Abstract: The Arnold Sommerfeld effect is an intriguing resonance capture and release series of events originally demonstrated in 1902. A single event is studied using a two degree of freedom mathematical model of a motor with imbalance mounted to laterally restricted spring connected cart. For a certain power supplied, in general the motor rotates at a speed consistent with a motor on a rigid base. However at speeds close to the natural frequency of the cart, it seemingly takes on extra oscillations where for a single rotation it both speeds up and then slows down. Therefore in a standard experimental demonstration of the effect, as the supplied torque force is increased or decreased, this may give the illusion that the stable operation of the motor is losing and gaining stability. This is not strictly the case, instead small oscillations always present in the system solution are amplified near the resonant frequency. The imbalance in the motor causes a single resonance curve to fold back on itself forming two fold bifurcations which leads to hysteresis and an asymmetry between increasing and decreasing the motor speed. Although as outlined the basic mechanism is due the interplay between two stable and one unstable limit cycles, a more complicated bifurcation scenario is observed for higher imbalances in the motor. The presence of a Z2 phase space symmetry tempers the dynamics and bifurcation picture.
Abstract: Acoustic metamaterials with specifically designed lattices can manipulate acoustic/elastic waves in unprecedented ways. Whereas there are many studies that focus on passive linear lattice, with nonreconfigurable structures. In this letter, we present the design, theory and experimental demonstration of an active nonlinear acoustic metamaterial, the dynamic properties of which can be modified instantaneously with reversibility. By incorporating active and nonlinear elements in a single unit cell, a realtime tunability and switchability of the band gap is achieved. In addition, we demonstrate a dynamic "editing" capability for shaping transmission spectra, which can be used to create the desired band gap and resonance. This feature is impossible to achieve in passive metamaterials. These advantages demonstrate the versatility of the proposed device, paving the way toward smart acoustic devices, such as logic elements, diode and transistor.
Abstract: A technique to optimize the stability of a general mechanical system is outlined. The method relies on decomposing the damping matrix into several component matrices, which may have some special structure or physical relevance. An optimization problem can then be formulated where the ratio of these are varied to either stabilize or make more stable the equilibrium state subject to sensible constraints. For the purpose of this study, we define a system to be more stable if its eigenvalue with largest real part is as negative as possible. The technique is demonstrated by applying it to an introduced nondimensionalized variant of a known minimal wobbling disc brake model. In this case, it is shown to be beneficial to shift some damping from the disc to the pins for a system optimized for stability.
Abstract Coupled systems of second order differential equations naturally arise in applied mechanics and mechanical engineering as the equations of motion for a mechanical device. For many engineering applications, as the avoidance of instabilities is sought, after a suitable canonical choice of coordinates and possible linearization around a central equilibrium, Lypunouv stability is determined via the eigenvalue of the Jacobian with maximal real part. This spectral stability test is well ingrained in the engineering industry with many commercial software packages focused on efficiently applying the technique computationally to systems of large degree. However in practise, application of the technique is often applied naively without due consideration for the limiting criteria.
This presentation will address second order linear differential equations with timeperiodic coefficients which often arise as the equations of motion for systems involving, e.g. rotors. Paradigmatic examples have been studied indepth by the dynamics and vibrations group of TU Darmstadt over several years and some ongoing projects will be summarised. It will be stressed that the above stability test is not applicable in these cases and emphasised that Floquet theory is the appropriate approach to determine stability. In this case, timeperiodicity is removed by a suitable coordinate transformation as specified by Floquet, with the eigenvalue of the monodromy matrix with largest absolute value, known as the maximal Floquet multiplier, determining stability. As a parameter is varied, e.g. the rotor speed, a bifurcation occurs and the system becomes unstable if a Floquet multiplier crosses the unit circle. Qualitatively different transitory dynamics results depending on whether the critical multiplier occurs at 1, 1, a multiple of 1, or as part of a complex conjugate pair. In addition, symmetries in the equation of motion can limit the type of bifurcations permitted and thereby the resultant dynamics.
A robust and efficient technique for calculating the Floquet multipliers numerically is therefore vital to understand realistic engineering models. Large stiff systems cause particular problems in obtaining reliable and accurate approximations of the monodromy matrix. For example, the computational cost of the direct integration method grows exponentially with size of the system as one is required to integrate over the period for each degree of freedom. Other, less well known, techniques exist which can make some computational savings by skipping monodromy matrix calculation and obtaining Floquet multipliers indirectly.
Preface: for the IUTAM proceedings.
