Nonlinear Dynamic Phenomena in Mechanics

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincare sections and Poincare maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincare sections.

If signal analyses were perfect without noise and clutters, then any transform can be equally chosen to represent the signal without any loss of information. However, if the analysis using Fourier transform (FT) happens to be a nonlinear dynamic phenomenon, the effect of nonlinearity must be postponed until a later time when a complicated mode-mode coupling is attempted without the assurance of any convergence. Alternatively, there exists a new paradigm of linear transforms called wavelet transform (WT) developed for French oil explorations. Such a WT enjoys the linear superposition principle, the computational efficiency, and the signal/noise ratio enhancement for a nonsinusoidal and nonstationary signal. Our extensions to a dynamic WT and furthermore to an adaptive WT are possible due to the fact that there exists a large set of square-integrable functions that are special solutions of the nonlinear dynamic medium and could be adopted for the WT. In order to analyze nonlinear dynamics phenomena in ocean, we are naturally led to the construction of a soliton mother wavelet. This common sense of 'pay the nonlinear price now and enjoy the linearity later' is certainly useful to probe any nonlinear dynamics. Research directions in wavelets, such as adaptivity, and neural network implementations are indicated, e.g., tailoring an active sonar profile for explorations.

We develop a continuous time model with heterogeneous fundamentalists, imitators, and discrete time delays. We show that nonlinear dynamic phenomena, such as coexistence of attractors and local and global bifurcations, occur due to the existence of a time gap in the process of adjustment of market prices.

Rub-impact and fatigue crack are two important rotor faults. Based on the crack theory, an improved switching crack model is presented. Dynamic characteristics of a rotor-bearing system with imbalance, rub-impact and transverse crack are attempted. Various nonlinear dynamic phenomena are analyzed using numerical method. The results reveal that unstable form of the rotor system with coupling faults is extremely complex as the rotating speed increases and there are some low frequencies with large amplitude. The influence of crack depth and angle on the dynamic behavior of a cracked rotor is important insomuch as the quasi-periodical and chaotic motions in system response will occur along with different parameters. It is indicated that this study can contribute to a further understanding of the nonlinear dynamics of such a rotor system with coupling faults of crack and rub-impact.

Pattern formation is a widespread phenomenon observed in physical, chemical and biological systems. During the past decades, powerful experimental and theoretical tools for the investigation of that phenomenon have been developed. Also, the set of pattern forming system was diversified. The present issue presents a panorama of pattern formation and other nonlinear dynamical phenomena in different natural and engineering systems. Special attention is paid to nonlinear dynamics of fluids. Non-equilibrium phenomena in chemical and quantum systems are also considered. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

We address two research applications in this methodological review: starting from an audio recording, the goal may be to characterize nonlinear phenomena (NLP) at the level of voice production or to test their perceptual effects on listeners. A crucial prerequisite for this work is the ability to detect NLP in acoustic signals, which can then be correlated with biologically relevant information about the caller and with listeners' reaction. NLP are often annotated manually, but this is labour-intensive and not very reliable, although we describe potentially helpful advanced visualization aids such as reassigned spectrograms and phasegrams. Objective acoustic features can also be useful, including general descriptives (harmonics-to-noise ratio, cepstral peak prominence, vocal roughness), statistics derived from nonlinear dynamics (correlation dimension) and NLP-specific measures (depth of modulation and subharmonics). On the perception side, playback studies can greatly benefit from tools for directly manipulating NLP in recordings. Adding frequency jumps, amplitude modulation and subharmonics is relatively straightforward. Creating biphonation, imitating chaos or removing NLP from a recording are more challenging, but feasible with parametric voice synthesis. We describe the most promising algorithms for analysing and manipulating NLP and provide detailed examples with audio files and R code in supplementary material.This article is part of the theme issue 'Nonlinear phenomena in vertebrate vocalizations: mechanisms and communicative functions'.

