Simultaneous clock recovery and data regeneration using a nonlinear optical loop mirror as an all-optical mixer

High-speed optical clock recovery, demultiplexing and data regeneration will be integral parts of any future photonic network based on high bit-rate OTDM. Much research has been conducted on devices that perform these functions, however to date each process has been demonstrated independently. A very promising method of all-optical switching is that of a semiconductor optical amplifier-based nonlinear optical loop mirror (SOA-NOLM). This has various advantages compared with the standard fiber NOLM, most notably low switching power, compact size and stability. We use the SOA-NOLM as an all-optical mixer in a classical phase-locked loop arrangement to achieve optical clock recovery, while at the same time achieving data regeneration in a single compact device

Nonlinear multicore waveguiding structures with balanced gain and loss

We study existence, stability, and dynamics of linear and nonlinear stationary modes propagating in radially symmetric multicore waveguides with balanced gain and loss. We demonstrate that, in general, the system can be reduced to an effective PT-symmetric dimer with asymmetric coupling. In the linear case, we find that there exist two modes with real propagation constants before an onset of the PT-symmetry breaking while other modes have always the propagation constants with nonzero imaginary parts. This leads to a stable (unstable) propagation of the modes when gain is localized in the core (ring) of the waveguiding structure. In the case of nonlinear response, we show that an interplay between nonlinearity, gain, and loss induces a high degree of instability, with only small windows in the parameter space where quasistable propagation is observed. We propose a novel stabilization mechanism based on a periodic modulation of both gain and loss along the propagation direction that allows bounded light propagation in the multicore waveguiding structures.

Local Bifurcations in a Nonlinear Model of a Bioreactor

This paper is partially supported by the Bulgarian Science Fund under grant Nr. DO 02– 359/2008.

Practical Stability and Vector-Lyapunov Functions for Impulsive Differential Equations with "Supremum"

Stability of nonlinear impulsive differential equations with "supremum" is studied. A special type of stability, combining two different measures and a dot product on a cone, is defined. Perturbing cone-valued piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential equations have been employed.

Nonlinear Waves: An Introduction

This book deals with equations of mathematical physics as the different modifications of the KdV equation, the Camassa-Holm type equations, several modifications of Burger's equation, the Hunter-Saxton equation, conservation laws equations and others. The equations originate from physics but are proposed here for their investigation via purely mathematical methods in the frames of university courses. More precisely, we propose classification theorems for the traveling wave solutions for a sufficiently large class of third order nonlinear PDE when the corresponding profiles develop different kind of singularities (cusps, peaks), existence and uniqueness results, etc. The orbital stability of the periodic solutions of traveling type for mKdV equations are also studied. Of great interest too is the interaction of peakon type solutions of the Camassa-Holm equation and the solvability of the classical and generalized Cauchy problem for the Hunter-Saxton equation. The Riemann problem for special systems of conservation laws and the corresponding -shocks are also considered. As it concerns numerical methods we apply the CNN approach. The book is addressed to a broader audience including graduate students, Ph.D. students, mathematicians, physicist, engineers and specialists in the domain of PDE.

Stability in Terms of Two Measures for Differential Equations with “Maxima”

Атанаска Георгиева, Стела Глухчева, Снежана Христова - Изследвана е устойчивостта на нелинейни диференциални уравнения с “максимуми” по отношение на две мерки. Приложени са две различни мерки за началните условия и за решението. Използван е методът на Разумихин, а също така и методът на сравнението на обикновени скаларни диференциални уравнения. Приложението на получените резултати и достатъчни условия за устойчивост е илюстрирано с пример.

Stable optical vortices in nonlinear multicore fibers

The multicore fiber (MCF) is a physical system of high practical importance. In addition to standard exploitation, MCFs may support discrete vortices that carry orbital angular momentum suitable for spatial-division multiplexing in high-capacity fiber-optic communication systems. These discrete vortices may also be attractive for high-power laser applications. We present the conditions of existence, stability, and coherent propagation of such optical vortices for two practical MCF designs. Through optimization, we found stable discrete vortices that were capable of transferring high coherent power through the MCF.