Nonlinear dynamics is a fascinating rapidly growing field in research and applications. Almost everywhere in nature one encounters beautiful dynamical phenomena which are challenging to understand. New phenomena have been found in physical experiments and in extensive numerical simulations. Engineering problems currently being investigated are for example selfexcited vibrations of rotor systems, dynamics of wind turbines, granular dynamics, fluidstructure interaction, vibroimpacts, elastic waves in nonlinear media and many other fields. Numerics and algorithmic tools can however only be effective when joined with qualitative insight to understand the underlying generic mechanisms. A wealth of analytical and semianalytical methods and techniques are available for the investigation of nonlinear systems. Many of them are classical, such as perturbation theory, others are of more recent origin. Examples of analytical methods for the investigation of ordinary and partial differential equations, which commonly arise in the context of nonlinear oscillations, are center manifold and normal form theory, nonlinear normal mode analysis and embedding techniques, such as Carleman linearization, among others. The major challenges are for example the study of bifurcations, the estimation of basins of attraction and the detection of chaotic behavior. The methods have to be adapted to different systems which may have slow and fast dynamics, as well as discontinuities yielding stiff differential equations.Abstract: We report on the analysis of dynamics of mutually coupled semiconductor lasers in photonic integrated circuits. Slotted Fabry Perot (SFP) lasers were integrated via waveguide sections of varying lengths to analyse the stability and properties of CW output, for use in advanced modulation formats.
Abstract: We theoretically investigate the dynamics of two mutually coupled identical singlemode semiconductor lasers. For small separation and large coupling between the lasers, symmetrybroken onecolour states are shown to be stable. In this case the light output of the lasers have significantly different intensities while at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable symmetrybroken twocolour states, where both lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. Using a five dimensional model we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetrybroken onecolour and twocolour states. Several of these states give rise to multistabilities and therefore allow for the design of alloptical memory elements on the basis of two coupled singlemode lasers. The switching performance of selected designs of optical memory elements is studied numerically.
Abstract: We report on the identification of 54 embedded clusters around 217 massive protostellar candidates of which 34 clusters are new detections. The embedded clusters are identified as stellar surface density enhancements in the 2 μm All Sky Survey (2MASS) data. Because the clusters are all associated with massive stars in their earliest evolutionary stage, the clusters should also be in an early stage of evolution. Thus the properties of these clusters should reflect properties associated with their formation rather than their evolution. For each cluster, we estimate the mass, the morphological type, the photometry and extinction. The clusters in our study, by their association with massive protostars and massive outflows, reinstate the notion that massive stars begin to form after the first generation of low mass stars have completed their accretion phase. Further, the observed high gas densities and accretion rates at the centers of these clusters is consistent with the hypothesis that high mass stars form by continuing accretion onto low mass stars.
Abstract: We theoretically investigate two laser diodes weakly coupled via a small amount of each laser's light entering the cavity of the other after a moderate delay. Using a well known rate equation description, eight stable dynamical states are categorised namely inphase and antiphase onecolour states, symmetry and symmetry broken undamped relaxation oscillations, symmetric and symmetrybroken quasiperiodic states, and symmetric and symmetrybroken chaotic dynamics.
Using a recently published method to remove singularities from the laser rate equations, it was possible to make a bifurcation diagram by varying the coupling strength and coupling phase. The extent of each dynamical state in terms of this parameter space is ascertained and the bifurcation transitions between them are studied. The Lyapunov exponent is calculated and used to delimit chaotic regions on the bifurcation diagram. Several routes to chaos are present and several areas of multistabilities are observed.
Abstract: The dynamics of two mutually coupled identical singlemode semiconductor lasers are theoretically investigated. For small separation and large coupling between the lasers, symmetrybroken onecolour states are shown to be stable. In this case the light output of the lasers have significantly different intensities whilst at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable twocolour states, where both singlemode lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. For low coupling but possibly large separation, the frequency of the relaxation oscillations of the freerunning lasers defines the dynamics. Chaotic and quasiperiodic states are identified and shown to be stable. For weak coupling undamped relaxation oscillations dominate where each laser is locked to three or more odd number of colours spaced by the relaxation oscillation frequency. It is shown that the instabilities that lead to these states are directly connected to the two colour mechanism where the change in the number of optical colours due to a change in the plane of oscillation.
At initial coupling, inphase and antiphase one colour states are shown to emerge from "on" uncoupled lasers using a perturbation method. Similarly symmetrybroken onecolour states come from considering one freerunning laser initially "on" and the other laser initially "off". The mechanism that leads to a bistability between inphase and antiphase onecolour states is understood. Due to an equivariant phase space symmetry of being able to exchange the identical lasers, a symmetric and symmetrybroken variant of all states mentioned above exists and is shown to be stable. Using a five dimensional model we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetrybroken onecolour, symmetric and symmetrybroken twocolour, symmetric and symmetrybroken undamped relaxation oscillations, symmetric and symmetrybroken quasiperiodic, and symmetric and symmetrybroken chaotic states. As symmetrybroken states always exist in pairs, they naturally give rise to bistability. Several of these states show multistabilities between symmetric and symmetrybroken variants and among states.
Three memory elements on the basis of bistabilities in one and two colour states for two coupled singlemode lasers are proposed. The switching performance of selected designs of optical memory elements is studied numerically.
Abstract: We theoretically investigate the dynamics of two mutually coupled identical singlemode semiconductor lasers. For small separation and large coupling between the lasers, symmetrybroken onecolour states are shown to be stable. In this case the light output of the lasers have significantly different intensities while at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable symmetrybroken twocolour states, where both lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. Using a five dimensional model we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetrybroken onecolour and twocolour states. Several of these states give rise to multistabilities and therefore allow for the design of alloptical memory elements on the basis of two coupled singlemode lasers. The switching performance of selected designs of optical memory elements is studied numerically.
