Abstract: Although the possibility of complex dynamical behaviors-limit cycles, quasiperiodic oscillations, and aperiodic chaos-has been recognized theoretically, most ecologists are skeptical of their importance in nature. In this paper we develop a meth- odology for reconstructing endogenous (or deterministic) dynamics from ecological time series. Our method consists of fitting a response surface to the yearly population change as a function of lagged population densities. Using the version of the model that includes two lags, we fitted time-series data for 14 insect and 22 vertebrate populations. The 14 insect populations were classified as: unregulated (1 case), exponentially stable (three cases), damped oscillations (six cases), limit cycles (one case), quasiperiodic oscillations (two cases), and chaos (one case). The vertebrate examples exhibited a similar spectrum of dynamics, although there were no cases of chaos. We tested the results of the response-surface meth- odology by calculating autocorrelation functions for each time series. Autocorrelation pat- terns were in agreement with our findings of periodic behaviors (damped oscillations, limit cycles, and quasiperiodicity). On the basis of these results, we conclude that the complete spectrum of dynamical behaviors, ranging from exponential stability to chaos, is likely to be found among natural populations.

Abstract: We show that the 1-dimensional Schrodinger equation with a quasiperiodic potential which is analytic on its hull admits a Floquet representation for almost every energyE in the upper part of the spectrum. We prove that the upper part of the spectrum is purely absolutely continuous and that, for a generic potential, it is a Cantor set. We also show that for a small potential these results extend to the whole spectrum.

Abstract: We review the results of the scaling and multifractal analyses for the spectra and wave-functions of the finite-difference Schrodinger equation: Here V is a function of period 1 and ω is irrational. For the Fibonacci model, V takes only two values (it is constant except for discontinuities) and the spectrum is purely singular continuous (critical wavefunctions). When V is a smooth function, the spectrum is purely absolutely continuous (extended wavefunctions) for λ small and purely dense point (localized wavefunctions) for λ large. For an intermediate λ, the spectrum is a mixture of absolutely continuous parts and dense point parts which are separated by a finite number of mobility edges. There is no singular continuous part. (An exception is the Harper model V (x) = cos (2πx), where the spectrum is always pure and the singular continuous one appears at λ = 2.)

Abstract: A discrete version of the complex Ginzburg-Landau equation is studied on a completely connected lattice of N sites. This can equivalently be described as a model of N identical globally coupled limit-cycle oscillators. The phase diagram is obtained by a combination of numerical and analytical techniques. A surprising variety of dynamical behaviors is found in the thermodynamic limit (N\ensuremath{\gg}1). Depending on the region of parameter space, one gets the following: (1) a simple homogeneous limit cycle; (2) a state with complete frequency locking but with no phase locking so that the global forcing term vanishes; (3) a breaking of the system into a few macroscopic clusters which can exhibit periodic or quasiperiodic dynamics; (4) surprisingly complex states where an individual oscillator behaves in a chaotic way but in a sufficiently coherent manner so that the average complex amplitude does not vanish in the thermodynamic limit. Moreover, in this last region, the dynamics of this natural order parameter is itself chaotic.

Abstract: We investigate cellular automata in four and five dimensions for which Chate and Manneville recently have found nontrivial collective behaviour. More precisely, though being fully deterministic, the average magnetization seems to be periodic respectively quasiperiodic, with superimposed noise whose amplitude decreases with system size. We confirm this behaviour on very large systems and over very large times. We analyse in detail the statistical properties of the “noise”. Systems on small lattices and/or subject to additional external noise are metastable. Arguments by Grinstein et al. suggest that in the periodic case the infinite deterministic systems should be metastable too. These arguments are generalized to quasiperiodic systems. We find evidence that they do indeed apply, but we find no direct evidence for metastability of large systems.

Abstract: The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.

Abstract: An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω3≈2ω2 and ω2≈2ω1 to a harmonic excitation of the third mode, where the ω m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.

Abstract: A new class of quasiperiodic superlattice structures called three-component Fibonacci structures has been studied both theoretically and experimentally. These structures with the characteristic irrational intervals A, B, and C can be produced by the substitution rule A\ensuremath{\rightarrow}AC, C\ensuremath{\rightarrow}B, and B\ensuremath{\rightarrow}A. The projection method is applied to deal with the pattern and index of their diffraction spectrum. The analytical results are compared with the experimental one from three-component Fibonacci Ta/Al superlattices. The experimental results are in good agreement with the numerical calculations using the model for compositionally modulated multilayers. Some possible applications of these structures are discussed.

