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Nonlinear Dynamics 1990: Volume 1 , Issue CONTENTS. Digitized from IA1631614-06 . Previous issue: sim_nonlinear-dynamics_1990_1_index . Next issue: sim_nonlinear-dynamics_1990_1_1 .

It was the purpose of the program to investigate some aspects of the fluid mechanics of LHeII, in particular nonlinear phenomena such as turbulence and shock waves and their effect on the critical conditions in HeII. The major accomplishments which resulted from the GALCIT research program are: The development of the first cryogenic shock tube for the production of strong shock waves. In this small (1 inch diameter) tube, shock Mach numbers of M =42 in Helium gas have been reached; the extension of the cryogenic tube techniques to work with liquid Helium II led to the first careful mapping of the nonlinear wave diagrams in He II, involving both first sound and second sound shock waves; ultr-second sound waves with frequencies up to 1 MHz have been used as a velocimeter for He II counterflow; a similarity rule was developed on the two-fluid equation with the additional Gorter-Mellink terms. This rule permits the reduction of all known critical heat flow experiments in capillaries and tubes to a single number reminiscent of a critical Reynolds number for laminar-turbulent transition in classical fluid mechanics. (Author)

External debt is indispensable, especially in developing countries which usually face budget deficits to cover up their saving-investment gap. However, the effect of external debt on inflation depends on whether it is increasing or decreasing. Hence, this study aims to examine the effect of external debt stocks on inflation using World Bank data from 1980 to 2020 in Cameroon. The study makes use of non-linear ARDL to examine the positive and negative changes in external debt stocks and their effects on inflation. The results indicate a long-run increasing and decreasing asymmetry effect of external debts on inflation. Only the coefficient of positive external debt stock on inflation is positive and significant in the long run while in the short run, positive and negative external debt stocks respectively have a negative and positive significant impact on inflation. The study recommends that the government should be mindful of increasing external debt as it will become inflationary in the long run.

We build on a previous statistical model for distributed systems and formulate it in a way that the deterministic and stochastic processes within the system are clearly separable. We show how internal fluctuations can be analysed in a systematic way using Van Kanpen's expansion method for Markov processes. We present some results for both stationary and time-dependent states. Our approach allows the effect of fluctuations to be explored, particularly in finite systems where such processes assume increasing importance.

A simple model of an atomic Bose-Einstein condensate in a box whose size varies with time is studied to determine the nature of adiabaticity in the nonlinear dynamics obtained within the Gross-Pitaevskii equation (the nonlinear Schr\"{o}dinger equation). Analytical and numerical methods are used to determine the nature of adiabaticity in this nonlinear quantum system. Criteria for validity of an adiabatic approximation are formulated.

This work proposes a model-reduction methodology that preserves Lagrangian structure (equivalently Hamiltonian structure) and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's `Lagrangian ingredients'---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy POD and reduced-basis sparsification (RBS). Results for a parameterized truss-structure problem demonstrate the importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.

We present a fully nonlinear theory for dust acoustic (DA) shocks and DA solitary pulses in a strongly coupled dusty plasma, which have been recently observed experimentally by Heinrich et al. [Phys. Rev. Lett. 103, 115002 (2009)], Teng et al. [Phys. Rev. Lett. 103, 245005 (2009)], and Bandyopadhyay et al. [Phys. Rev. Lett. 101, 065006 (2008)]. For this purpose, we use a generalized hydrodynamic model for the strongly coupled dust grains, accounting for arbitrary large amplitude dust number density compressions and potential distributions associated with fully nonlinear nonstationary DA waves. Time-dependent numerical solutions of our nonlinear model compare favorably well with the recent experimental works (mentioned above) that have reported the formation of large amplitude non-stationary DA shocks and DA solitary pulses in low-temperature dusty plasma discharges.

We present an analytical and numerical study of a nonlinear diffusion model which describes density relaxation of loosely packed particles under gravity and weak random (thermal) vibration, and compare the results with Monte Carlo simulations of a lattice gas under gravity. The dynamical equation can be thought of as a local density functional theory for a class of lattice gases used to model slow relaxation of glassy and granular materials. The theory predicts a jamming transition line between a low density fluid phase and a high density glassy regime, characterized by diverging relaxation time and logarithmic or power-law compaction according to the specific form of the diffusion coefficient. In particular, we show that the model exhibits history dependent properties, such as quasi reversible-irreversible cycle and memory effects -- as observed in recent experiments, and dynamical heterogeneities.

