Order and chaos in a new 3D dynamical model describing motion in non-axially symmetric galaxies

We present a new dynamical model describing 3D motion in non-axially symmetric galaxies. The model covers a wide range of galaxies from a disk system to an elliptical galaxy by suitably choosing the dynamical parameters. We study the regular and chaotic character of orbits in the model and try to connect the degree of chaos with the parameter describing the deviation of the system from axial symmetry. In order to obtain this, we use the Smaller ALingment Index (SALI) method to extensive samples of orbits obtained by integrating numerically the equations of motion, as well as the variational equations. Our results suggest that the influence of the deviation parameter on the portion of chaotic orbits strongly depends on the vertical distance z from the galactic plane of the orbits. Using different sets of initial conditions, we show that the chaotic motion is dominant in galaxy models with low values of z, while in the case of stars with large values of z the regular motion is more abundant, both in elliptical and disk galaxy models.     

A new method for detecting non-linear damping and restoring forces in non-linear oscillation systems from transient data

The purpose of this study is to recover the functional form of both non-linear damping and non-linear restoring forces in the non-linear oscillatory motions of an autonomous system. Using two sets of measured motion response data of the system, an inverse problem is formulated for recovering (or identification): the differential equation of motion is transformed into an equivalent integral equation of motion. The identification, which is non-linear, is shown to be one-to-one. However, the inverse problem formulated herein is concerned with the Volterra-type of non-linear integral equation of the first kind. This leads to numerical instability: solutions of the inverse problem lack stability properties. In order to overcome the difficulty, a regularization method is applied to the identification process. In addition, an L-curve criterion, combined with regularization, is introduced to find an optimal choice for the regularization parameter (i.e., the number of iterations), in the presence of noisy data. The workability of the identification is investigated for simultaneously recovering the functional form of the non-linear damping and the non-linear restoring forces through a numerical experiment.     

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory

Summary Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.

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Sommario Da diversi anni gli esponenti caratteristici di Lyapunov sono divenuti di notevole interesse nello studio dei sistemi dinamici al fine di caratterizzare quantitativamente le proprietà di stocasticità, legate essenzialmente alla divergenza esponenziale di orbite vicine. Si presenta dunque il problema del calcolo esplicito di tali esponenti, già risolto solo per il massimo di essi. Nel presente lavoro si dà un metodo per il calcolo di tutti tali esponenti, basato sul calcolo degli esponenti di ordine maggiore di uno, legati alla crescita di volumi. A tal fine si dà un teorema che mette in relazione gli esponenti di ordine uno con quelli di ordine superiore. Il metodo numerico e alcune applicazioni saranno date in un sucessivo articolo.

     

Nonlinear closure modeling in reduced order models for turbulent flows: a dynamical system approach

Nonlinear Dynamics - Reduced-order models (ROM) of structurally dominated fluid flows have significant applications in science and engineering, such as design, control, and optimization....     

A new method for Fourier series expansions: Applications in rotor-seal systems

Complex nonlinearities of rotor-seal systems make it difficult to implement some widely used techniques for nonlinear vibrations. This is partly due to the fact that it is rather cumbersome to expand the nonlinear terms into Fourier series. In this paper, a novel Fourier series expansion method is proposed to circumvent this difficulty. The incremental harmonic balance method is utilized to obtain the solutions of a rotor-seal system, where the complicated nonlinearities are handled by the expansion method. Periodic, double-periodic and triple-periodic solutions are obtained in excellent agreement with numerical results, which shows the validity and efficacy of the proposed solution procedures.     

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

Summary The present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).

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Sommario Questo articolo, insieme con il precedente (Parte 1: Teoria, pubblicato in questa stessa rivista) è inteso a fornire un metodo esplicito per il calcolo di tutti gli esponenti caratteristici di Lvapunov per un sistema dinamico. Dopo la teoria generale su tali esponenti sviluppata nella prima parte, qui si illustra il metodo di calcolo (Capitolo A) e si danno esempi numerici per applicazioni di varietà in sè e per sistemi Hamiltoniani (Capitolo B).

     

The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold

This paper discusses stability conditions and a chaotic behavior of the Lorenz dynamical system involving the Caputo fractional derivative of orders between 0 and 1. Contrary to some existing results on the topic, we study these problems with respect to a general (not specified) value of the Rayleigh number as a varying control parameter. Such a bifurcation analysis is known for the classical Lorenz system; we show that analysis of its fractional extension can yield different conclusions. In particular, we theoretically derive (and numerically illustrate) that nontrivial equilibria of the fractional Lorenz system become locally asymptotically stable for all values of the Rayleigh number large enough, which contradicts the behavior known from the classical case. As a main proof tool, we derive the optimal Routh–Hurwitz conditions of fractional type, i.e., necessary and sufficient conditions guaranteeing that all zeros of the corresponding characteristic polynomial are located inside the Matignon stability sector. Beside it, we perform other bifurcation investigations of the fractional Lorenz system, especially those documenting its transition from stability to chaotic behavior.     

Synchronization of a novel fractional order stretch-twist-fold (STF) flow chaotic system and its application to a new authenticated encryption scheme (AES)

In this paper, a new fractional order stretch-twist-fold (STF) flow dynamical system is proposed. The stability analysis of the proposed system equilibria is accomplished and we establish that the system is exhibited chaos even for order less than 3. The active control method is applied to enquire the hybrid phase synchronization between two identical fractional order STF flow chaotic systems. These synchronized systems are applied to formulate an authenticated encryption scheme newly for message (text and image) recovery. It is widely applied in the field of secure communication. Numerical simulations are presented to validate the effectiveness of the proposed theory.     

