Abstract: A free-floating plate is introduced in a Benard convection cell with an open surface. It partially covers the cell and distorts the local heat flux, inducing a coherent flow that in turn moves the plate. Remarkably, the plate can be driven to a periodic motion even under the action of a turbulent fluid. The period of the oscillation depends on the coverage ratio, and on the Rayleigh number of the convective system. The plate oscillatory behavior observed in this experiment may be related to a geological model, in which continents drift in a quasiperiodic fashion. (c) 2000 The American Physical Society.

Abstract: In this paper the monotonic twist theorem is extended to the quasiperiodic case and applied to establish regularity of motion in a system of a particle bouncing elastically between two quasiperiodically moving walls. It is shown that the velocity of the particle is uniformly bounded in time if the frequencies satisfy a Diophantine inequality. This answers a question recently asked in Levi and Zehnder (1995 SIAM J. Math. Anal. 26 1233-56).

Abstract: We consider the underdamped dynamics of a chain of atoms subject to a dc driving force and a quasiperiodic substrate potential. The system has three inherent length scales which we take to be mutually incommensurate. We find that when the length scales are related by the spiral mean (a cubic irrational) there exists a value of the interparticle interaction strength above which the static friction is zero. When the length scales are related by the golden mean (a quadratic irrational) the static friction is always nonzero. >From considerations based on the connection of this problem to standard map theory, we postulate that zero static friction is generally possible for incommensurate ratios of the length scales involved. However, when the length scales are quadratic irrationals, or have some commensurability with each other, the static friction will be nonzero for all choices of interaction parameters. We also comment on the nature of the depinning mechanisms and the steady states achieved by the moving chain.

Abstract: We introduce a spatially localized inhomogeneity into the two-dimensional complex Ginzburg-Landau equation. We observe that this can produce two types of target wave patterns: stationary and breathing. In both cases, far from the target center, the field variables correspond to an outward propagating periodic traveling wave. In the breathing case, however, the region in the vicinity of the target center experiences a periodic temporal modulation at a frequency, in addition to that of the wave frequency of the faraway outward waves. Thus at a fixed point near the target, the breathing case yields a quasiperiodic time variation of the field. We investigate the transition between stationary and breathing targets, and note the existence of hysteresis. We also discuss the competition between the two types of target waves and spiral waves.

Abstract: This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.

Abstract: It is well-known that the dynamics of the Arnold circle map is phase-locked in regions of the parameter space called Arnold tongues. If the map is invertible, the only possible dynamics is either quasiperiodic motion, or phase-locked behavior with a unique attracting periodic orbit. Under the influence of quasiperiodic forcing the dynamics of the map changes dramatically. Inside the Arnold tongues open regions of multistability exist, and the parameter dependency of the dynamics becomes rather complex. This paper discusses the bifurcation structure inside the Arnold tongue with zero rotation number and includes a study of nonsmooth bifurcations that happen for large nonlinearity in the region with strange nonchaotic attractors.

Abstract: A very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element, exhibiting a rich variety of dynamical features, is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua (MLC). By constructing a two-parameter phase diagram in the (F - ω) plane, corresponding to the forcing amplitude (F) and frequency (ω), we identify, besides the familiar period-doubling scenario to chaos, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, Farey sequences, and so on. The chaotic dynamics is verified by both experimental as well as computer simulation studies including PSPICE.

Abstract: A very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element, exhibiting a rich variety of dynamical features, is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua(MLC). By constructing a two-parameter phase diagram in the $(F-\omega)$ plane, corresponding to the forcing amplitude (F) and frequency $(\omega)$, we identify, besides the familiar period-doubling scenario to chaos, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, Farey sequences, and so on. The chaotic dynamics is verified by both experimental as well as computer simulation studies including PSPICE.

Abstract: This paper discusses the dynamics of the circus movement around a one-dimensional continuous and uniform loop of model cardiac cells. The membrane ionic currents are represented by a modified Beeler–Reuter formulation. The description of the quasiperiodic regimes initiated in a previous paper is completed and their equivalence with those predicted by an integral-delay model is discussed. The two models differ in the number of quasiperiodic modes of reentry that can be sustained and in the type of bifurcation by which they appear. These differences are the consequences of the modulation of the repolarization phase by diffusion in the ionic model. A modified version of the integral-delay model that includes a spatial averaging in the calculation of the action potential duration is shown to suppress the discrepancies. © 2000 Biomedical Engineering Society.

Abstract: A general solution of the elastic fields in 1D hexagonal quasicrystals with point groups 6mm, 62h2h, m2h and 6/mhmm is given in terms of four `harmonic' functions Fi (i = 1,2,3,4). Then we consider the problem of a circular crack embedded in an infinite 1D hexagonal quasicrystal of point group 6mm. The results obtained in this paper automatically reduce to those in the classical elasticity theory when the phason field is absent.

