Search

We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state-dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three contain only constant delays. We implemented this expansion and the computation of the normal form coeffcients in Matlab using symbolic differentiation and the resulting code HHnfDDEsd is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena. [ABSTRACT FROM AUTHOR]

Unstable periodic orbits occur naturally in many nonlinear dynamical systems. They can generally not be observed directly, but a number of control schemes have been suggested to stabilize them. One such scheme is that by Pyragas [Phys. Lett. A, 170 (1992), pp. 421-428], which uses time-delayed feedback to target a specific unstable periodic orbit of a given period and stabilize it. This paper considers the global effect of applying Pyragas control to a nonlinear dynamical system. Specifically, we consider the standard example of the subcritical Hopf normal form subject to Pyragas control, which is a delay differential equation (DDE) that models how a generic unstable periodic orbit is stabilized. Our aim is to study how this DDE model depends on its different parameters, including the phase of the feedback and the imaginary part of the cubic coefficient, over their entire ranges. We show that the delayed feedback control induces infinitely many curves of Hopf bifurcations, from which emanate infinitely many periodic orbits that, in turn, have further bifurcations. Moreover, we show that, in addition to the stabilized target periodic orbit, there are possibly infinitely many stable periodic orbits. We compactify the parameter plane to show how these Hopf bifurcation curves change when the 2π-periodic phase of the feedback is varied. In particular, the domain of stability of the target periodic orbit changes in this process, and, at certain parameter values, it disappears completely. Overall, we present a comprehensive global picture of the dynamics induced by Pyragas control. [ABSTRACT FROM AUTHOR]

A method of investigating quasi-static mechanisms is presented and applied to an overcenter mechanism and to a nose landing gear mechanism. The method uses static equilibrium equations along with equations describing the geometric constraints in the mechanism. In the spirit of bifurcation analysis, solutions to these steady-state equations are then continued numerically in parameters of interest. Results obtained from the bifurcation method agree with the equivalent results obtained from two overcenter mechanism dynamic models (one state-space and one multibody dynamic model), while a considerable computation time reduction is demonstrated with the overcenter mechanism. The analysis performed with the nose landing gear model demonstrates the flexibility of the continuation approach, allowing conventional model states to be used as continuation parameters without a need to reformulate the equations within the model. This flexibility, coupled with the computation time reductions, suggests that the bifurcation approach has potential for analyzing complex landing gear mechanisms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the geometry of slow manifolds of a dynamical system with one fast and two slow variables. Specifically, we study the dynamics near a folded-node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dynamics. For example, canard solutions are associated with mixed-mode oscillations, where they organize regions with different numbers of small oscillations. We perform a numerical study of the geometry of two-dimensional slow manifolds in the normal form of a folded node in ℝ3. Namely, we view the part of a slow manifold that is of interest as a one-parameter family of orbit segments up to a suitable cross-section. Hence, it is the solution of a two-point boundary value problem, which we solve by numerical continuation with the package AUTO. The computed family of orbit segments is used to obtain a mesh representation of the manifold as a surface. With this approach we show how the attracting and repelling slow manifolds change in dependence on the eigenvalue ratio μ associated with the folded-node singularity. At μ = 1 two primary canards bifurcate and secondary canards are created at odd integer values of μ. We compute 24 secondary canards to investigate how they spiral more and more around one of the primary canards. The first sixteen secondary canards are continued in μ to obtain a numerical bifurcation diagram. [ABSTRACT FROM AUTHOR]

Abstract: We consider a multi-transverse-mode vertical-cavity surface-emitting laser (VCSEL) subject to optical feedback. The system is modeled by a partial differential equation for the spatial carrier population, which is coupled to delay differential equations for the electric fields of the participating transverse modes that are subject to external optical feedback. We consider here the case that the VCSEL supports the two basic, rotationally symmetric, linearly polarized optical modes LP01 and LP02. In our model each LP mode receives feedback not only from itself but also from the other LP mode; the amount of cross-feedback can be controlled by a parameter. Specifically, we use numerical continuation techniques to present a detailed analysis of the steady state, external cavity mode (ECM) structure in dependence on the feedback strength, the feedback phase and the amount of cross-feedback. This shows that the case of zero cross-feedback is degenerate and changes quite dramatically even in the presence of small feedback from the other transverse mode. On the other hand, in an intermediate range of cross-feedback the ECM structure does not change qualitatively in a physically relevant range of feedback strength. We consider the entire transition from zero cross-feedback to zero self-feedback, in which we identify the key changes in the ECM structure. [Copyright &y& Elsevier]

