their Floquet stability without building the monodromy matrix. The solution and its perturbation are discretised in time either using finite differences, Fourier-Galerkin or Chebyshev expansions. The resulting nonlinear set of equations describing the periodic orbit is solved using a Newton-Raphson algorithm. The linearised equations determining the stability lead to a generalised eigenvalue problem. Unlike the Fourier-Galerkin method, the use of Chebyshev polynomials or finite differences has the advantage that the relevant Floquet exponents are directly given without the well known issue of having to sort out the eigenvalues. The speed of convergence of these three methods is illustrated with examples from the Lorenz system, the Langford system and a two-dimensional thermal convection flow inside a differentially heated cavity. This last example demonstrates the potential of the newly proposed Chebyshev expansion for large-scale problems arising from the discretisation of the incompressible Navier-Stokes equations.