In the fields of light manipulation and localization, quasiperiodic photonic crystals, or quasicrystals (PQs), are causing an upsurge in research because their rotational symmetry long-range orientation transverse lattice arrays, as they lack translational symmetry. It allows for optimization well-established propagation properties has introduced new guiding features. Therefore, a class, crysta...

This is a brief survey of recent results on the KAM stability of quasiperiodic dynamics using renormalization of vector fields.

Energy level spacing statistics are discussed for the octagonal tiling, a two dimensional quasiperiodic structure. A recursion relation is written for the probability distributions of variables defined on finite size approximants to this quasiperiodic tiling, using their property of similarity under inflation. Three types of distribution functions are introduced and determined by a combination ...

Speech is a complex acoustic signal showing a quasiperiodic structure at several timescales. Integrated neural signals recorded in the cortex also show periodicity at different timescales. In this chapter we outline the neural mechanisms that potentially allow the auditory cortex to segment and encode continuous speech. This chapter focuses on how the human auditory cortex uses the temporal str...

Quasipatterns (two-dimensional patterns that are quasiperiodic in any spatial direction) remain one of the outstanding problems of pattern formation. As with problems involving quasiperiodicity, there is a small divisor problem. In this paper, we consider 8-fold, 10-fold, 12-fold, and higher order quasipattern solutions of the Swift–Hohenberg equation. We prove that a formal solution, given by ...

We investigate localization phenomena and stability properties of quasiperiodic oscillations in N degree of freedom Hamiltonian systems and N coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of N coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), wh...

We consider travelling periodical and quasiperiodical waves in single mode fibres, with weak birefringence and under the action of cross-phase modulation. The problem is reduced to the " 1:2:1 " integrable case of the two-particle quartic potential. A general approach for finding elliptic solutions is given. New solutions which are associated with two-gap Treibich-Verdier potentials are found. ...