A Harrod modell strukturális stabilitása (Structural stability of the Harrod model)

In this study it is shown that the nontrivial hyperbolic fixed point of a nonlinear dynamical system, which is formulated by means of the adaptive expectations, corresponds to the unstable equilibrium of Harrod. We prove that this nonlinear dynamical (in the sense of Harrod) model is structurally stable under suitable economic conditions. In the case of structural stability, small changes of the functions (C1-perturbations of the vector field) describing the expected and the true time variation of the capital coefficients do not influence the qualitative properties of the endogenous variables, that is, although the trajectories may slightly change, their structure is the same as that of the unperturbed one, and therefore these models are suitable for long-time predictions. In this situation the critique of Lucas or Engel is not valid. There is no topological conjugacy between the perturbed and unperturbed models; the change of the growth rate between two levels may require different times for the perturbed and unperturbed models.

Soliton based DPSK encoded sequences:stability and dynamical properties

We study the dynamical properties of the RZ-DPSK encoded sequences of bits, focusing on the instabilities in the train leading to the bit stream corruption. The problem is studied within the framework of the complex Toda chain model for optical solitons. We show how the bit composition of the pattern affects the initial stage of the train dynamics and explain the general mechanisms of the appearance of unstable collective soliton modes. Then we discuss the nonlinear regime using the asymptotic properties of the pulse stream at large propagation distances and analyze the dynamical behavior of the train elucidating different scenarios for the pattern instabilities. ©2010 Crown.

The Helmholtzian Equation: Stabilizing Mechanisms for Quadratic Nonlinear Fields

Hexagonal Resonant Triad patterns are shown to exist as stable solutions of a particular type of nonlinear field where no cubic field nonlinearity is present. The zero ‘dc’ Fourier mode is shown to stabilize these patterns produced by a pure quadratic field nonlinearity. Closed form solutions and stability results are obtained near the critical point, complimented by numerical studies far from the critical point. These results are obtained using a neural field based on the Helmholtzian operator. Constraints on structure and parameters for a general pure quadratic neural field which supports hexagonal patterns are obtained.

Nonlinear development of matrix-converter instabilities

Matrix converters convert a three-phase alternating-current power supply to a power supply of a different peak voltage and frequency, and are an emerging technology in a wide variety of applications. However, they are susceptible to an instability, whose behaviour is examined herein. The desired “steady-state” mode of operation of the matrix converter becomes unstable in a Hopf bifurcation as the output/input voltage transfer ratio, q, is increased through some threshold value, qc. Through weakly nonlinear analysis and direct numerical simulation of an averaged model, we show that this bifurcation is subcritical for typical parameter values, leading to hysteresis in the transition to the oscillatory state: there may thus be undesirable large-amplitude oscillations in the output voltages even when q is below the linear stability threshold value qc.

Nonlinear Approaches to Attitude Control Using Magnetic and Mechanical Actuation

This thesis argues the attitude control problem of nanosatellites, which has been a challenging issue over the years for the scientific community and still constitutes an active area of research. The interest is increasing as more than 70% of future satellite launches are nanosatellites. Therefore, new challenges appear with the miniaturisation of the subsystems and improvements must be reached. In this framework, the aim of this thesis is to develop novel control approaches for three-axis stabilisation of nanosatellites equipped with magnetorquers and reaction wheels, to improve the performance of the existent control strategies and demonstrate the stability of the system. In particular, this thesis is focused on the development of non-linear control techniques to stabilise full-actuated nanosatellites, and in the case of underactuation, in which the number of control variables is less than the degrees of freedom of the system. The main contributions are, for the first control strategy proposed, to demonstrate global asymptotic stability derived from control laws that stabilise the system in a target frame, a fixed direction of the orbit frame. Simulation results show good performance, also in presence of disturbances, and a theoretical selection of the magnetic control gain is given. The second control approach presents instead, a novel stable control methodology for three-axis stabilisation in underactuated conditions. The control scheme consists of the dynamical implementation of an attitude manoeuvre planning by means of a switching control logic. A detailed numerical analysis of the control law gains and the effect on the convergence time, total integrated and maximum torque is presented demonstrating the good performance and robustness also in the presence of disturbances.