Preference-Free Option Pricing with Path-Dependent Volatility: A Closed-Form Approach

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation A. C.‐L. Chian, F. A. Borotto, E. L. Rempel; Onset of Alfvén Turbulence via Boundary Crisis. AIP Conf. Proc. 11 June 2003; 669 (1): 715–718. https://doi.org/10.1063/1.1594030 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

We present modeling, analysis, and experimental investigation for dynamic instabilities and bifurcations in electrostatically actuated resonators. These instabilities are induced by exciting a microstructure with a nonlinear forcing composed of a DC parallel-plate electrostatic load superimposed to an AC harmonic load. Because of the dominant effect of the electrostatic nonlinearity, several resonances and nonlinear phenomena are induced. Examples of these are the excitation of secondary-resonances, superharmonic and subharmonic, at half and twice the natural frequency of the microstructure. Also, local bifurcations, such as saddle-node and pitchfork, and global bifurcations, such as the escape phenomenon and the homoclinic tangling may occur. These lead to undesirable jumps, hysteresis, and dynamic pull-in instabilities in MEMS devices and structures. The present work represents an attempt to explore these topics in more depth. The first part of this paper is focused on analyzing and studying the nonlinear dynamics of a capacitive device both theoretically and experimentally with a focus on the case of primary-resonance excitation (near the fundamental natural frequency of the structure). The device is made up of two cantilever beams with a proof mass attached to their tips. A nonlinear spring-mass-damper model is utilized, which accounts for squeeze-film damping. Long-time integration for the equation of motion is used to compare with the obtained experimental data. Then, global dynamic analysis is conducted using a finite difference method (primary resonance) and shooting method (subharmonic resonance) combined with the Floquet theory to capture periodic orbits and analyze their stability. The domains of attraction (basins of attraction) for selected data are calculated numerically. Experimental data revealing primary and sub-harmonic resonances, dynamic pull-in, and the escape-from-a-potential-well phenomenon are shown and compared with the theoretical results.

Analysis of nonlinear dynamics of fully submerged payload hanging from offshore crane vessel

Nonlinear phenomena play an important role in the finite dynamics of elastic structural systems with initial curvature, such as suspended cables and arches. In this paper, we document their richness and analyze some of them with reference to two different multidegree-of-freedom models of an elastic cable, a system which can easily exhibit multiple internal resonance conditions with ensuing accretion of the nonlinear and chaotic coupling phenomena: an experimental cable/mass system and a discrete theoretical model of a continuous system. Attention is devoted to multimodal interaction, coexistence and competition of various response classes, transition mechanisms, nonstationary motions, reconstruction of attractors in pseudo-phase space, and quantitative characterization of chaotic responses.KeywordsChaotic MotionInternal ResonanceQuantitative CharacterizationMultimodal InteractionChaotic ResponseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Spatially localized excitation of a one-dimensional magnetic system in an external high-frequency field is studied analytically (collective variable method) and numerically. The dependences of the amplitude of local quasi-soliton states on the amplitude of this field for different frequencies and damping in the system are found for the case of direct application of a localized magnetic field. The field dependence is found to have hysteresis and “instability windows” in the stationary states. Within these instability regions, the excitations become nonstationary: they are accompanied by additional periodic nutation of the magnetization vector. Here such nonlinear dynamics phenomena as period doubling and transitions to chaos are observed.

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drift and mooring systems. We were able to reveal new phenomena based on the design methodology developed at the University of Michigan for autonomous mooring systems and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or non-periodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes 2–3 orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

Sloshing is a complex nonlinear dynamic phenomenon. It is difficult and challenging to model the coupling between the sloshing and the motion of the tank body. The sloshing has significant influence on the stability of the system. Hence, it is essential and necessary to study the dynamic characteristics of sloshing system. In this study, a wave height function and a potential function are used to describe sloshing phenomenon and the moving boundary conditions between the fluid and the structure. Nonlinear dynamics are established to model the coupled dynamic equations based on Hamilton principle with the assumption of ideal fluid. It is shown that the unity of fluid motion and rigid body motion can be realized by the application of Hamilton principle. The obtained equations are solved via Garlerkin's method to yield approximate numerical solutions. Finally, the theoretical analysis is verified by comparing the numerical solutions with the dynamic response of the FEM software.