Damping models in the truncated derivative nonlinear Schrödinger equation

Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schrödinger equation, has been analyzed; wave 1 is linearly unstable with growth rate , and waves 2 and 3 are stable with damping 2 and 3, respectively. The dependence of gross dynamical features on the damping model as characterized by the relation between damping and wave-vector ratios, 2 /3, k2 /k3, and the polarization of the waves, is discussed; two damping models, Landau k and resistive k2, are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist as against flow contraction just requiring.In the case of right-hand RH polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if 2+3/2.

Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation

In previous papers, the type-I intermittent phenomenon with continuous reinjection probability density (RPD) has been extensively studied. However, in this paper type-I intermittency considering discontinuous RPD function in one-dimensional maps is analyzed. To carry out the present study the analytic approximation presented by del Río and Elaskar (Int. J. Bifurc. Chaos 20:1185-1191, 2010) and Elaskar et al. (Physica A. 390:2759-2768, 2011) is extended to consider discontinuous RPD functions. The results of this analysis show that the characteristic relation only depends on the position of the lower bound of reinjection (LBR), therefore for the LBR below the tangent point the relation {Mathematical expression}, where {Mathematical expression} is the control parameter, remains robust regardless the form of the RPD, although the average of the laminar phases {Mathematical expression} can change. Finally, the study of discontinuous RPD for type-I intermittency which occurs in a three-wave truncation model for the derivative nonlinear Schrodinger equation is presented. In all tests the theoretical results properly verify the numerical data

Modelling relativistic solitary wave interactions in over-dense plasmas: a perturbed nonlinear Schröndinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Chaos in non lineal Alfven waves using the DNLS equations

The electro-dynamical tethers emit waves in structured denominated Alfven wings. The Derivative Nonlineal Schrödinger Equation (DNLS) possesses the capacity to describe the propagation of circularly polarized Alfven waves of finite amplitude in cold plasmas. The DNLS equation is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In this article is presented a theoretical and numerical analysis when the growth rate of the unstable wave is next to zero considering two damping models: Landau and resistive. The DNLS equation presents a chaotic dynamics when is consider only three wave truncation. The evolution to chaos possesses three routes: hard transition, period-doubling and intermittence of type I.

Nonlinear analysis of a simple model of temperature evolution in a satellite

We analyze a simple model of the heat transfer to and from a small satellite orbiting round a solar system planet. Our approach considers the satellite isothermal, with external heat input from the environment and from internal energy dissipation, and output to the environment as black-body radiation. The resulting nonlinear ordinary differential equation for the satellite’s temperature is analyzed by qualitative, perturbation and numerical methods, which prove that the temperature approaches a periodic pattern (attracting limit cycle). This approach can occur in two ways, according to the values of the parameters: (i) a slow decay towards the limit cycle over a time longer than the period, or (ii) a fast decay towards the limit cycle over a time shorter than the period. In the first case, an exactly soluble average equation is valid. We discuss the consequences of our model for the thermal stability of satellites.

Hard transition to chaotic dynamics in Alfven wave fronts

The derivative nonlinear Schrodinger DNLS equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase, no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand LH polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about unstable wave frequency 2/4 x ion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model different dampings of daughter waves,four-dimensional flow; both models differ in significant phase-space features but keep common features essential for the transition.

A large strain anisotropic elastoplastic continuum theory for nonlinear kinematic hardening and texture evolution

In this paper we present a continuum theory for large strain anisotropic elastoplasticity based on a decomposition of the modified plastic velocity gradient into energetic and dissipative parts. The theory includes the Armstrong and Frederick hardening rule as well as multilayer models as special cases even for large strain anisotropic elastoplasticity. Texture evolution may also be modelled by the formulation, which allows for a meaningful interpretation of the terms of the dissipation equation

Reduced three-wave model to study the hard transition to chaotic dynamics in Alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model (equal damping of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase), no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic dynamics that is absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralelling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable.

Fully 3-wave model to study the hard transition to chaotic dynamics in alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. No matter how small the growth rate of the unstable wave, the four-dimensional flow for the three wave amplitudes and a relative phase, with both resistive damping and linear Landau damping, exhibits chaotic relaxation oscillations that are absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable. The parameter domain developing chaos is much broader than the corresponding domain in a reduced 3-wave model that assumes equal dampings of the daughter waves

Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method of freeform optics

The Monge–Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L2 Monge-Kantorovich (LMK) theory, and introduce an efficient approach for finding the optimal mapping of the LMK problem. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.