In this work we characterize the unusual nonlinear dynamics of the resonance response, exhibited by our NbN superconducting microwave resonators, using different operating conditions. The nonlinear dynamics, occurring at relatively low input powers (2-4 orders of magnitude lower than Nb), and which include among others, bifurcations in the resonance curve, hysteresis loops and resonance frequency shift, are measured herein using varying temperature, applied magnetic field, white noise and rapid frequency sweeps. Based on these measurement results, we consider a hypothesis according to which Josephson junctions forming weak links at the boundaries of the NbN grains are responsible for the observed behavior, and we show that most of the experimental results are qualitatively consistent with such hypothesis.

Within of the framework of semiclassical approximation a general formalism for deriving an effective current generating bremsstrahlung of arbitrary number of soft gluons (longitudinal or transverse ones) in scattering of higher-energy parton off thermal parton in hot quark-gluon plasma with subsequent extension to two and more scatterers, is obtained. For the case of static color centers an expression for energy loss induced by usual bremsstrahlung of lowest-order with allowance for an effective temperature-induced gluon mass and finite mass of the projectile (heavy quark), is derived. The detailed analysis of contribution to radiation energy loss associated with existence of effective three-gluon vertex induced by hot QCD medium, is performed. It is shown that in general, the bremsstrahlung associated with this vertex have no sharp direction (as in the case of usual bremsstrahlung) and therefore here, we can expect an absence of suppression effect due to multiple scattering. For the case of two color static scattering centers it was shown that the problem of calculation of bremsstrahlung induced by four-gluon hard thermal loop (HTL) vertex correction can be reduced to the problem of the calculation of bremsstrahlung induced by three-gluon HTL correction. It was shown that for limiting value of soft gluon occupation number $N_{\bf k}\sim 1/\alpha_s$ all higher processes of bremsstrahlung of arbitrary number of soft gluons become of the same order in coupling, and the problem of resummation of all relevant contributions to radiation energy loss of fast parton, arises. An explicit expression for matrix element of two soft gluon bremsstrahlung in small angles approximation is obtained.

We demonstrate experimentally an unusual behavior of the parametric polariton scattering in semiconductor microcavity under a strong cw resonant excitation. The maximum of the scattered signal above the threshold of stimulated parametric scattering does not shift along the microcavity lower polariton branch with the change of pump detuning or angle of incidence but is stuck around the normal direction. We show theoretically that such a behavior can be modelled numerically by a system of Maxwell and nonlinear Schroedinger equations for cavity polaritons and explained via the competition between the bistability of a driven nonlinear MC polariton and the instabilities of parametric polariton-polariton scattering.

Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.

Consider the nonlinear Schr\"odinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.

The behavior of two-layer, quasi-geostrophic flow ill a channel, which is subject to baroclinic instability, is investigated using a high-resolution numerical model. Solutions are obtained for both free-slip sidewalls (which allow tangential velocities but zero stress) and rigid sidewalls (which enforce zero velocity). Results for the slippery model are presented first, and the physics underlying the observed behavior is examined. As the Froude number F is increased, the system exhibits a transition from steady flow to periodic, quasi-periodic, and finally chaotic behavior. As F is increased to about five times the linear critical value, the motion becomes chaotic, and for even larger values of F it moves toward a "geostrophic turbulence" regime. The route to chaos is determined to be the breakdown of a two-torus.

In 1995 we reported experimental results on chaos suppression (control of chaotic dynamics) in a nonlinear buckling elastic structure as well as anti-control of chaos in the vibrations of structure with dry friction. In 1996-1997 we have been able to achieve the suppression of chaotic dynamics for an impacting mechanical oscillator using a method based on the ideas in the ogy control of chaos theory. Impact in loose structural connections as well as in machine components such as gears have been shown to be a source of deterministic noise in these systems. The control of the impact chaos was effected using a digitally pulsed magnetic actuator.

Experimental and theoretical studies of the nonlinear dynamics and control of aeroelastic systems including airfoils and wings have been conducted. Nonlinear geometric stiffness and freeplay have been considered specifically and good theoretical / experimental correlation obtained. Active control systems have been designed, built and tested to increase flutter speed and reduce limit cycle oscillations (LCO). Our most recent work includes theoretical studies of aerodynamic nonlinearities in the transonic range and their effect on flutter and LCO.

