Autonomous coupled oscillators with hyperbolic strange attractors

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.     

Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors

We study numerically the complex domains of validity for KAM theory in generalized standard mappings. We compare methods based on Padé approximants and methods based on the study of periodic orbits.     

Statistics of strange attractors by generalized cell mapping

It is proposed in this paper to use the generalized cell mapping to locate strange attractors of dynamical systems and to determine their statistical properties. The cell-to-cell mapping method is based upon the idea of replacing the state space continuum by a large collection of state space cells and of expressing the evolution of the dynamical system in terms of a cell-to-cell mapping. This leads to a Markov chain which in turn allows us to compute all the statistical properties as well as the invariant distribution. After a general discussion, the method is applied in this paper to strange attractors of a variety of systems governed either by point mappings or by differential equations. The results indicate that it is a viable, effective and attractive method. Some comments on this method in comparison with the method of direct iteration will also be made.     

A Feigenbaum sequence of bifurcations in the Lorenz model

For some high values of the Rayleigh numberr, the Lorenz model exhibits laminar behavior due to the presence of a stable periodic orbit. A detailed numerical study shows that, forr decreasing, the turbulent behavior is reached via an infinite sequence of bifurcations, whereas forr increasing, this is due to a collapse of the stable orbit to a hyperbolic one. The infinite sequence of bifurcations is found to be compatible with Feigenbaum's conjecture.     

Evidence of Strange Attractors in Class C Amplifier with Single Bipolar Transistor: Polynomial and Piecewise-Linear Case

This paper presents and briefly discusses recent observations of dynamics associated with isolated generalized bipolar transistor cells. A mathematical model of this simple system is considered on the highest level of abstraction such that it comprises many different network topologies. The key property of the analyzed structure is its bias point since the transistor is modeled via two-port admittance parameters. A necessary but not sufficient condition for the evolution of autonomous complex behavior is the nonlinear bilateral nature of the transistor with arbitrary reason that causes this effect. It is proved both by numerical analysis and experimental measurement that chaotic motion is miscellaneous, robust, and it is neither numerical artifact nor long transient motion.     

Dissipative finite degrees of freedom dynamical system and description of optical systems with saturable amplification, saturable losses and filtering

Single-mode fiber optical system with saturable amplification, saturable losses and spectral filtering as proposed by Rozanov and Fedorov (1998) [10] is studied. The system of ordinary differential equations (ODE’s) that can help investigation of the original physical system is proposed. It allows calculation of linear and nonlinear fixed points as well as the study of their stability, so it can be used for analysis of coherent structures and their classification. Derived system of ODE’s extends the earlier one proposed by van Saarloos and Hohenberg (1992) [2], for the analysis of coherent structures of the qubic-quintic Ginzburg-Landau equation, by including additionally the temporal dependences of the gain and losses. In order to verify it, it was applied to the earlier considered cases of fast and slow changes in the amplification and losses. Earlier obtained localized structures namely pulses, have been observed via numerical solution of the proposed system. In addition, new families of fronts have been identified.     

On the weak-noise limit of Fokker-Planck models

The weak-noise limit of Fokker-Planck models leads to a set of nonlinear Hamiltonian canonical equations. We show that the existence of a nonequilibrium potential in the weak-noise limit requires the existence of whiskered tori in the Hamiltonian system and, therefore, the complete integrability of the latter. A specific model is considered, where the Hamiltonian system in the weak-noise limit is not integrable. Two different perturbative solutions are constructed: the first solution describes analytically the breakdown of the whiskered tori due to the appearance of wild séparatrices; the second solution allows the analytic construction of an approximate nonequilibrium potential and an asymptotic expression for the probability density in the steady state.On leave from Institute for Theoretical Physics, Eötvös University, Budapest, Hungary.     

Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2-torus

We study generic piecewise linear hyperbolic automorphisms of the 2-torus. We explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations.     

A rigorous partial justification of Greene's criterion

We prove several theorems that lend support to Greene's criterion for the existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modification that can work in the case that the Aubry-Mather sets have nonzero Lyapunov exponents. The latter is based on a closing lemma for sets with nonzero Lyapunov exponents, which may have several other applications.     