Nonlinear seismic behaviour of concrete arch bridges

Arch bridge structural solution has been known for centuries, in fact the simple nature of arch that require low tension and shear strength was an advantage as the simple materials like stone and brick were the only option back in ancient centuries. By the pass of time especially after industrial revolution, the new materials were adopted in construction of arch bridges to reach longer spans. Nowadays one long span arch bridge is made of steel, concrete or combination of these two as "CFST", as the result of using these high strength materials, very long spans can be achieved. The current record for longest arch belongs to Chaotianmen bridge over Yangtze river in China with 552 meters span made of steel and the longest reinforced concrete type is Wanxian bridge which also cross the Yangtze river through a 420 meters span. Today the designer is no longer limited by span length as long as arch bridge is the most applicable solution among other approaches, i.e. cable stayed and suspended bridges are more reasonable if very long span is desired. Like any super structure, the economical and architectural aspects in construction of a bridge is extremely important, in other words, as a narrower bridge has better appearance, it also require smaller volume of material which make the design more economical. Design of such bridge, beside the high strength materials, requires precise structural analysis approaches capable of integrating the combination of material behaviour and complex geometry of structure and various types of loads which may be applied to bridge during its service life. Depend on the design strategy, analysis may only evaluates the linear elastic behaviour of structure or consider the nonlinear properties as well. Although most of structures in the past were designed to act in their elastic range, the rapid increase in computational capacity allow us to consider different sources of nonlinearities in order to achieve a more realistic evaluations where the dynamic behaviour of bridge is important especially in seismic zones where large movements may occur or structure experience P - _ effect during the earthquake. The above mentioned type of analysis is computationally expensive and very time consuming. In recent years, several methods were proposed in order to resolve this problem. Discussion of recent developments on these methods and their application on long span concrete arch bridges is the main goal of this research. Accordingly available long span concrete arch bridges have been studied to gather the critical information about their geometrical aspects and properties of their materials. Based on concluded information, several concrete arch bridges were designed for further studies. The main span of these bridges range from 100 to 400 meters. The Structural analysis methods implemented in in this study are as following: Elastic Analysis: Direct Response History Analysis (DRHA): This method solves the direct equation of motion over time history of applied acceleration or imposed load in linear elastic range. Modal Response History Analysis (MRHA): Similar to DRHA, this method is also based on time history, but the equation of motion is simplified to single degree of freedom system and calculates the response of each mode independently. Performing this analysis require less time than DRHA. Modal Response Spectrum Analysis (MRSA): As it is obvious from its name, this method calculates the peak response of structure for each mode and combine them using modal combination rules based on the introduced spectra of ground motion. This method is expected to be fastest among Elastic analysis. Inelastic Analysis: Nonlinear Response History Analysis (NL-RHA): The most accurate strategy to address significant nonlinearities in structural dynamics is undoubtedly the nonlinear response history analysis which is similar to DRHA but extended to inelastic range by updating the stiffness matrix for every iteration. This onerous task, clearly increase the computational cost especially for unsymmetrical buildings that requires to be analyzed in a full 3D model for taking the torsional effects in to consideration. Modal Pushover Analysis (MPA): The Modal Pushover Analysis is basically the MRHA but extended to inelastic stage. After all, the MRHA cannot solve the system of dynamics because the resisting force fs(u; u_ ) is unknown for inelastic stage. The solution of MPA for this obstacle is using the previously recorded fs to evaluate system of dynamics. Extended Modal Pushover Analysis (EMPA): Expanded Modal pushover is a one of very recent proposed methods which evaluates response of structure under multi-directional excitation using the modal pushover analysis strategy. In one specific mode,the original pushover neglect the contribution of the directions different than characteristic one, this is reasonable in regular symmetric building but a structure with complex shape like long span arch bridges may go through strong modal coupling. This method intend to consider modal coupling while it take same time of computation as MPA. Coupled Nonlinear Static Pushover Analysis (CNSP): The EMPA includes the contribution of non-characteristic direction to the formal MPA procedure. However the static pushovers in EMPA are performed individually for every mode, accordingly the resulted values from different modes can be combined but this is only valid in elastic phase; as soon as any element in structure starts yielding the neutral axis of that section is no longer fixed for both response during the earthquake, meaning the longitudinal deflection unavoidably affect the transverse one or vice versa. To overcome this drawback, the CNSP suggests executing pushover analysis for governing modes of each direction at the same time. This strategy is estimated to be more accurate than MPA and EMPA, moreover the calculation time is reduced because only one pushover analysis is required. Regardless of the strategy, the accuracy of structural analysis is highly dependent on modelling and numerical integration approaches used in evaluation of each method. Therefore the widely used Finite Element Method is implemented in process of all analysis performed in this research. In order to address the study, chapter 2, starts with gathered information about constructed long span arch bridges, this chapter continuous with geometrical and material definition of new models. Chapter 3 provides the detailed information about structural analysis strategies; furthermore the step by step description of procedure of all methods is available in Appendix A. The document ends with the description of results and conclusion of chapter 4.