Nonlinear Time Evolution Research Articles

Nonlinear rheological models for structured interfaces

The GENERIC formalism is a formulation of nonequilibrium thermodynamics ideally suited to develop nonlinear constitutive equations for the stress–deformation behavior of complex interfaces. Here we develop a GENERIC model for multiphase systems with interfaces displaying nonlinear viscoelastic stress–deformation behavior. The link of this behavior to the microstructure of the interface is described by including a scalar and a tensorial structural variable in the set of independent surface variables. We derive an expression for the surface stress tensor in terms of these structural variables, and a set of general nonlinear time evolution equations for these variables, coupling them to the deformation field. We use these general equations to develop a number of specific models, valid for application near equilibrium, or valid for application far beyond equilibrium.

The use of a hybrid model to compute the nonlinear acoustic performance of silencers for the finite amplitude acoustic wave

In the present study, a hybrid method is proposed for predicting the acoustic performance of a silencer for a nonlinear wave. This method is developed by combining two models: (i) a frequency-domain model for the computation of sound attenuation due to a silencer in a linear regime and (ii) a wavenumber space model for the prediction of the nonlinear time-evolution of finite amplitudes of the acoustic wave in a uniform duct of the same length as the silencer. The present method is proposed under the observation that the physical process of the nonlinear sound attenuation phenomenon of a silencer may be decoupled into two distinct mechanisms: (a) a linear acoustic energy loss that owes to the mismatch in the acoustic impedance between reactive elements and/or the sound absorption of acoustic liners in a silencer; (b) a nonlinear acoustic energy loss that is due to the energy-cascade phenomenon that arises from the nonlinear interaction between components of different frequencies. To establish the validity of the present model for predicting the acoustic performance of silencers, two model problems are considered. First, the performance of simple expansion mufflers with nonlinear incident waves has been predicted. Second, proposed method is applied for computing nonlinear acoustic wave propagation in the NASA Langley impedance duct configuration with ceramic tubular liner (CT57). Both results obtained from the hybrid models are compared with those from computational aero-acoustic techniques in a time–space domain that utilize a high-order finite-difference method. Through these comparisons, it is shown that there are good agreements between the two predictions. The main advantage of the present method is that it can effectively compute the nonlinear acoustic performance of silencers in nonlinear regimes without time–space domain calculations that generally entail a greater computational burden.

Synchronization properties of coupled electrochemical bursters

In the present paper the non-linear time evolution of the electrodissolution/passivation of iron in sulfuric acid–sodium chloride solution is explored. The working electrode is an assembly of two iron disk electrodes, interacting through the electrolyte, and the time evolution of the electric current has the form of spontaneous bursting oscillations. The main aim of this work is to characterize the modes of synchrony and more specifically the synchronization of electric bursts and the synchronization of electric spikes. The characterization is based on the calculation of the firing rates from the experimental time series and the extraction of the slow dynamics by low-pass filtering. The analysis is performed both for elliptic and square wave electrochemical bursting and the results are compared with neural oscillations.

Low dimensional behavior of large systems of globally coupled oscillators

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.

Output feedback stabilization for a class of nonlinear time-evolution systems

A variety of problems in nonlinear time-evolution systems such as communication networks, computer networks, manufacturing, traffic management, etc., can be modelled as min–max-plus systems in which operations of min, max and addition appear simultaneously. Systems with only maximum (or minimum) constraints can be modelled as max-plus system and handled by max-plus algebra which changes the original nonlinear system in the traditional sense into linear system in this framework. Min-max-plus systems are extensions of max-plus systems and nonlinear even in the max-plus algebra view. Output feedback stabilization for min–max-plus systems with min–max-plus inputs and max-plus outputs is considered in this paper. Max-plus projection representation for the closed-loop system with min–max-plus output feedback is introduced and the formula to calculate the cycle time is presented. Stabilization of reachable systems with at least one observable state and a further result for reachable and observable systems are worked out, during which max-plus output feedbacks are used to stabilize the systems. The method based on the max-plus algebra is constructive in nature.