Abstract: A nonlinear dynamo model for solar activity which includes the feedback of the helicity upon the mean magnetic field has been investigated. The qualitative behavior of a seven-dimensional system of ordinary differential equations obtained by truncation of that model has been studied numerically. It has been compared with results from a sixth order system derived from the seventh order system by a special polar coordinate transformation. Depending on characteristic parameters, the seven-dimensional model exhibits periodic, quasiperiodic (on T2 and T3) and chaotic behavior where a route to chaos via the transition T2⇒T3⇒T2⇒ chaos has been found to be typical. In contrast to that, no chaotic state occurs in the reduced system due to a nonregularity of the coordinate transformation.

Abstract: The structural properties of a pentagonal quasiperiodic tiling obtained by new deflation and matching rules are described. Two new types of vertices occur that cannot be found in generalized Penrose patterns. Tile and vertex frequencies as well as average edge valencies of vertices are enumerated. The geometry underlying the acceptance domain involves self-similar fractals. Two methods of constructing the fractally shaped acceptance domain have been found.

Abstract: A new class of 1D quasiperiodic lattices, for which the substitution rules are B → BA, and A → BAB, has been studied in several aspects The high-dimensional projection method for obtaining the quasilattice is presented A multifractral spectral behavior and gap labeling properties have been found, which display the perfect quasiperiodicity of the studied model

Abstract: We have studied the dynamics of regular two-dimensional Josephson-junction arrays subjected to electromagnetic radiation at frequencies comparable to the individual junction's characteristic frequency. The junctions are described using the resistively-shunted-junction model including capacitance with the plasma frequency also comparable to the characteristic frequency. The dynamical behavior falls into several different general classes, namely, periodic, quasiperiodic, and chaotic, depending on the particular characteristics of the junctions, the input currents, and the amplitude and frequency of the radiation. Detailed examples of each of these types of behavior are given. Current-voltage characteristics are examined and related to the dynamical behavior and the junction's properties. The effect of finite temperatures, included by means of a Langevin noise current, is also discussed, as is the stability of various types of dynamical states.

Abstract: In this paper, we consider the following system of differential equations: θω+Θ(θ, z), z=Az+f(θ, z), where θ∈C m , ω=(ω 1 ,..., ω m )∈R m , z∈C n , A is a diagonalizable matrix, f and Θ are analytic functions in both variables and 2π-periodic in each component of the vector θ, Θ=O(|z|) and f=O(|z| 2 ) as z→0. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations θ=ω, z=Az by an analytic transformation provided that the eigenvalues of A and the frequency ω satisfy certain small-divisor conditions

Abstract: We review our recent approximation scheme to calculate the normal-superconducting phase boundary, T c(H), of a superconducting wire network in a magnetic field in terms of interacting loop currents. The theory is based on the London approximation of the linearized Ginzburg-Landau equation. An approximate general formula is derived for any two-dimensional space-filling lattice comprising tiles of two shapes. We provide many examples illustrating the use of this method with a particular emphasis on the fluxoid distribution. In addition to periodic lattices, we also discuss quasiperiodic lattices and fractal Sierpinski gaskets.

Abstract: In spectral form the 2D incompressible Navier–Stokes equations in a square periodic region will be represented by 430 complex Fourier amplitudes which correspond to isotropic truncation of the upper wave number 16. For small viscosity, we have found five equilibrium states I–V in the entire range of forcing; I—fixed point, II—circle, III—closed orbit, IV—torus, and V—chaos. The fixed‐point equilibrium state is the laminar flow. As the forcing passes through a critical value, the fixed point evolves directly to equilibrium state III under a typical multimode forcing. The chaotic transition takes place on a 2‐torus‐like manifold (equilibrium state IV) which is the product space of a circle and the closed orbit of equilibrium state III, similar to the quasiperiodic 2‐torus of Ruelle and Takens. For sufficiently large forcing, the evolution of equilibrium state V is nothing but a simulation of quasistationary 2D turbulence. From the Lyapunov exponents of turbulent flows, we have evaluated the constants in the theoretical results of Foias and his colleagues, which relate the determining mode and fractal dimension with the enstrophy dissipation wave number of 2D turbulence.

Abstract: We present a real-space renormalization-group method for evaluating the exact dynamic structure factor S (q,ω) of a quasiperiodic Fibonacci chain. Contrary to earlier work that takes account only of the global aspects of the symmetry of the chain, our method additionally takes care of the local environmental aspects of the symmetry by separating the original lattice into a finite number of self-similar interpenetrating sublattices, followed by elimination of the coupling between them. Our method also yields correctly the positions of the Bragg peaks of the Fibonacci chain

