Kolmogorov-Arnol’d-Moser Theory classically was mainly developed for conservative systems, establishing persistence results for quasi-periodic invariant tori in nearly integrable systems. In this survey we focus on dissipative systems, where similar results hold. In non-conservative settings often parameters are needed for the persistence of invariant tori. When considering families of such dyn...

We consider 1D completely resonant nonlinear wave equations of the type vtt − vxx = −v +O(v) with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, de...

We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of b frequencies, b ≤ d + 2, in arbitrary dimension d and for arbitrary non integrable algebraic nonlinearity p. This reflects the conservation of d momenta, energy and L norm. In 1d, we prove the existence of quasi-periodic solutions with arbit...

For every prime p and integer n > 3 we explicitly construct an abelian variety A/Fpn of dimension n such that for a suitable prime l the group of quasi-isogenies of A/Fpn of l-power degree is canonically a dense subgroup of the n-th Morava stabilizer group at p. We also give a variant of this result taking into account a polarization. This is motivated by the recent construction of topological ...

Quasi-phase matching schemes employing quasi-periodic or random spatial modulations, previously applied to perturbative nonlinear optics, are demonstrated theoretically for the extreme nonlinear optical process of high harmonic generation. We show that quasi-periodic quasi-phase matching of high harmonic generation can be used for simultaneous enhancement of arbitrarily chosen spectral regions....

We consider travelling periodic and quasi-periodic wave solutions of a set of coupled nonlinear Schrödinger equations. In fibre optics these equations can be used to model single mode fibers with strong birefingence and two-mode optical fibers. Recently these equations appear as model, which describe pulse-pulse interaction in wavelength-division-multiplexed channels of optical fiber transmissi...

Coexistence The special case when all the independent solutions of a linear, T -periodic ODE are T -periodic. Hill’s equation A second order ODE of the form ẍ + p(t)x = 0, with p(t) T -periodic. Instability pockets Finite domains, usually intersections of instability tongues, where the trivial solution of linear, T -periodic ODEs is unstable. Instability tongues Domains in parameter space where...

In this paper, we investigate the transitional dynamics and quasi-periodic solution appearing after the Saddle-Node (SN) bifurcation of a periodic solution in an inductor-coupled asymmetrical van der Pol oscillators with hard-type nonlinearity. In particular, we elucidate, by investigating global bifurcation of unstable manifold (UM) of saddles, that transitional dynamics and quasi-periodic sol...