Noise-induced kinetic model for autonomous motion of the contact line in oil-water systems with chemical reactions

A model for autonomous motion of the contact line of the oil-water interface along a solid surface is proposed. The present model is inspired by the spontaneous wave generation and the contact line motion of an oil-water interface composed of cationic surfactant and oil-soluble anions. The motion is created through wetting by an adsorption of surfactant followed by an autocatalytic process with a chemical reaction and also dewetting due to the desorption of the remaining monolayer. The wetting process is accelerated by the contact line motion itself through the convection-enhanced transport of reaction constituents, which is autocatalytic in nature. These processes are expressed by nonlinear time-evolution equations for the velocity and the amplitude. Following the model, the autonomous motion is essentially excitable, and hence the spatiotemporal pattern is of the noise-induced type. The present model explains well the diverse features of the experimental results with a few parameters.

25143 Itokawa: direct detection of the current decelerating spin state due to YORP effect

Aims. The Hayabusa mission revealed fundamental physical properties of the small near-Earth asteroid 25143 Itokawa, such as shape and mass, during its rendezvous with the asteroid in 2005. Resulting from this, the YORP-induced change in the asteroid’s spin state has been predicted theoretically. The purpose of this study is to investigate the YORP effect for Itokawa directly from an observational perspective. Methods. A long-term campaign of ground-based photometric observations of Itokawa were performed from March 2001 to December 2006. The observed asteroid lightcurves were compared with numerical modeling using the detailed shape model and global surface photometric properties derived from the Hayabusa mission. Results. As a non-linear time evolution of rotational phase lag is shown, we found that Itokawa has been decreasing its spin rate by u ω = −1.2(±0.2) × 10 −17 rad s −2 . The detected deceleration rate is almost consistent with, but slightly smaller than, the theoretically predicted value due to the YORP effect.

Nonlinear dynamos in laminar, helical pipe flow

Steady, pressure-driven incompressible laminar flow of an electrically conducting fluid down a helically symmetric pipe is known to be capable of sustaining a dynamo at a fairly low hydrodynamic Reynolds number. The nonlinear time evolution of such a dynamo is investigated. Both the fluid motion and the magnetic field are assumed to be helically symmetric. As the magnetic Reynolds number is increased, the dynamo first saturates with a steady field. Subsequently, time-periodic solutions occur exhibiting field reversal. Eventually a-periodic behavior is found, with quasiperiodic bursts and field reversals. Occasionally the field is all but totally quenched, reestablishing itself on the long time scale of the linear eigenfunction.

Weak Noise Approach to the Logistic Map

Using a nonperturbative weak noise approach we investigate the interference of noise and chaos in simple 1D maps. We replace the noise-driven 1D map by an area-preserving 2D map modelling the Poincare sections of a conserved dynamical system with unbounded energy manifolds. We analyze the properties of the 2D map and draw conclusions concerning the interference of noise on the nonlinear time evolution. We apply this technique to the standard period-doubling sequence in the logistic map. From the 2D area-preserving analogue we, in addition to the usual period-doubling sequence, obtain a series of period doubled cycles which are elliptic in nature. These cycles are spinning off the real axis at parameters values corresponding to the standard period doubling events.

Vortex formation in an electron plasma with a sheared flow

The formation of vortex structures in an electron plasma with a sheared flow is investigated. The electron fluid is drifting in a self-electric field generated by an unshielded electron population. This setting is linearly unstable and an instability of diocotron (slipping-stream) type occurs. The time scale of the dynamics is assumed to be much shorter than the ion plasma and ion gyroperiods. Consequently, the ions do not respond to the wave potential and serve only as a neutralizing background. An equation determining the nonlinear evolution of the electrostatic potential in a plane perpendicular to an external magnetic field is derived within the drift approximation. The governing equation is then analyzed for the case with a localized shear in the electron fluid velocity. Possible final states of the diocotron instability are investigated analytically and solutions in the form of a tripolar vortex, a zonal flow, and a vortex street are found. The nonlinear time evolution of the diocotron instability is investigated by solving the governing equation numerically. In particular, the dynamics of nonlinearly saturated states and the formation of such states are discussed. Numerical solutions show a vortex street structure in a saturated state. The relevance of our investigation for space and laboratory plasmas is discussed.