Experimental and theoretical studies of the nonlinear dynamics and control of aeroelastic systems including airfoils and wings have been conducted. Nonlinear geometric stiffness and freeplay have been considered specifically and good theoretical / experimental correlation obtained. Active control systems have been designed, built and tested to increase flutter speed and reduce limit cycle oscillations (LCO). Our most recent work includes theoretical studies of aerodynamic nonlinearities in the transonic range and their effect on flutter and LCO.

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Controlling the translational motion of cold atoms using optical lattice potentials is of both theoretical and experimental interest. By designing two on-resonance time sequences of kicking optical lattice potentials, a novel connection between two paradigms of nonlinear mapping systems, i.e., the kicked rotor model and the kicked Harper model, is established. In particular, it is shown that Hofstadter's butterfly quasi-energy spectrum in periodically driven quantum systems may soon be realized experimentally, with the effective Planck constant tunable by varying the time delay between two sequences of control fields. Extensions of this study are also discussed. The results are intended to open up a new generation of cold-atom experiments of quantum nonlinear dynamics

Controlling the translational motion of cold atoms using optical lattice potentials is of both theoretical and experimental interest. By designing two on-resonance time sequences of kicking optical lattice potentials, a novel connection between two paradigms of nonlinear mapping systems, i.e., the kicked rotor model and the kicked Harper model, is established. In particular, it is shown that Hofstadter's butterfly quasi-energy spectrum in periodically driven quantum systems may soon be realized experimentally, with the effective Planck constant tunable by varying the time delay between two sequences of control fields. Extensions of this study are also discussed. The results are intended to open up a new generation of cold-atom experiments of quantum nonlinear dynamics

A spectral operator theoretic perspective was taken for modeling nonlinear dynamics subjected to external inputs, due to prescribed forcing or an environment. Under this Directors Research Initiative (DRI), we focused on spectral properties (eigenfunctions, eigenvalues) of the Koopman operator (i.e., an infinite dimensional linear operator that can describe the full dynamics of a nonlinear system). We showed how external forcing could be treated in a Koopman decomposition-based framework. Furthermore, during the course of our study using Koopman decompositions of periodically excited Hopf bifurcation systems, we characterized a new attractor that had never been described in the literature or texts. We called this attractor the quasi-periodic intermittency attractor and showed that it may be a fundamental building block of a variety of multi-scale dynamics systems. Perhaps the most far reaching impact of this DRI will be a contribution that was not planned in the original proposal. This contribution has to do with the generalization of Koopman decompositions using a fractional calculus perspective on complexity. By using a combination of Koopman operator theory and fractional calculus, we showed that our generalized spectral decompositions are better suited for complex systems influenced by an external environment in which long-term memory is introduced.

For spacecraft conducting on-orbit operations, changes to the structure of the spacecraft are not uncommon. These planned or unanticipated changes in inertia properties couple with the spacecraft's attitude dynamics and typically require estimation. For systems with time-varying inertia parameters, multiple model adaptive estimation (MMAE) routines can be utilized for parameter and state estimates. MMAE algorithms involve constructing a bank of recursive estimators, each assuming a different hypothesis for the systems dynamics. This research has three distinct, but related, contributions to satellite attitude dynamics and estimation. In the first part of this research, MMAE routines employing parallel banks of unscented attitude filters are applied to analytical models of spacecraft with time-varying mass moments of inertia (MOI), with the objective of estimating the MOI and classifying the spacecraft's behavior. New adaptive estimation techniques were either modified or developed that can detect discontinuities in MOI up to 98 % of the time in the specific problem scenario. Second, heuristic optimization techniques and numerical methods are applied to Wahba's single-frame attitude estimation problem, decreasing computation time by an average of nearly 67%. Finally, this research poses MOI estimation as an ODE parameter identification problem, achieving successful numerical estimates through shooting methods and exploiting the polhodes of rigid body motion with results, on average, to be within 1 % to 5 % of the true MOI values.

Under AFOSR grant F49620-92-J-0498, Drs Barcilon and Nathan are studying the impact of steady, zonally varying vorticity forcing on topographic instability and finite amplitude, low frequency oscillations. The model consists of a viscous, barotropic fluid on a zonally periodic, midlatitude B-plane channel, bounded above by a flat, rigid boundary, below by a bottom topography, and laterally by sidewalls at y = o,TT Topographically induced low frequency oscillations in the presence of a steady, zonally varying vorticity source have been studied. The source has been shown to interact with the pertubation field to produce zonally rectified wave fluxes that change the zonal momentum balance of the system. This change provides a mechanism for coupling the topographic and vorticity forcings, leading to a dramatic alteration in the Hopf bifurcation from stationary to low frequency, finite amplitude oscillation.