This paper presents a method that is able to continue periodic orbits in systems where only output of the evolution over a given time period is available, which is the typical situation in an experiment. The starting point of our paper is an analysis of time-delayed feedback control, a method to stabilize periodic orbits experimentally that is popular among physicists. We show that the well-known topological limitations of this method can be overcome by an embedding into a pseudo-arclength continuation and prove that embedded time-delayed feedback control is able to stabilize periodic orbits that have at most one unstable Floquet multiplier sufficiently close to the unit circle. In the second part we introduce preconditioning into the time-delayed feedback control. In this way, we extract a nonlinear system of equations from time profiles, which we solve using Newton iterations. We demonstrate the feasibility of our method by continuing periodic orbits in a laser model through folds, and by computing the family of canard orbits of the classical stiff Van der Pol system with constant forcing. [ABSTRACT FROM AUTHOR]

The Lorenz manifold is an intriguing two-dimensional surface that illustrates chaotic dynamics in the well-known Lorenz system. While it is not possible to find an explicit analytic expression for the Lorenz manifold, we have developed a method for calculating a numerical approximation that builds the surface up as successive geodesic level sets. The resulting mesh approximation can be read as crochet instructions, which means that we are able to generate a three-dimensional model of the Lorenz manifold. The crocheted model directly motivated us to investigate the curvature properties of the Lorenz manifold by means of determining an approximation to the Gaussian curvature from our geodesic mesh representation. We then translate this information to colour, which leads to a new visualization of the geometry of the Lorenz manifold. This colouring enhances the aesthetics of the visualization by revealing a striking pattern of regions of positive and negative Gaussian curvature. [ABSTRACT FROM AUTHOR]

Abstract: A method is presented for the automatic construction of an experimental bifurcation diagram of an optically injected solid state laser. From measured time series of laser output intensity, different identifiers of aspects of the dynamics are derived. Combinations of these identifiers are then used to characterize different possible bifurcations. The resulting experimental bifurcation diagram in the plane of injection strength versus detuning includes saddle-node, Hopf, period-doubling and torus bifurcations. It is shown to agree well with a theoretical bifurcation analysis of a corresponding rate equation model. [Copyright &y& Elsevier]

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell. [ABSTRACT FROM AUTHOR]

We present an algorithm for computing one-dimensional stable and unstable manifolds of saddle periodic orbits in a Poincaré section. The computation is set up as a boundary value problem by restricting both end points of orbit segments to the section. Starting from the periodic orbit itself, we use collocation routines from {\sc Auto} to continue the solutions of the boundary value problem such that one end point of the orbit segment varies along a part of the manifold that was already computed. In this way, the other end point of the orbit segment traces out a new piece of the manifold. As opposed to standard methods that use shooting to compute the Poincaré map as the kth return map, our approach defines the Poincaré map as the solution of a boundary value problem. This enables us to compute global manifolds through points where the flow is tangent to the section---a situation that is typically encountered unless one is dealing with a periodically forced system. Another major advantage of our approach is that it deals effectively with the problem of extreme sensitivity of the Poincaré map to its argument, which is a typical feature in the important class of slow-fast systems. We illustrate and test our algorithm by computing stable and unstable manifolds for three examples: the forced Van der Pol oscillator, a model of a semiconductor laser with optical injection, and a slow-fast chemical oscillator. All examples are accompanied by animations demonstrating how the manifolds grow during the computation. [ABSTRACT FROM AUTHOR]