Three-Dimensional Lumped Mass/Lumped Spring Modeling and Nonlinear Behavior of a Levitated Droplet

The nonlinear dynamics time evolution of an electromagnetically levitated (EML) droplet is considered in this study. The droplet is modeled as a three dimensional system with lumped masses and elastic springs. A new expression for the spring elastic constants in a global stiffness matrix has been derived, and equations of motion presented. The chaotic behavior of the system is analyzed for different coil configurations. The stability of the motion is studied using the Lyapunov exponents. Computations were performed for droplets of aluminum and copper.

An efficient numerical scheme for simulating particle acceleration in evolving cosmic-ray modified shocks

We have developed a new, very efficient numerical scheme to solve the CR diffusion–convection equation that can be applied to the study of the nonlinear time evolution of CR modified shocks for arbitrary spatial diffusion properties. The efficiency of the scheme derives from its use of coarse-grained finite momentum volumes. This approach has enabled us, using ∼10–20 momentum bins spanning nine orders of magnitude in momentum, to carry out simulations that agree well with results from simulations of modified shocks carried out with our conventional finite difference scheme requiring more than an order of magnitude more momentum points. The coarse-grained, CGMV scheme reduces execution times by a factor approximately half the ratio of momentum bins used in the two methods. Depending on the momentum dependence of the diffusion, additional economies in required spatial and time resolution can be utilized in the CGMV scheme, as well. These allow a computational speed-up of at least an order of magnitude in some cases.

Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear Damped Attachment

Linear oscillator coupled to damped strongly nonlinear attachment with small mass is considered as a model design for nonlinear energy sink (NES). Damped nonlinear normal modes of the system are considered for the case of 1:1 resonance by combining the invariant manifold approach and multiple scales expansion. Special asymptotical structure of the model allows a clear distinction between three time scales. These time scales correspond to fast vibrations, evolution of the system toward the nonlinear normal mode and time evolution of the invariant manifold, respectively. Time evolution of the invariant manifold may be accompanied by bifurcations, depending on the exact potential of the nonlinear spring and value of the damping coefficient. Passage of the invariant manifold through bifurcations may bring about destruction of the resonance regime and essential gain in the energy dissipation rate.

Nonlinear behavior of resistive pressure driven modes in stellarator/heliotron plasmas with vacuum magnetic islands

Effects of externally applied vacuum magnetic islands (VMIs) on nonlinear saturation levels of resistive pressure driven modes in stellarator/heliotron plasmas are examined with the use of three-dimensional nonlinear time-evolution simulations for cylindrical plasmas. The simulation code is based on reduced magnetohydrodynamics equations and the equations are discretized by the finite element method in the poloidal cross section and Fourier expansion in the toroidal direction. It is shown that the growth of magnetic islands in a plasma can be controlled by an external perturbational magnetic field that generates VMIs.

Structure of multiphoton quantum optics. I. Canonical formalism and homodyne squeezed states

We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local-oscillator angle; for n-mode systems they depend on n heterodyne mixing angles. The canonical formalism realizes nontrivial mixing of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems, and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamiltonians of degenerate multiphoton processes and define a class of non-Gaussian, nonclassical multiphoton states that exhibit properties of coherence and squeezing. We show that such homodyne multiphoton squeezed states are generated by unitary operators with a nonlinear time evolution that realizes the homodyne mixing of a pair of conjugate quadratures. Tuning of the local-oscillator angle allows us to vary at will the statistical properties of such states. We discuss the relevance of the formalism for the study of degenerate (up-)down-conversion processes. In a companion paper [F. Dell'Anno, S. De Siena, and F. Illuminati, 69, 033813 (2004)], we provide the extension of the nonlinear canonical formalism to multimode systems, we introduce the associated heterodyne multiphoton squeezed states, and we discussmore » their possible experimental realization.« less

Film rupture in the diffuse interface model coupled to hydrodynamics.

The process of dewetting of a thin liquid film is usually described using a long-wave approximation yielding a single evolution equation for the film thickness. This equation incorporates an additional pressure term-the disjoining pressure-accounting for the molecular forces. Recently a disjoining pressure was derived coupling hydrodynamics to the diffuse interface model [L. M. Pismen and Y. Pomeau, Phys. Rev. E 62, 2480 (2000)]. Using the resulting evolution equation as a generic example for the evolution of unstable thin films, we examine the thickness ranges for linear instability and metastability for flat films, the families of stationary periodic and localized solutions, and their linear stability. The results are compared to simulations of the nonlinear time evolution. From this we conclude that, within the linearly unstable thickness range, there exists a well defined subrange where finite perturbations are crucial for the time evolution and the resulting structures. In the remainder of the linearly unstable thickness range the resulting structures are controlled by the fastest flat film mode assumed up to now for the entire linearly unstable thickness range. Finally, the implications for other forms of disjoining pressure in dewetting and for spinodal decomposition are discussed.