The author has studied the qualitative behaviour of nonlinear waves for hyperbolic conservation laws with or without the effects of dissipations, discretization, or nonlinear resonance. The fundamental problem of well-posedness theory for hyperbolic conservation laws is being resolved. It is shown that no physical law, beyond the second law of thermodynamics, is needed. The shock waves for finite difference schemes are shown to have slow decaying tails due to the effect of small divisor. Physical degenerate dissipation matrix is shown to give rise to rich nonlinear wave phenomena. Nonlinear waves for non-strictly hyperbolic system are shown to behave sensitively as a functional of the dissipation matrix. The ideas of wave tracing and pointwise estimates introduced by the author play the central role in the analysis of these problems.

Progress is reported in measuring the nonlinear optical properties and picosecond dynamics of carriers in multiple quantum wells and other bulk and Molecular Beam Epitaxially grown structures. Discussions of our progress are divided into six parts; (1) fabrication and characterization, (2) picosecond time-resolved transient absorption, (3) picosecond time-resolved photoluminescence, (4) theory of dressed excitons, (5) picosecond photorefractive effects, and (6) picosecond and femtosecond laser development. Keywords: Nonlinear Optics; Optical Devices; Ultrafast Phenomena; Multiple Quantum Wells.

The dynamical properties of road traffic time series from North-Rhine Westphalian motorways are investigated. The article shows that road traffic dynamics is well described as a persistent stochastic process with two fixed points representing the freeflow (non-congested) and the congested state regime. These traffic states have different statistical properties, with respect to waiting time distribution, velocity distribution and autocorrelation. Logdifferences of velocity records reveal non-normal, obviously leptocurtic distribution. Further, linear and nonlinear phase-plane based analysis methods yield no evidence for any determinism or deterministic chaos to be involved in traffic dynamics on shorter than diurnal time scales. Several Hurst-exponent estimators indicate long-range dependence for the free flow state. Finally, our results are not in accordance to the typical heuristic fingerprints of self-organized criticality. We suggest the more simplistic assumption of a non-critical phase transition between freeflow and congested traffic.

Session topics include: Amplitude equations; Shear flow; Bifurcations and transitions to chaos; Direct numerical simulation; Dynamic systems and turbulence; Vorticity dynamics.

In these lectures we derive (mostly) the equations of viscous �uid dynamics. We then show how they may be generalized to the problem of Rayleigh-Benard convection�the problem of a �uid heated from below. Later we show how the RB problem itself may be reduced to the famous Lorenz equations. The highlights of these lectures are as follows: � Navier-Stokes equations of �uid dynamics (mass and momentum conser_vation). � Reynolds number � Phenomenology of RB convection � Rayleigh number � Equations of RB convection. Thus far we have dealt almost exclusively with the temporal behavior of a few variables. In these lectures we digress, and discuss the evolution of a continuum.

In these lectures we derive (mostly) the equations of viscous �uid dynamics. We then show how they may be generalized to the problem of Rayleigh-Benard convection�the problem of a �uid heated from below. Later we show how the RB problem itself may be reduced to the famous Lorenz equations. The highlights of these lectures are as follows: � Navier-Stokes equations of �uid dynamics (mass and momentum conser_vation). � Reynolds number � Phenomenology of RB convection � Rayleigh number � Equations of RB convection. Thus far we have dealt almost exclusively with the temporal behavior of a few variables. In these lectures we digress, and discuss the evolution of a continuum.

We consider a linear Schr\"odinger equation with a small nonlinear perturbation in $R^3$. Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial data is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schr\"odinger equation and its asymptotic profile can have two different types of decay: 1. The resonance dominated solutions decay as $t^{-1/2}$. 2. The radiation dominated solutions decay at least like $t^{-3/2}$.

The forces and moments produced by a vehicle's aerodynamic control surfaces are often nonlinear functions of control surface deflection. This phenomenon limits the accuracy of state-of-the-art control allocation algorithms since all of the approaches are based on the assumption that the control variable rates are linear functions of the surface deflections and that control variable rate increments are not produced for zero deflections. The errors introduced by this assumption are currently mitigated by the robustness resulting from feedback control laws. A method for improving the performance of the feedback control/control allocation system is presented that directly attacks the inaccuracies introduced by these linear assumptions.

