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This paper presents a study of how different vibration modes contribute to the dynamics of an inclined cable that is parametrically excited close to a 2:1 internal resonance. The behaviour of inclined cables is important for the design and analysis of cable-stay bridges. In this work the cable vibrations are modelled by a four-mode model. This type of model has been used previously to study the onset of cable sway motion caused by internal resonances which occur due to the nonlinear modal coupling terms. A bifurcation study is carried out with numerical continuation techniques applied to the scaled and averaged modal equations. As part of this analysis, the amplitudes of the cable vibration response to support inputs is computed. These theoretical results are compared with experimental measurements taken from a 5.4 m long inclined cable with a vertical support input at the lower end. In general this comparison shows a very high level of agreement.

We study the structure of the multistable continuous wave emission region of a semiconductor laser subject to coherent optical feedback from a Fabry-Perot filter. Key parameters organizing the degree of multistability are uncovered, and they include the feedback phase and the frequency detuning.

We investigate the dynamical properties of two mutually delay-coupled semiconductor lasers that are coupled via their optical fields. Because a semiconductor laser is an oscillator that features strong coupling between its amplitude and phase, this system serves as a prototype model of coupled amplitude-phase oscillators. Our main interest here is in the dynamics near and within the locking region where the two lasers emit light of the same frequency. We present experimental observations that give evidence for four qualitatively different dynamical regimes: stable continuous wave emission, oscillations at the laser's characteristic relaxation oscillation frequency, oscillation related to the frequency difference between the two lasers, and chaotic dynamics. We characterise and identify these dynamical regimes and analyse them by means of a bifurcation analysis of the corresponding rate equation model with delay. Specifically, we present the underlying bifurcation structure, where the detuning and the pump current are the main bifurcation parameter. The combination of experiment and bifurcation analysis shows how changes of the dynamics arise from the presence of local and global bifurcations near the locking region.

During ground manoeuvres a loss of lateral stability due to the saturation of the main landing gear tyres can cause the aircraft to enter a skid or a spin. The lateral stability is governed not only by aspects of the gear design, such as its geometry and tyre characteristics, but also by operational parameters, for example, the weather and taxiway condition. In this paper we develop an improved understanding and new presentation of the dynamics of an aircraft manoeuvring on the ground, ultimately aimed at optimisation and automa- tion of ground operations. To investigate turning manoeuvres we apply techniques from dynamical systems theory to a modified version of a nonlinear computer model of an A320 passenger aircraft developed by the Landing Gear Group at Airbus UK. Specifically, we present a bifurcation analysis of the underlying solution structure that governs the dynamics of turning manoeuvres with dependence on the steering angle and thrust level. Furthermore, a detailed study of the behaviour when lateral stability is lost focuses on how the tyre saturation at different wheel sets lead to qualitatively different types of overall behaviour. The presented bifurcation diagrams identify parameter regions for which undesirable behaviour is avoidable and, thus, they form a foundation for defining the safe operating limits during turning manoeuvres.

Hybrid experiments, which couple mechanical experiments and computer simulations bidirectionally and in real-time, are a promising experimental technique in engineering. A fundamental problem of this technique are delays in the coupling between simulation and experiment. We discuss this issue for a simple prototype hybrid experiment: a mechanical pendulum that is parametrically excited by coupling it to a simulated linear mass-spring-damper system. Under realistic conditions a small delay in the coupling can give rise to an essential instability. Namely, the linearization has infinitely many unstable eigenvalues for arbitrarily small delay. This type of instability is impossible to compensate for with any of the standard compensation techniques known in engineering. We introduce an approach based on feedback control and Newton iterations and show that it is able to overcome the essential instability. The basic idea consists of two parts. First, we change the bidirectional coupling between experiment and computer simulation to a unidirectional coupling and stabilize the experiment with a feedback loop. Second, we place the modified hybrid experiment into a Newton iteration scheme. If the iteration converges then the hybrid experiment behaves just as the original emulated system (within the experimental accuracy). Using path-following, oscillations and their bifurcations can be tracked systematically without knowledge of an underlying model for the experiment.

We demonstrate a method for tracking oscillations and their stability boundaries (bifurcations) in nonlinear systems. Our method does not require an underlying model of the dynamical system but instead relies on feedback stabilizability. This gives the approach the potential to transfer the full power of numerical bifurcation analysis techniques from the purely computational domain to real-life experiments.