True damping and frequency prediction for aeroelastic systems: The PP method

This paper presents a numerical scheme for stability analysis of the aeroelastic systems in the Laplace domain. The proposed technique, which is called the PP method, is proposed for when the aerodynamic model is represented in the Laplace domain and includes complicated transcendental expressions in terms of the Laplace variable. This method utilizes a matrix iterative procedure to find the eigenvalues of the system and generalizes the other methods such as the P and PK methods for prediction of the flutter conditions. The major advantage of this technique over the other approximate methods is true prediction of subcritical damping and frequency values of the aeroelastic modes. To examine the present technique for stability analysis, some typical examples are used which illustrate its applicability and advantages.     

Hamiltonian structure and relative equilibria of an axisymmetric rigid body in a second degree and order gravity field: cylindrical and generalized hyperbolic equilibria

The full dynamics of an axisymmetric rigid body in a uniformly rotating second degree and order gravity field are investigated, where orbit and attitude motions of the body are coupled through the gravity. Compared with the classical orbital dynamics with the body considered as a point mass, the full dynamics is a higher-precision model in close proximity of the central body where the gravitational orbit–attitude coupling is significant, such as a spacecraft about a small asteroid or an irregular-shaped natural satellite about a planet. The full dynamics are modeled by using the non-canonical Hamiltonian structure, in terms of variables expressed in the frame fixed with the central body. A Poisson reduction is carried out by means of the axial symmetry of the body, and a reduced system with lower dimension, as well as its non-canonical Hamiltonian structure and equations of motion, is obtained through the reduction process. With the second-order potential, three types of relative equilibria are found to be possible: cylindrical equilibria, generalized hyperbolic equilibria, and conic equilibria, which are counterparts to cylindrical equilibria, hyperbolic equilibria, and conic equilibria of an axisymmetric rigid body in a spherical gravity field, respectively. The geometrical properties and existence of the cylindrical equilibria and generalized hyperbolic equilibria are investigated in detail. It has been found that compared with the classical results in a spherical gravity field, the relative equilibria in this study are more complicated and diverse. The most significant difference is that the non-spherical gravity field enables the existence of non-Lagrangian hyperbolic equilibria, called generalized hyperbolic, which cannot exist in a spherical gravity.     

The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

The purpose of this paper is to provide a brief summary of the various analytical predictive criteria in order for strange phenomena to occur in a class of softening nonlinear oscillators, oscillators which may exhibit escape from a potential well. Implications of Melnikov's criteria are discussed first and transient chaos in the twin-well potential oscillator is illustrated. Three different heuristic criteria for steady state chaos or escape solution, proposes by F. Moon, G. Schmidt and W. Szempliskia-Stupnicka, are then presented and compared to computer simulation results.     

The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

Optical plume velocimetry: a new flow measurement technique for use in seafloor hydrothermal systems

Evidence suggests that fluid flow rates in mid-ocean ridge hydrothermal systems may be strongly influenced by mechanical forces such as ocean tidal loading. However, long time-series measurements of flow have not been collected in these environments. We develop a non-invasive method, called optical plume velocimetry (OPV), suitable for obtaining fluid flow rates through black smoker vents based on image analysis of effluent video. We use video from laboratory flows to evaluate three different methods for estimating the image-velocity field that are based on region-based matching, spectral-analysis of Hovmöller diagrams, and temporal cross-correlation of adjacent pixel values. We find that OPV is most sensitive and least biased when the cross-correlation method is used and conclude that OPV should not be applied to flows that are transitioning between jet-like and plume-like behavior.     

The hybrid method, a new method for accurate heat transfer predictions in channels with axially variable heat flux

A new computation method is presented for the accurate determination of temperature fields and heat transfer characteristics for steady forced convection flow in coolant channels with an arbitrary prescribed wall heat flux. The essential feature of this method is that the fluid region is discretized in the section normal to the flow direction. Energy balances set up for each of the elements yield a system of linear first order differential equations with as a sole independent variable an axial distance parameter. This system of equations is solved analytically using a matrix method. The method is illustrated for the case of laminar flow in flat and circular ducts.     

On a new extended finite element method for dislocations: Core enrichment and nonlinear formulation

A recently developed finite element method for the modeling of dislocations is improved by adding enrichments in the neighborhood of the dislocation core. In this method, the dislocation is modeled by a line or surface of discontinuity in two or three dimensions. The method is applicable to nonlinear and anisotropic materials, large deformations, and complicated geometries. Two separate enrichments are considered: a discontinuous jump enrichment and a singular enrichment based on the closed-form, infinite-domain solutions for the dislocation core. Several examples are presented for dislocations constrained in layered materials in 2D and 3D to illustrate the applicability of the method to interface problems.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

A nonlinear dynamical systems analysis of child emotion displays in relation to family context and child adjustment: a cox hazard approach

This report examines how the relative attractor strengths of children's display of three emotion states, anger, sadness/fear, and neutral-engaged, are associated with exposure to maternal negative affect and care giving disruptions, and to child antisocial behavior and depression. Exposure to negative maternal affect was associated with a weaker attractor state for sadness or fear displays relative to those for anger and neutral-engaged displays. Exposure to care giving disruptions was associated with stronger attractor strength for anger and sadness/fear relative to that for neutral-engaged. Overt and covert antisocial behaviors were associated with weaker attractor states for sadness/fear displays relative that for the neutral-engaged displays. Overt antisocial behavior was associated with a stronger attractor state for anger displays relative to that for neutral-engaged displays, and covert antisocial behavior with a weaker attractor state for fear/sadness displays relative to that for neutral-engaged displays. Child depressive symptoms were marginally associated with a stronger attractor state for fear/sadness displays relative to neutral-engaged. The data suggest the attractor strengths for emotion display states are affected by social experience and that between-individual risk for various forms of psychopathology is related to the relative intra-individual attractor strength of various emotion displays in a multi-state emotion display system.