We consider adaptive control problem in presence of nonlinear parametrization of uncertainties in the model. It is shown that despite traditional approaches require for domination in the control loop during adaptation, it is not often necessary to use such energy inefficient compensators it in wide range of applications. In particular, we show that recently introduced adaptive control algorithms in finite form which are applicable to monotonic parameterized systems can be extended to general smooth non-monotonic parametrization. These schemes do not require any damping or domination in control inputs.

Controllers are often designed based on a reduced or simplified model of the plant dynamics. In this context, we investigate whether it is possible to synthesize a stabilizing event-triggered feedback law for networked control systems (NCS) which have two time-scales, based only on an approximate model of the slow dynamics. We follow an emulation-like approach as we assume that we know how to solve the problem in the absence of sampling and then we study how to design the event-triggering rule under communication constraints. The NCS is modeled as a hybrid singularly perturbed system which exhibits the feature to generate jumps for both the fast variable and the error variable induced by the sampling. The first conclusion is that a triggering law which guarantees the stability and the existence of a uniform minimum amount of time between two transmissions for the slow model may not ensure the existence of such a time for the overall system, which makes the controller not implementable in practice. The objective of this contribution is twofold. We first show that existing event-triggering conditions can be adapted to singularly perturbed systems and semiglobal practical stability can be ensured in this case. Second, we propose another technique that combines event-triggered and time-triggered results in the sense that transmissions are only allowed after a predefined amount of time has elapsed since the last transmission. This technique has the advantage, under an additional assumption, to ensure a global asymptotic stability property and to allow the user to directly tune the minimum inter-transmission interval. We believe that this technique is of its own interest independently of the two-time scale nature of the addressed problem. The results are shown to be applicable to a class of globally Lipschitz systems.

We examine the motions of particles in quadrupole ion traps as a function of damping and trapping forces, including cases where nonlinear damping or nonlinearities in the electric field geometry play significant roles. In the absence of nonlinearities, particles are either damped to the trap center or ejected, while their addition brings about a rich spectrum of stable closed particle trajectories. In three-dimensional (3D) quadrupole traps, the extended orbits are typically confined to the trap axis, and for this case we present a 1D analysis of the relevant equation of motion. We follow this with an analysis of 2D quadrupole traps that frequently show diamond-shaped closed orbits. For both the 1D and 2D cases we present experimental observations of the calculated trajectories in microparticle ion traps. We also report the discovery of a new collective behavior in damped 2D microparticle ion traps, where particles spontaneously assemble into a remarkable knot of overlapping, corotating diamond orbits, self-stabilized by air currents arising from the particle motion.

We construct two optimal Newton-Secant like iterative methods for solving non-linear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.

Identifying governing equations from data is a critical step in the modeling and control of complex dynamical systems. Here, we investigate the data-driven identification of nonlinear dynamical systems with inputs and forcing using regression methods, including sparse regression. Specifically, we generalize the sparse identification of nonlinear dynamics (SINDY) algorithm to include external inputs and feedback control. This method is demonstrated on examples including the Lotka-Volterra predator--prey model and the Lorenz system with forcing and control. We also connect the present algorithm with the dynamic mode decomposition (DMD) and Koopman operator theory to provide a broader context.

Identifying governing equations from data is a critical step in the modeling and control of complex dynamical systems. Here, we investigate the data-driven identification of nonlinear dynamical systems with inputs and forcing using regression methods, including sparse regression. Specifically, we generalize the sparse identification of nonlinear dynamics (SINDY) algorithm to include external inputs and feedback control. This method is demonstrated on examples including the Lotka-Volterra predator--prey model and the Lorenz system with forcing and control. We also connect the present algorithm with the dynamic mode decomposition (DMD) and Koopman operator theory to provide a broader context.

Electron-positron clusters are studied using a quantum hydrodynamic model that includes Coulomb and exchange interactions. A variational Lagrangian method is used to determine their stationary and dynamical properties. The cluster static features are validated against existing Hartree-Fock calculations. In the linear response regime, we investigate both dipole and monopole (breathing) modes. The dipole mode is reminiscent of the surface plasmon mode usually observed in metal clusters. The nonlinear regime is explored by means of numerical simulations. We show that, by exciting the cluster with a chirped laser pulse with slowly varying frequency (autoresonance), it is possible to efficiently separate the electron and positron populations on a timescale of a few tens of femtoseconds.

