Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbi...

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of ...

We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically and exactly, state by state, by means of periodic orbit expansions. We prove analytically that the periodic orbit series exist and converge to the correct spec...

Let f be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector ( q , r q ). Then f has a topologically monotone periodic orbit with the same

We introduce a general procedure for obtaining a low-dimensional linear time-periodic model from a very high-dimensional nonlinear system that has an asymptotically stable periodic orbit. Our goal is to develop models that are suitable for designing feedback controllers for fluids systems with periodic orbits, such as periodically shedding wakes, or flow control problems where periodic actuatio...

In this paper we introduce a general methodology for computing numerically the normal form around a periodic orbit of an autonomous analytic Hamiltonian system The process follows two steps First we expand the Hamiltonian in suit able coordinates around the orbit and second we perform a standard normal form scheme based on the Lie series method This scheme is carried out up to some nite order a...

In this paper we introduce a general methodology for computing (numerically) the normal form around a periodic orbit of an autonomous analytic Hamiltonian system. The process follows two steps. First, we expand the Hamiltonian in suitable coordinates around the orbit and second, we perform a standard normal form scheme, based on the Lie series method. This scheme is carried out up to some nite ...

— We consider the Klein-Gordon equation (KG) on a Riemannian surface M ∂ t u−∆u−mu+ u = 0, p ∈ N, (t, x) ∈ R×M, which is globally well-posed in the energy space. Viewed as a first order Hamiltonian system in the variables (u, v ≡ ∂tu), the associated flow lets invariant the two dimensional space of (u, v) independent of x. It turns out that in this invariant space, there is a homoclinic orbit t...

— We consider the Klein-Gordon equation (KG) on a Riemannian surface M ∂ t u−∆u−mu+ u = 0, p ∈ N, (t, x) ∈ R×M, which is globally well-posed in the energy space. Viewed as a first order Hamiltonian system in the variables (u, v ≡ ∂tu), the associated flow lets invariant the two dimensional space of (u, v) independent of x. It turns out that in this invariant space, there is a homoclinic orbit t...

Recently found unstable time-periodic solutions to the incompressible Navier-Stokes equation are reviewed to discuss their relevance to plane Couette turbulence and isotropic turbulence. It is shown that the periodic motion embedded in the Couette turbulence exhibits a regeneration cycle of near-wall coherent structures, which consists of formation and breakdown of streamwise vortices and low-v...