Abstract: The electrical transport in quasiperiodic systems at zero temperature is studied by means of the KuboGreenwood formula within a tight-binding model. Their dc conductivity is compared with that obtained from the Landauer formula. Special attention is paid to the transparent states, whose transmittance is unity. The ac conductivity of these states shows a rapid diminution as a function of the frequency, in comparison with that of periodic systems. Minima in these conduction spectra are observed, which are located at ( N21)\v 54nputu for the periodic case. Finally, the localization of the eigenstates is analyzed by looking at the Lyapunov exponent and the participation ratio. The latter is shown to be an inappropriate quantity to characterize the critically localized states.

Abstract: For the engineering of mechanical systems with a complex interplay of regular and chaotic behavior it is important to know the forces involved. It is shown how they can be computed and their time developntent evaluated. Characteristic features of periodic, quasiperiodic, and chaotic motion are identified. Classical methods such as Fourier transform and various statistics are used and compared to a redundant version of wavelet analysis. The latter is proposed as the most informative coherent representation of the distribution of times, frequencies, and amplitudes.

Abstract: We present a method for constructing models of weakly self-gravitating, finite dispersion eccentric stellar disks around central black holes. The disk is stationary in a frame rotating at a constant precession speed. The stars populate quasiperiodic orbits whose parents are numerically integrated periodic orbits in the total potential. We approximate the quasiperiodic orbits by distributions of Kepler orbits dispersed in eccentricity and orientation, using an approximate phase space distribution function written in terms of the Kepler integrals of motion. We show an example of a model with properties similar to those of the double nucleus of M31. The properties of our models are primarily determined by the behavior of the periodic orbits. Self-gravity in the disk causes these orbits to assume a characteristic radial eccentricity profile, which gives rise to distinctive multi-peaked line-of-sight velocity distributions (LOSVDs) along lines of sight near the black hole. The multi-peaked features should be observable in M31 at the resolution of STIS. These features provide the best means of identifying an eccentric nuclear disk in M31, and can be used to constrain the disk properties and black hole mass.

Abstract: Consider a linear autonomous Hamiltonian system with a time periodic bound state solution. In this paper we study the structural instability of this bound state ^M relative to time almost periodic perturbations which are small, localized and Hamiltonian. This class of perturbations includes those whose time dependence is periodic, but encompasses a large class of those with finite (quasiperiodic) or infinitely many non-commensurate frequencies. Problems of the type considered arise in many areas of application including ionization physics and the propagation of light in optical fibers in the presence of defects. The mechanism of instability is radiation damping due to resonant coupling of the bound state to the continuum modes by the time-dependent perturbation. This results in a transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. These results generalize those of A. Soffer and M.I. Weinstein, who treated localized time-periodic perturbations of a particular form. In the present work, new analytical issues need to be addressed in view of (i) the presence of infinitely many frequencies which may resonate with the continuum as well as (ii) the possible accumulation of such resonances in the continuous spectrum. The theory is applied to a general class of Schr\"odinger operators.

Abstract: Generic symmetry and transport properties of near separatrix motion in $1\frac{1}{2}$-degree-of-freedom Hamiltonian systems are studied. First the rescaling invariance of motion near saddle points, with respect to the transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}\ensuremath{\lambda}\ensuremath{\epsilon},$ $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\chi}}\ensuremath{\chi}+\ensuremath{\pi}$ of the amplitude $\ensuremath{\epsilon}$ and phase $\ensuremath{\chi},$ of the time-periodic perturbation, is recalled. The rescaling parameter $\ensuremath{\lambda}$ depends only on the frequency of the perturbation, and the behavior of an unperturbed Hamiltonian near a saddle point. Additional rescaling symmetry of the motion with respect to transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\epsilon}}{\ensuremath{\lambda}}^{1/2}\ensuremath{\epsilon},$ $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\chi}}\ensuremath{\chi}\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ is found for some Hamiltonian systems possessing symmetry in the phase space. It is shown that these rescaling invariance properties of motion lead to strong periodic (or quasiperiodic) dependencies of all statistical characteristics of the chaotic motion near the separatrix on ${\mathrm{log}}_{10}\ensuremath{\epsilon}$ with the period ${\mathrm{log}}_{10}\ensuremath{\lambda}.$ These properties are examined for different models of chaotic motion by direct numerical integrations of equations of motion, as by well as using a computationally efficient method of the separatrix mapping.

Abstract: A cell dynamical system model for deterministic chaos enables precise quantification of the round-off error growth,i.e., deterministic chaos in digital computer realizations of mathematical models of continuum dynamical systems. The model predicts the following: (a) The phase space trajectory (strange attractor) when resolved as a function of the computer accuracy has intrinsic logarithmic spiral curvature with the quasiperiodic Penrose tiling pattern for the internal structure. (b) The universal constant for deterministic chaos is identified as the steady-state fractional round-off error k for each computational step and is equal to 1 /sqr(tau) (=0.382) where tau is the golden mean. (c) The Feigenbaum's universal constants a and d are functions of k and, further, the expression 2(a**2) = (pie)*d quantifies the steady-state ordered emergence of the fractal geometry of the strange attractor. (d) The power spectra of chaotic dynamical systems follow the universal and unique inverse power law form of the statistical normal distribution.

