Quasi-Periodic Bifurcations and Chaos

Interactions of two or more nonlinear oscillators generally produce complex dynamics with periodic solutions, tori, tori doubling, and chaos. A natural feature in real-life models is that the interactions are quasi-periodic, which means that the individual components have frequencies that are incommensurable. For a discussion of diophantine frequency vectors and the measure of Cantor sets in two-frequency systems, see [1], with many references also inspired by the conservative setting leading to families of invariant tori; we mention in this setting the basic paper [2]. Our focus is different from KAM theory and, with some exceptions, also from dissipa tive KAM theory, as we are especially interested in the practical context where we start with dissipative systems that are subsequently perturbed and lead to complex behaviour. This can produce bifurcations of isolated tori and their qualitative changes, whereas classical KAM theory is involved with conservative, usually Hamiltonian, systems where tori arise in dense sets, as well as families with positive measure. As we shall see, we will consider for our analysis a system with damping and thermostatic control, where the combination of two of such systems adds forcing and bifurcation phenomena. Multifrequency oscillations arise in many applications of various disciplines such as mechanical engineering, laser systems, and electronic circuits; for a useful list of such applications in many fields, see [3] (in particular their references 1-23) and [4]. See also [5,6]. In [3,4], the emphasis is on the construction of charts of Lyapunov exponents for interacting self-excited systems. Such numerically obtained charts yield enormous inspiration for further analysis, but in our approach, we combine both analytic methods (averaging) and numerical bifurcation theory. This hybrid approach will be fruitful. A number of recent papers on quasi-perioidc bifurcations is concerned with maps; such an approach produces important general insight, but the relation with ODEs has yet to be established.