The animation shows the control-based continuation of the family of unstable periodic orbits for the idealized setup of the dry friction oscillator. It is based on data obtained in a computer simulation

We introduce a method for tracking nonlinear oscillations and their bifurcations in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system nor the ability to set its initial conditions. Instead it relies on feedback stabilizability, which makes the approach applicable in an experiment. This is demonstrated with a proof-of-concept computer experiment of the classical autonomous dry friction oscillator, where we use a fixed time step simulation and include noise to mimic experimental limitations. For this system we track in one parameter a family of unstable nonlinear oscillations that forms the boundary between the basins of attraction of a stable equilibrium and a stable stick-slip oscillation. Furthermore, we track in two parameters the curves of Hopf bifurcation and grazing-sliding bifurcation that form the boundary of the bistability region. An accompanying animation further visualizes the action of the controller during the tracking process

We study the spectral and dynamical behavior of two identical, mutually delay-coupled semiconductor lasers. We concentrate on the short coupling-time regime where the number of basic states of the system, the compound laser modes (CLMs), is small so that their individual behavior can be studied both experimentally and theoretically. Specifically, for small spectral detuning we find several stable CLMs of the coupled system where both lasers lock onto a common frequency and emit continuous wave output. A bifurcation analysis of the CLMs in the full rate equation model with delay reveals the structure of stable and unstable CLMs that are organized by saddle-node and pitchfork bifurcations. We find a characteristic bifurcation scenario as a function of the detuning and the coupling phase between the two lasers that explains experimentally observed multistabilities and mode jumps in the locking region.

This is a chapter for a book edited by Deb Kane and Alan Shore. It explains and reviews the bifurcation analysis of the full DDE model of a given laser system. The main examples are a semiconducor receiving opical feedback from (i) a conventional mirror, and (ii) from a phase-conjugate mirror.

This work concentrates on the lateral oscillations in vehicles, also called shimmy, with a particular emphasis on aircraft. A mathematical model of a nose landing gear is discussed with geometric detail that has been mostly neglected in the past research. Stability criteria for the shimmy-free operation of the landing gear are derived using linear stability analysis. Nonlinear analysis is used not only to study the qualitative behaviour of the Hopf bifurcation but also to analyze the system beyond the Hopf bifurcation. The manuscript concludes with suggestions for future research.

We demonstrate a method for tracking oscillations and their stability boundaries (bifurcations) in nonlinear systems. Our method does not require an underlying model of the dynamical system but instead relies on feedback stabilizability. Our method allows one to determine bifurcations of the dynamical system without the need to observe the transient oscillations for a long time to determine their decay or growth. Moreover, in the context of hybrid experiments, which couple experiments and computer simulations bidirectionally and in real-time, our method is able to overcome the presence of coupling delays (or, more generally, unknown actuator dynamics), which is a fundamental problem that is currently limiting the use of hybrid testing. We illustrate the basic ideas with a computer simulation (including coupling delays and noise) of a prototype nonlinear hybrid experiment: a real pendulum coupled at its pivot to a computer simulation of a vertically excited mass-spring-damper system.

Inclined cables are important structural elements of cable-stayed bridges. When the bridge deck oscillates, large amplitude cable vibrations can arise in various modes as a result of the low cable damping, parametric excitation or non-linear modal coupling. The resulting vibrations are undesirable and potentially damaging to the long-term performance of the bridge. The phenomena can be modelled considering internal resonances between in-plane and out-of-plane modes of vibration of the cable. % Here they have been studied using a four-mode model that represents the response of an inclined cable vertically excited at the lower end (i.e. from the deck) with a frequency that is close to the natural frequency of the second cable mode in each plane and twice the frequency of the first mode in each plane. The modal equations of the model are investigated using the software package AUTO for the numerical continuation of solutions of a system of ODEs. This allows the identification of the important solution branches in the cable model responsible for unwanted vibration behaviour. Here they have been studied using a four-mode model that represents the response of an inclined cable vertically excited at the lower end (i.e. from the deck) with a frequency that is close to the natural frequency of the second cable mode in each plane and twice the frequency of the first mode in each plane. The modal equations of the model are investigated using the software package AUTO for the numerical continuation of solutions of a system of ODEs. This allows the identification of the important solution branches in the cable model responsible for unwanted vibration behaviour. The result of our analysis is that we identify amplitudes of excitation above which modes other than the directly excited mode (the second in-plane mode) start contributing to the response of the cable. In addition, we show that the response amplitudes in these additional modes is of similar magnitude to the amplitudes in the directly excited mode, which could be considered an issue in the design of cable-stayed bridges. In summary, by using a numerical continuation technique we predict when the response of the cable will change from a single in-plane mode to coupled responses in two or more modes, in-plane or in both planes, and the modal amplitudes involved in these coupled responses.

We develop and study a model of an aircraft nose landing gear with torsional, lateral and longitudinal degrees of freedom. The corresponding three modes are coupled in a nonlinear fashion via the geometry of the landing gear in the presence of a nonzero rake angle, as well as via the nonlinear tyre forces. Their interplay may lead to different types of shimmy oscillations as a function of the forward velocity and the vertical force on the landing gear. Methods from nonlinear dynamics, especially numerical continuation of equilibria and periodic solutions, are used to asses how the three modes contribute to different types of shimmy dynamics. We conclude that the longitudinal mode does not actively participate in the nose landing gear dynamics over the entire range of forward velocity and vertical force.