Abstract: The circular differential equation (2) is studied thoroughly in this paper. Some inspiring methods are developed to treat this equation and the corresponding Poincare maps are constructed exactly. Depending on the control parameter β the equation may exhibit limit cycles, subharmonic or quasiperiodic motions. When β varies from β>1 to β m≔ 1−β 2 =q/p is a rational or an irrational number, and for the latter it is dense in S 1 .

Abstract: After establishing the method of constructing a class of one-dimensional (1D) Fibonacci-class quasiperiodic (FC(n)) ferroelectric domains system, we have studied the properties of the electric field of the second harmonic generation (SHG) by means of the small-signal approximation in the case of the vertically transmission. It was found that only the second harmonic light (SHL) peaks which were indexed by two special integers q and p would be the brightest and the spectra whose positions were decided by successive FC(n) integers \(\) and \(\) were perfect self-similar without considering the dispersive effect of the refractive index on SHL. The effect of the vacancies for some special spectral lines was also studied generally. The analytic results were confirmed by the numerical simulations.

Abstract: We studied statistical properties of the energy spectra of two-dimensional quasiperiodic tight-binding models. The multifractal nature of the eigenstates of these models is corroborated by the scaling of the participation numbers with the system size. Hence one might have expected ‘critical’ or ‘intermediate’ statistics for the level-spacing distributions as observed at the metal-insulator transition in the three-dimensional Anderson model of disorder. However, our numerical results are in perfect agreement with the universal level-spacing distributions of the Gaussian orthogonal random matrix ensemble, including the distribution of spacings between second, third, and fourth neighbour energy levels.

Abstract: We propose a structure model for the ${\mathrm{Al}}_{72}{\mathrm{Ni}}_{20}{\mathrm{Co}}_{8}$ decagonal quasicrystals based on its ${\ensuremath{\tau}}^{2}$-inflated ${\mathrm{Al}}_{13}{\mathrm{Co}}_{4}$ approximant phase: Applying a ${10}_{5}$ screw operation on a monolayer obtained from the approximant reproduces almost all features of 2-nm clusters seen in atomic-resolution Z-contrast images of ${\mathrm{Al}}_{72}{\mathrm{Ni}}_{20}{\mathrm{Co}}_{8}$ decagonal quasicrystals. The exception is the central ring, where the symmetry is broken due to chemical ordering of Al and transition metals. By restricting possible overlaps, this enforces the perfect quasiperiodic tiling.

Abstract: With a generic model excitable system, we have investigated the spatio-temporal dynamics of a spiral tip in the presence of an extrinsic localized periodic modulation. The tip of a modulated spiral does exhibit a variety of different trajectories depending on the strength and the frequency of the modulation. Its motion can be quasiperiodic on a 2-torus, quasiperiodic or mode locked on a 3-torus, or fully chaotic. Various bifurcations, including hard Hopf bifurcations and saddle-node bifurcations at strong resonances and period-doubling bifurcations of a mode-locked 3-torus, are revealed. In particular, the phenomenon of the period-doubling cascade of a resonant spiral tip trajectory is reported.

Abstract: The low energy behavior of the S = 1/2 antiferromagnetic Heisenberg chains with precious mean quasiperiodic exchange modulation is studied y the density matrix method. Based on the scaling behavior of the energy gap distribution, it is found that the ground state of this model belongs to a universality class different from that of the XY chain for which the precious mean exchange modulation is marginal. This result is consistent with the recent bosonization analysis of Vidal, Mouhanna and Giamarchi.

Abstract: We investigate zero-field Ising models on periodic approximants of planar quasiperiodic tilings by means of partition function zeros and high-temperature expansions These are obtained by employing a determinant expression for the partition function The partition function zeros in the complex temperature plane yield precise estimates of the critical temperature of the quasiperiodic model Concerning the critical behaviour, our results are compatible with Onsager universality, in agreement with the Harris–Luck criterion based on scaling arguments

Abstract: The geometrical relationships between quasiperiodic structures and the cubic β-phase (CsCl-type) formed in a continuous phase transformation is discussed on the examples of decagonal Al–Co–Me (Me=Cu, Ni) and icosahedral Al–Mn–Pd. For these purposes, first the alternative higher-dimensional embedding of decagonal Al–Co–Ni as incommensurately modulated structure is introduced. By projection onto the physical space, a periodic average structure of the decagonal quasicrystal can be obtained. It is demonstrated that the lattice of the periodic average structure of the quasiperiodic phases matches very well the lattice of the cubic β-phase. In this way, the experimentally observed orientational order and registry of the cubic β-phase with regard to the quasiperiodic structure can be nicely explained.