Poincaré's program for the global analysis of a dynamical system starts by considering
simple solutions, such as equilibria and periodic orbits, together with their corresponding …

This paper develops a Chebyshev–Taylor spectral method for studying stable/unstable
manifolds attached to periodic solutions of differential equations. The work exploits the …

In this paper, we present and illustrate a general methodology to apply KAM theory in
particular problems, based on an a posteriori approach. We focus on the existence of real …

In this work we develop a computer-assisted technique for proving existence of periodic
solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit …

In this work we develop some automatic procedures for computing high order polynomial
expansions of local (un) stable manifolds for equilibria of differential equations. Our method …

We study polynomial expansions of local unstable manifolds attached to equilibrium
solutions of parabolic partial differential equations. Due to the smoothing properties of …

We present numerical results and computer assisted proofs of the existence of periodic
orbits for the Kuramoto--Sivashinky equation. These two results are based on writing down …

We develop a theoretical framework for computer-assisted proofs of the existence of
invariant objects in semilinear PDEs. The invariant objects considered in this paper are …

This lecture describes validated numerical tools which are used for global analysis of
nonlinear systems. The main focus is dynamics near and between equilibrium solutions of …

We perform a numerical study of the breakdown of hyperbolicity of quasi-periodic attractors
in the dissipative standard map. In this study, we compute the quasi-periodic attractors …