Phase Transitions in a Driven Lattice Gas with Anisotropic Interactions

The Ising lattice gas, with its well known equilibrium properties, displays a number of surprising phenomena when driven into nonequilibrium steady states. We study such a model with anisotropic interparticle interactions (J _||J ), using both Monte Carlo simulations and high temperature series techniques. Under saturation drive, the shift in the transition temperature can be both positive and negative, depending on the ratio J _||/J ! For finite drives, both first- and second-order transitions are observed. Some aspects of the phase diagram can be predicted by investigating the two-point correlation function at the first nontrivial order of a high-temperature series expansion.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Periodic orbits in magnetic billiards

We propose a simple method to calculate periodic orbits in two-dimensional systems with no symbolic dynamics. The method is based on a line by line scan of the Poincaré surface of section and is particularly useful for billiards. We have applied it to the Square and Sinai's billiards subjected to a uniform orthogonal magnetic field and we obtained about 2000 orbits for both systems using absolutely no information about their symbolic dynamics. Received 21 September 1999 and Received in final form 13 April 2000     

On the Hénon-Pomeau attractor

The behavior of the iterates of the mapT(x, y) = (1+y–ax ²,bx) can be useful for the understanding of turbulence. In this study we fix the value ofb at 0.3 and allowa to take values in a certain range. We begin with the study of the casea=1.4, for which we determine the existence of a strange attractor, whose region of attraction and Hausdorff dimension are obtained. As we changea, we study numerically the existence of periodic orbits (POs) and strange attractors (SAs), and the way in which they evolve and bifurcate, including the computation of the associated Lyapunov numbers. Several mechanisms are proposed to explain the creation and disappearance of SAs, the basin of attraction of POs, and the cascades of bifurcations of POs and of SAs for increasing and decreasing values ofa. The role of homoclinic and heteroclinic points is stressed.     

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [–1, 1][–1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as grows beyond some critical value _c. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.     

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.     

On the detection of closed orbits in time-dependent systems by using double-pulse lasers

In the photodetachment of atoms or negative ions by a double-pulse laser, the first pulse of the double-pulse laser generates waves and the delayed second pulse may detect them. The phenomenon of the excitation and detection of waves by a double-pulse laser can be used to identify the closed orbits in the system. We demonstrate this phenomenon with a negative hydrogen ion (H⁻) by analyzing the total population excited by a double-pulse laser in a time-dependent field for different physical parameters. By analyzing the total excited population using a double-pulse laser, we can uncover all the closed orbits existing in the system. We demonstrate that this can be realized by scanning the first pulse position and the time delay between the two pulses.     

A collective variable approach for optical solitons in the cubic-quintic complex Ginzburg-Landau equation with third-order dispersion

Considering the theory of electromagnetic, especially from the Maxwell equations, a basic equation modeling the propagation of ultrashort optical solitons in optical fibers is derived, namely a cubic-quintic complex Ginzburg-Landau equation (CQGLE) with third-order dispersion (TOD). Considering this one-dimensional CQGLE, we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fiber optic-links. Equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance. A fully numerical simulation of the CQGLE finally tests the results of the CV theory. It appears chaotic pulses, attenuate pulses and stable pulses under some parameter values.     

Transitions between secondary structures in isolated polyalanines

Monte Carlo simulations of gas-phase polyalanine peptides have been carried out with the Amber ff96 force field. A low-temperature structural transition takes place between the α-helix stable conformation and β-sheet structures, followed by the unfolding phase change. The transition state ensembles connecting the helix and sheet conformations are investigated by sampling the energy landscape along specific geometric order parameters as putative reaction coordinates, namely the electric dipole μ, the end-to-end distance d, and the gyration radius R_g. By performing series of shooting trajectories, the committor probabilities and their distributions are obtained, revealing that only the electric dipole provides a satisfactory transition coordinate for the α↔β interconversion. The nucleus at the transition is found to have a high helical content.     

Truncated Navier-stokes equations: continuous transition from a five-mode to a seven-mode model

A two-parameter family of nonlinear differential equations x=F(x, R, ) is studied, which allows one to connect continuously, as varies from zero to one, the different phenomenologies exhibited by a model of 5-mode truncated Navier-Stokes equations and by a 7-mode one extending it. A critical value is found for, at which the most significant phenomena of the 5-mode system either vanish or go to infinity. New phenomena arise then, leading to the 7-mode model.Supported by G.N.F.M., C.N.R.