Stability issues in racing motorcycles: an in-depth analysis of the chatter vibration

Racing motorcycles are prone to an unstable oscillatory motion of the swingarm and rear wheel, commonly known as ‘chatter’. This vibration mode typically has a frequency of 17 Hz to 22 Hz and typically occurs during heavy braking manoeuvres. The appearance of chatter can cause reduced rider confidence, and therefore lead to longer lap times during races and the increased risk of crashing. This thesis looks to further the understanding of this mode. It includes the development of a simplified model to explore the effects roll angle and lateral dynamics have on the chatter mode using linear analysis. The mechanisms of instability and parameter sensitivities are also examined. The effects of the nonlinearities present in the minimal model equations of motion are examined, including the identification of limit cycles and their stability, inspecting individual nonlinear terms and their effects, and introducing tyre relaxation and determining the effect it has on the dynamics. Finally, an exploratory study of the mid-corner region of a typical racing manoeuvre is performed in hopes to better understand if any high frequency tyre induced instabilities like chatter can occur.

Effect Of Platform Connection And Abutment Material On Stress Distribution In Single Anterior Implant-supported Restorations: A Nonlinear 3-dimensional Finite Element Analysis.

Although various abutment connections and materials have recently been introduced, insufficient data exist regarding the effect of stress distribution on their mechanical performance. The purpose of this study was to investigate the effect of different abutment materials and platform connections on stress distribution in single anterior implant-supported restorations with the finite element method. Nine experimental groups were modeled from the combination of 3 platform connections (external hexagon, internal hexagon, and Morse tapered) and 3 abutment materials (titanium, zirconia, and hybrid) as follows: external hexagon-titanium, external hexagon-zirconia, external hexagon-hybrid, internal hexagon-titanium, internal hexagon-zirconia, internal hexagon-hybrid, Morse tapered-titanium, Morse tapered-zirconia, and Morse tapered-hybrid. Finite element models consisted of a 4×13-mm implant, anatomic abutment, and lithium disilicate central incisor crown cemented over the abutment. The 49 N occlusal loading was applied in 6 steps to simulate the incisal guidance. Equivalent von Mises stress (σvM) was used for both the qualitative and quantitative evaluation of the implant and abutment in all the groups and the maximum (σmax) and minimum (σmin) principal stresses for the numerical comparison of the zirconia parts. The highest abutment σvM occurred in the Morse-tapered groups and the lowest in the external hexagon-hybrid, internal hexagon-titanium, and internal hexagon-hybrid groups. The σmax and σmin values were lower in the hybrid groups than in the zirconia groups. The stress distribution concentrated in the abutment-implant interface in all the groups, regardless of the platform connection or abutment material. The platform connection influenced the stress on abutments more than the abutment material. The stress values for implants were similar among different platform connections, but greater stress concentrations were observed in internal connections.

Comparison Of Postoperative Stability Of Three Rigid Internal Fixation Techniques After Sagittal Split Ramus Osteotomy For Mandibular Advancement.

The aim of this investigation was to compare the skeletal stability of three different rigid fixation methods after mandibular advancement. Fifty-five class II malocclusion patients treated with the use of bilateral sagittal split ramus osteotomy and mandibular advancement were selected for this retrospective study. Group 1 (n = 17) had miniplates with monocortical screws, Group 2 (n = 16) had bicortical screws and Group 3 (n = 22) had the osteotomy fixed by means of the hybrid technique. Cephalograms were taken preoperatively, 1 week within the postoperative care period, and 6 months after the orthognathic surgery. Linear and angular changes of the cephalometric landmarks of the chin region were measured at each period, and the changes at each cephalometric landmark were determined for the time gaps. Postoperative changes in the mandibular shape were analyzed to determine the stability of fixation methods. There was minimum difference in the relapse of the mandibular advancement among the three groups. Statistical analysis showed no significant difference in postoperative stability. However, a positive correlation between the amount of advancement and the amount of postoperative relapse was demonstrated by the linear multiple regression test (p < 0.05). It can be concluded that all techniques can be used to obtain stable postoperative results in mandibular advancement after 6 months.