Nonlinear evolution of plasmoid structure

The plasmoid observed in the Earth’s magnetotail shows a wide variety of the complicated plasma structures that are not simply described by the standard Petschek reconnection model. The interaction of the plasmoid propagating tailward with the surrounding plasmas of the plasma sheet at rest may be important to understand the plasma sheet structure and the plasma heating observed in the magnetotail. The nonlinear time evolution of the plasmoid is studied by using a large-scale, high-resolution, two dimensional MHD simulation code. Several discontinuities/shocks are formed in association with magnetic reconnection: 1) a pair of the standard Petscheck-type slow shock waves emanating from the X-type neutral point, 2) the tangential discontinuity inside the plasmoid that separates the accelerated plasmas from the original plasma sheet plasmas, 3) the slow shock with a “crab-hand” structure surrounding the front-side of the plasmoid, 4) the intermediate shocks in the edge of the plasma sheet inside the plasmoid, and 5) the contact discontinuity inside the plasma sheet that separates the shock-heated plasmas from the Joule heated plasmas by the magnetic diffusion at the X-type neutral point. We also discuss how those plasma discontinuities/shocks structures are affected by the lobe/mantle plasma condition.

Bulk-to-surface-wave self-conversion in optically induced ionization processes.

Nonlinear time evolution of a p-polarized wave mode with inhomogeneous transverse structure producing tunnel ionization of a gas is investigated by numerical simulation and theoretical analysis. A phenomenon of trapping of electromagnetic radiation via its adiabatic conversion into surface waves guided by the field-created plasma structure is found out numerically. This process is accompanied by significant frequency downshifting of the electromagnetic radiation. The underlying physical mechanism is explained using a simple theoretical model. The described phenomena may play significant role in the self-channeling and frequency tuning of intense (approximately 10(14)-10(18) W/cm(2)) laser pulses in dense gases.

Spin squeezing in nonlinear spin-coherent states

We introduce the nonlinear spin coherent state via its ladder operator formalism and propose a type of nonlinear spin coherent state by the nonlinear time evolution of spin coherent states. By a new version of spectroscopic squeezing criteria we study the spin squeezing in both the spin coherent state and nonlinear spin coherent state. The results show that the spin coherent state is not squeezed in the x, y, and z directions, and the nonlinear spin coherent state may be squeezed in the x and y directions.

Two-step phase separation of a polymer mixture. II. Time evolution of structure factor

Nonlinear time evolution of phase-separating structures in the two-step phase separation process was investigated for a deuterated polybutadiene–polyisoprene mixture by using a time-resolved light scattering technique. The mixture studied has a lower critical solution temperature type phase diagram with a spinodal temperature of 36 °C. The first-step phase separation via spinodal decomposition (SD) was conducted by a temperature jump (T-jump) from 23 °C to 42 °C, and to the late stage of the SD for varying time periods, t0, in order to develop phase-separated domains with varying characteristic size Λm,1. This phase separation was followed by the second-step T-jump to a higher temperature of 70 °C so that each phase-separated domain is again quenched into thermodynamically unstable region. Nonlinear time evolution processes of phase-separating structures after the second-step SD were explored as a function of size of the initial structures Λm,1. We found the following intriguing effects of the initial structures on further evolution of phase-separating structure via the second-step SD: (1) When Λm,1≫Λm,0 (characteristic length of composition fluctuations developed in the early stage SD after quenching the system from a single-phase state to 70 °C), small domains were evolved within the initial domains (defined as large domains) developed during the first-step SD process, while (2) when Λm,1⩽Λm,0, the small domains were not developed, but only the large domains grew at a growth rate larger than that at 42 °C. In the former case (1), we succeeded in separating the scattering due to the small domains and that due to large domains from the observed scattering profile. The separation allows us to investigate a coupling of the time evolution of the large and small domains and nonlinear pathways for the system to achieve a new equilibrium structure after the second-step SD process.