Abstract: Over the last three decades, laser dynamics has provided physical demonstrations for many fascinating theoretical
nonlinear phenomena. Lasers subject to feedback, electrical modulation or optical injection were extensively studied
by systems of low dimensional ordinary differential equations. These systems are similar to a predatorprey model,
whereby the photons act as a predator which feeds off carriers which are continually pumped. In addition, as there
are several orders of magnitude difference between lifetimes of a photon and a carrier, the physics within lasers are
a paradigmatic example of a slowfast system.
Here we consider two identical mutually coupled single mode laser diodes in the small delay limit. We introduce
a fivedimensional model which overcomes singularities encountered in conventional reduced models. These new
coordinates allowed us to study the dynamical features using a bifurcation diagram, obtained by varying the coupling
strength and the coupling phase between the lasers. Analysis reveals a large region of multistability which is organised
by three codimension two bifurcations.
The dynamics in the multistable region is a direct consequence of the slowfast nature of the system. Both symmetric
and symmetrybroken two colour states exists, where the magnitude of the each laser's electric field, the fast variable,
oscillates at a frequency which is too high for the carriers, the slow variable. In these states the intensity of the single
mode lasers are constant with two optical frequencies. Due to the Z2 symmetry of being able to exchange both
lasers, symmetrybroken states always exist in pairs. When stable regions of symmetrybroken and symmetric states
overlap, this creates regions of tristability.
Finally with new knowledge of the slowfast dynamical behaviour, bifurcation boundaries, and multistabilities, we
purpose several all optical memory element and all optical logic gate designs.
Abstract:
Multistability and synchronisation are phenomena, which play an important role in many nonlinear dynamical systems. Often their underlying origins are higher order codimension bifurcations and are therefore of broad interest.
Here we consider a simple system of two identical mutually coupled single mode laser diodes in the small delay limit. By varying the coupling strength and the coupling phase we observe a number of distinct regions of multistability.
In particular it is shown that a symmetrybroken state is stable for large coupling. In this case the intensity of each laser is significantly different, but the lasers are synchronised (locked) to the same optical frequency. As the system is invariant under the exchange of the lasers, a complimentary symmetrybroken state exists and is stable. Both symmetry broken states coexist with a stable symmetric state for which both lasers are mutually locked and emit light at equal intensity and frequency. This creates a region of tristability between locked states.
For weaker coupling the intensities of the symmetry broken states oscillate with a new frequency leading to an extensive region of bistability between limit cycles.
The observed phenomena are explained by a five dimensional model. This model uses a set of coordinates which avoids the singularities encountered in conventional phaseamplitude models. A bifurcation analysis reveals that the multistability is organised by three codimension two bifurcations.
Abstract: We present a new approach to a previously studied problem [1,2]. Two lasers are mutually coupled via a small percentage of each lasers light entering the other after a delay. Any minuscule change in the distance between the two lasers would results in a large change in the phase of incoming light. The input phase therefore becomes a free (sweep) parameter. This infinite dimensional problem is modelled with a modified version of the LangKobayashi equations [3].
The problem can be made finite dimensional by assuming an infinite speed of light. To reduce the dimensions further we take the relative phase difference between the lasers and so constant frequency waves (phasors) become the fixed points of the system. Standard fixed point analysis is conducted. Solutions with different inversions (carrier densities) in the two lasers are shown to be unstable for nondetuned lasers. Solutions with the same inversions have outputs which are either inphase or antiphase. These solutions are stable for a range of the free parameter.
After a forwardshift of the input phase, the constant phase (balanced) solutions are shown to lose stability via a pitchfork bifurcation. This joins the inphase and antiphase via mixed phase (unbalanced) solutions which confirms analytically a principle result of [2] in the zero delay limit. Backshifted, it loses stability via a Hopf bifurcation. A limit cycle is born which after a further small change in input phase undergoes a pitchfork bifurcation of limit cycles. Analytically expressions for these fixed and critical points are obtained. The results are then transferred back to the infinite dimensional problem by noting parametric similarities between the finite and infinite cases. To have a continuous transition between these two cases all eigenvalues must remain unchanged for at least a finite increase in delay. We obtain transferred results for the Hopf and pitchfork bifurcations. New bifurcations of two saddlenodes exist for relatively small delay. These effect the stability of the phasors as the delay is increased. Complete analytical expressions are obtained for this. This work may have interesting applications utilizing bistabilities in limit cycles.
REFERENCES
[1] Harmutt Erzgraeber, Bernd Krauskopf, Lenstra Daan Compound laser modes of mutually delaycoupled
lasers. SIAM J. Applied Dynamical Systems Vol 5 No 1 pp 3065
[2] A. Gravrieldes, V. Kovanis, T. Erneux Analytical stability boundaries for a semicondictor laser subject to
optical injection Optics Communcations 136 (1997) 253256
[3] R. Lang and K. Kobayashi External optical feedback effects on semiconductor injection laser
properties. IEEE J. Quantum Electron., (1980) pp. 347353
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