Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.Comment: 17 pages, 2 figure

Phononic canonical quasicrystalline waveguides

The dynamic behavior of the class of periodic waveguides whose unit cells are generated through a quasicrystalline sequence can be interpreted geometrically in terms of a trace map that embodies the recursive rule obeyed by traces of the transmission matrices. We introduce the concept of canonical quasicrystalline waveguides, for which the orbits predicted by the trace map at specific frequencies, called canonical frequencies, are periodic. In particular, there exist three families of canonical waveguides. The theory reveals that for those (i) the frequency spectra are periodic and the periodicity depends on the canonical frequencies, (ii) a set of multiple periodic orbits exists at frequencies that differ from the canonical ones, and (iii) perturbation of the periodic orbit and linearization of the trace map define a scaling parameter, linked to the golden ratio, which governs the self-similar structure of the spectra. The periodicity of the waveguide responses is experimentally verified on finite specimens composed of selected canonical unit cells

Phonon Localization in One-Dimensional Quasiperiodic Chains

Quasiperiodic long range order is intermediate between spatial periodicity and disorder, and the excitations in 1D quasiperiodic systems are believed to be transitional between extended and localized. These ideas are tested with a numerical analysis of two incommensurate 1D elastic chains: Frenkel-Kontorova (FK) and Lennard-Jones (LJ). The ground state configurations and the eigenfrequencies and eigenfunctions for harmonic excitations are determined. Aubry's "transition by breaking the analyticity" is observed in the ground state of each model, but the behavior of the excitations is qualitatively different. Phonon localization is observed for some modes in the LJ chain on both sides of the transition. The localization phenomenon apparently is decoupled from the distribution of eigenfrequencies since the spectrum changes from continuous to Cantor-set-like when the interaction parameters are varied to cross the analyticity--breaking transition. The eigenfunctions of the FK chain satisfy the "quasi-Bloch" theorem below the transition, but not above it, while only a subset of the eigenfunctions of the LJ chain satisfy the theorem.Comment: This is a revised version to appear in Physical Review B; includes additional and necessary clarifications and comments. 7 pages; requires revtex.sty v3.0, epsf.sty; includes 6 EPS figures. Postscript version also available at http://lifshitz.physics.wisc.edu/www/koltenbah/koltenbah_homepage.htm

Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasi-periodic edge of chaos

We investigate the probability density of rescaled sum of iterates of sine-circle map within quasi-periodic route to chaos. When the dynamical system is strongly mixing (i.e., ergodic), standard Central Limit Theorem (CLT) is expected to be valid, but at the edge of chaos where iterates have strong correlations, the standard CLT is not necessarily to be valid anymore. We discuss here the main characteristics of the central limit behavior of deterministic dynamical systems which exhibit quasi-periodic route to chaos. At the golden-mean onset of chaos for the sine-circle map, we numerically verify that the probability density appears to converge to a q-Gaussian with q<1 as the golden mean value is approached.Comment: 7 pages, 7 figures, 1 tabl

Lubricated friction between incommensurate substrates

This paper is part of a study of the frictional dynamics of a confined solid lubricant film - modelled as a one-dimensional chain of interacting particles confined between two ideally incommensurate substrates, one of which is driven relative to the other through an attached spring moving at constant velocity. This model system is characterized by three inherent length scales; depending on the precise choice of incommensurability among them it displays a strikingly different tribological behavior. Contrary to two length-scale systems such as the standard Frenkel-Kontorova (FK) model, for large chain stiffness one finds that here the most favorable (lowest friction) sliding regime is achieved by chain-substrate incommensurabilities belonging to the class of non-quadratic irrational numbers (e.g., the spiral mean). The well-known golden mean (quadratic) incommensurability which slides best in the standard FK model shows instead higher kinetic-friction values. The underlying reason lies in the pinning properties of the lattice of solitons formed by the chain with the substrate having the closest periodicity, with the other slider.Comment: 14 pagine latex - elsart, including 4 figures, submitted to Tribology Internationa

325 MHz VLA Observations of Ultracool Dwarfs TVLM 513-46546 and 2MASS J0036+1821104

We present 325 MHz (90 cm wavelength) radio observations of ultracool dwarfs TVLM 513-46546 and 2MASS J0036+1821104 using the Very Large Array (VLA) in June 2007. Ultracool dwarfs are expected to be undetectable at radio frequencies, yet observations at 8.5 GHz (3.5 cm) and 4.9 GHz (6 cm) of have revealed sources with > 100 {\mu}Jy quiescent radio flux and > 1 mJy pulses coincident with stellar rotation. The anomalous emission is likely a combination of gyrosynchrotron and cyclotron maser processes in a long-duration, large-scale magnetic field. Since the characteristic frequency for each process scales directly with the magnetic field magnitude, emission at lower frequencies may be detectable from regions with weaker field strength. We detect no significant radio emission at 325 MHz from TVLM 513-46546 or 2MASS J0036+1821104 over multiple stellar rotations, establishing 2.5{\sigma} total flux limits of 795 {\mu}Jy and 942 {\mu}Jy respectively. Analysis of an archival VLA 1.4 GHz observation of 2MASS J0036+1821104 from January 2005 also yields a non-detection at the level of < 130 {\mu}Jy . The combined radio observation history (0.3 GHz to 8.5 GHz) for these sources suggests a continuum emission spectrum for ultracool dwarfs which is either flat or inverted below 2-3 GHz. Further, if the cyclotron maser instability is responsible for the pulsed radio emission observed on some ultracool dwarfs, our low-frequency non-detections suggest that the active region responsible for the high-frequency bursts is confined within 2 stellar radii and driven by electron beams with energies less than 5 keV.Comment: 11 pages, 5 figures, submitted to A

Solitons and exact velocity quantization of incommensurate sliders

We analyze in some detail the recently discovered velocity quantization phenomena in the classical motion of an idealized one-dimensional solid lubricant, consisting of a harmonic chain interposed between two periodic sliders. The ratio w = v_cm/v_ext of the chain center-of-mass velocity to the externally imposed relative velocity of the sliders is pinned to exact ``plateau'' values for wide ranges of parameters, such as sliders corrugation amplitudes, external velocity, chain stiffness and dissipation, and is strictly determined by the commensurability ratios alone. The phenomenon is caused by one slider rigidly dragging the density solitons (kinks/antikinks) that the chain forms with the other slider. Possible consequences of these results for some real systems are discussed.Comment: 12 pages 6 figures. Small fixup after Referee's comments. In print in J. Phys.: Condens. Matte