Abstract: 1 Introductory Remarks.- Problems.- 2 Semiconductor Physics.- 2.1 Fundamentals of Nonlinear Dynamics.- 2.1.1 Historical Remarks.- 2.1.2 Plasma Ansatz.- 2.1.3 Negative Differential Conductivity.- 2.1.4 Transport Mechanisms.- 2.2 Recent Experimental Progress.- 2.3 Model Experimental System.- 2.3.1 Material Characterization.- 2.3.2 Experimental Set-up.- 2.4 Experimental Results.- 2.4.1 Static Current-Voltage Characteristics.- 2.4.2 Temporal Instabilities.- 2.4.3 Spatial Structures.- 2.4.4 Spatio-Temporal Behavior.- Problems.- 3 Nonlinear Dynamics.- 3.1 Basic Ideas and Definitions.- 3.2 Fixed Points.- 3.2.1 Fundamental Bifurcations.- 3.2.2 Catastrophe Theory.- 3.2.3 Experiments.- 3.3 Periodic Oscillations.- 3.3.1 The Periodic State.- 3.3.2 Bifurcations to Periodic States.- 3.3.3 Experiments.- 3.4 Quasiperiodic Oscillations.- 3.4.1 The Quasiperiodic State.- 3.4.2 Bifurcations to Quasiperiodic States.- 3.4.3 Influence of Nonlinearities.- 3.4.4 Experiments.- 3.5 Chaotic Oscillations and Hierarchy of Dynamical States.- 3.5.1 The Chaotic State.- 3.5.2 Characterization Methods.- 3.5.3 Bifurcations to Chaotic States.- 3.6 Spatio-Temporal Dynamics.- Problems.- 4 Mathematical Background.- 4.1 Basic Concepts in the Theory of Dynamical Systems.- 4.1.1 Dissipative Dynamical Systems and Attractors.- 4.1.2 Invariant Probability Measures.- 4.1.3 Invariant Manifolds.- 4.1.4 Chaos.- 4.2 Scaling Behavior of Attractors of Dissipative Dynamical Systems.- 4.2.1 Scale Invariance.- 4.2.2 Symbolic Dynamics.- 4.2.3 Analogy with Statistical Mechanics.- 4.2.4 Partition Function for Chaotic Attractors of Dissipative Dynamical Systems.- 4.2.5 Discussion of the Partition Function.- 4.3 Generalized Dimensions, Lyapunov Exponents, Entropies.- 4.3.1 Definition of the Scaling Functions for Sampling Processes.- 4.3.2 Relation Between the Scaling Functions and a Thermodynamical Formalism.- 4.3.3 Relation Between the Scaling of the Support and the Scaling of the Measure (Generic Case).- 4.3.4 Discussion of Nonanalyticities.- 4.3.5 Evidence of Phase-Transition-Like Behavior in Experimental Observations.- 4.4 Evaluation of Experimental Systems.- 4.4.1 Embedding of a Time Series.- 4.4.2 Lyapunov Exponents from the Dynamical Equations and from Time Series.- 4.4.3 Results and Stability of Results.- 4.4.4 Comparison with Other Methods.- 4.5 High-Dimensional Systems.- 4.5.1 Singular-Value Decomposition.- 4.5.2 The Modified Approach.- 4.5.3 Results on Simulated Data.- 4.6 Lyapunov Exponents, Rotation Numbers and the Degree of Mappings.- 4.6.1 Characterization of Solutions of Dynamical Systems via the Degree of Mapping.- 4.6.2 Riemannian Motions on Manifolds of Constant Negative Curvature.- 4.7 Conclusions.- Problems.- References.

Abstract: We introduce the concept of exponential attractor for non-autonomous systems. Then we prove the existence and finite dimensionality of the attractor for the model equation where K and f are quasiperiodic in time.

Abstract: A scheme for obtaining the exact wave functions of an electron on a quasiperiodic lattice is presented It is shown that the trace map plays a very important role for construction of the infinite‐dimensional Riemann theta function in terms of which the wave functions can be represented

Abstract: We prove that all the non-negative Lyapunov exponents of difference Schrodinger equation $$ - y_{n + 1} + Q_n y_n - y_{n - 1} = 0, - \infty< n< + \infty $$ are strictly positive. Herey n eR m andQ n is a symmetricm×m matrix whose offdiagonal elements do not depend onn, and the diagonal elements are quasiperiodic functions $$q_{nj} (\theta ) = \lambda f_j (e^{2\pi i(\theta + n\alpha )} ) - E$$ with allf i non-constant analytic functions, λ sufficiently large, and α any irrational number.

Abstract: Hopf bifurcation on a rotating square lattice is considered. As many as seven primary solution branches are found to exist. Five of these are periodic in time and two are quasiperiodic with two independent frequencies. The stability properties of the solutions are established. A structurally stable attracting heteroclinic cycle connecting four rolls travelling in four orthogonal directions is also found. The relevance of the analysis to overstable convection in a plane rotating layer is discussed.

Abstract: A dynamical map is obtained from a class of quasiperiodic discrete Schrodinger equations in one dimension which include the Fibonacci system. The potentials are constant except for steps at special points.