An ultracold gas is considered, loaded into a lattice, each site of which is formed by a double-well potential. Initial conditions, after the loading, correspond to a nonequilibrium state. The nonlinear dynamics of the system, starting with a nonequilibrium state, is analysed in the local-field approximation. The importance of taking into account attenuation, caused by particle collisions, is emphasized. The presence of this attenuation dramatically influences the system dynamics.

We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schr\"odinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion. The main focus of the paper is chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential or fly ballistically with weak chaotic oscillations of its momentum or wander in the optical lattice changing the direction of motion in a chaotic way. In the regime of chaotic wandering atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion -- Poincar\'e mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. We find common features with typical non-hyperbolic Hamiltonian systems -- chains of resonant islands of different sizes embedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of sticking of atomic trajectories to boundaries of regular islands, that should have a great influence to atomic transport in optical lattices, is found and demonstrated numerically.

We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schr\"odinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion. The main focus of the paper is chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential or fly ballistically with weak chaotic oscillations of its momentum or wander in the optical lattice changing the direction of motion in a chaotic way. In the regime of chaotic wandering atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion -- Poincar\'e mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. We find common features with typical non-hyperbolic Hamiltonian systems -- chains of resonant islands of different sizes embedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of sticking of atomic trajectories to boundaries of regular islands, that should have a great influence to atomic transport in optical lattices, is found and demonstrated numerically.

We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schr\"odinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion. The main focus of the paper is chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential or fly ballistically with weak chaotic oscillations of its momentum or wander in the optical lattice changing the direction of motion in a chaotic way. In the regime of chaotic wandering atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion -- Poincar\'e mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. We find common features with typical non-hyperbolic Hamiltonian systems -- chains of resonant islands of different sizes embedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of sticking of atomic trajectories to boundaries of regular islands, that should have a great influence to atomic transport in optical lattices, is found and demonstrated numerically.

2 v. (xii, 717 p.) : 22 cm

Methods of dynamical system's theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean -- a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle's scattering and transport. A numerical analysis reveals a nonattracting chaotic invariant set $\Lambda$ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle's coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a jet current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time.

Methods of dynamical system's theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean -- a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle's scattering and transport. A numerical analysis reveals a nonattracting chaotic invariant set $\Lambda$ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle's coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a jet current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time.

Methods of dynamical system's theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean -- a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle's scattering and transport. A numerical analysis reveals a nonattracting chaotic invariant set $\Lambda$ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle's coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a jet current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time.

Nonlinear Dynamics is an active publication that covers all nonlinear dynamic phenomena. It was established in 1990 and is published 16 times a year by Springer Dordrecht. The journal focuses on the study of nonlinear dynamic phenomena in various fields such as mechanical, structural, civil, aeronautical, ocean, electrical, and control systems. The scope encompasses a wide range of topics, including but not limited to chaos theory, bifurcations, stability analysis, control and synchronization of chaotic systems, nonlinear vibration and dynamics, and nonlinear phenomena in materials and structures. Nonlinear Dynamics can be found online at: https://www.springer.com/journal/11071

The nonlinear differential equations of motion for a general parachute-riser-payload system are developed. The resulting math model is then applied for analyzing the descent dynamics and stability characteristics of both the drogue stabilization phase and the main descent phase of the space shuttle solid rocket booster (SRB) recovery system. The formulation of the problem is characterized by a minimum number of simplifying assumptions and full application of state-of-the-art parachute technology. The parachute suspension lines and the parachute risers can be modeled as elastic elements, and the whole system may be subjected to specified wind and gust profiles in order to assess their effects on the stability of the recovery system.

Semidirect methods are discussed, their present role, as well as some developments for their application in computational fluid dynamics. A semidirect method is a computational scheme that uses a fast, direct, elliptic solver as the driving algorithm for the iterative solution of finite difference equations. Specific subtopics include: (1) direct Cauchy Riemann solvers for first order elliptic equations; (2) application of the semidirect method to the mixed elliptic hyperbolic problem of steady, inviscid transonic flow; and (3) the treatment of interior conditions, such as those on an airfoil or wing, in semidirect methods.