Real-time dynamic substructuring is an experimental technique for testing the dynamic behaviour of complex structures. It involves creating a hybrid model of the entire structure by combining an experimental test piece-the substructure-with a numerical model describing the remainder of the system. The technique is useful when it is impractical to experimentally test the entire structure or complete numerical modelling is insufficient. In this paper, we focus on the influence of delay in the system, which is generally due to the inherent dynamics of the transfer systems (actuators) used for structural testing. This naturally gives rise to a delay differential equation (DDE) model of the substructured system. With the case of a substructured system consisting of a single mass-spring oscillator we demonstrate how a DDE model can be used to understand the influence of the response delay of the actuator. Specifically, we describe a number of methods for identifying the critical time delay above which the system becomes unstable. Because of the low damping in many large structures a typical situation is that a substructuring test would operate in an unstable region if additional techniques were not implemented in practice. We demonstrate with an adaptive delay compensation technique that the substructured mass-spring oscillator system can be stabilized successfully in an experiment. The approach of DDE modelling also allows us to determine the dependence of the critical delay on the parameters of the delay compensation scheme. Using this approach we develop an over-compensation scheme that will help ensure stable experimental testing from initiation to steady state operation. This technique is particularly suited to stiff structures or those with very low natural damping as regularly encountered in structural engineering. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A pendulum can be stabilized in its upright position by proportional-plus-derivative (PD) feedback control only if the latency in the control loop is smaller than a certain critical delay. This critical delay is determined by the presence of a fully symmetric triple-zero eigenvalue singularity, a bifurcation of codimension three. We investigate three possible modifications of the PD scheme with the aim of extending the range of permissible delays. Effectively, these modifications introduce another parameter. This additional parameter can be used to continue the triple-zero singularity in four parameters until it gains a higher-order degeneracy imposing a new limit on the permissible delay. It turns out that the most effective modification is to feed back the value of the position with a small (intentional) additional delay on top of the control loop latency. [ABSTRACT FROM AUTHOR]

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus, where the number of preimages changes, is made up of so-called critical curves, that are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of preimages changes by two. We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of preimages, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently. We take a bifurcation theory point of view and consider the two basic codimension-one interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps. [ABSTRACT FROM AUTHOR]

The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system. [ABSTRACT FROM AUTHOR]

We present an algorithm to compute the one-dimensional stable manifold of a saddle point for a planar map. In contrast to current standard techniques, here it is not necessary to know the inverse or approximate it, for example, by using Newton's method. Rather than using the inverse, the manifold is grown starting from the linear eigenspace near the saddle point by adding a point that maps back onto an earlier segment of the stable manifold. The performance of the algorithm is compared to other methods using an example in which the inverse map is known explicitly. The strength of our method is illustrated with examples of noninvertible maps, where the stable set may consist of many different pieces, and with a piecewise-smooth model of an interrupted cutting process. The algorithm has been implemented for use in the DsTool environment and is available for download with this paper. [ABSTRACT FROM AUTHOR]

Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape. [ABSTRACT FROM AUTHOR]

Pyragas time-delayed feedback is a control scheme designed to stabilize unstable periodic orbits, which occur naturally in many nonlinear dynamical systems. It has been successfully implemented in a number of applications, including lasers and chemical systems. The control scheme targets a specific unstable periodic orbit by adding a feedback term with a delay chosen as the period of the unstable periodic orbit. However, in an experimental or industrial environment, obtaining the exact period or setting the delay equal to the exact period of the target periodic orbit may be difficult. This could be due to a number of factors, such as incomplete information on the system or the delay being set by inaccurate equipment. In this paper, we evaluate the effect of Pyragas control on the prototypical generic subcritical Hopf normal form when the delay is close to but not equal to the period of the target periodic orbit. Specifically, we consider two cases: first, a constant, and second, a linear approximation of the period. We compare these two cases to the case where the delay is set exactly to the target period, which serves as the benchmark case. For this comparison, we construct bifurcation diagrams and determine any regions where a stable periodic orbit close to the target is stabilized by the control scheme. In this way, we find that at least a linear approximation of the period is required for successful stabilization by Pyragas control. [ABSTRACT FROM AUTHOR]

Abstract: 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. As such it constitutes a prototype example of delay-coupled laser systems, which play an important role, e.g., in telecommunication. 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. 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. [Copyright &y& Elsevier]

A study is made into the occurrence of shimmy oscillations in a dual-wheel main landing gear. Nonlinear equations of motion are developed for the system, and various effects are considered, including gyroscopic coupling, nonlinear tire properties, geometric nonlinearities, and fluid shock damping. Of particular interest in this study is the presence of freeplay: this is introduced as a lateral play at the apex of the torque link joints. Using bifurcation analysis methods, the dynamics of this system are explored as the forward velocity and loading force acting on the gear are varied. For the zero freeplay case, the system is found to be stable over its physical operating range with shimmy oscillations appearing only for extreme loading forces and speed. However, with the introduction of freeplay, shimmy may be observed over more typical operating conditions, and the resulting oscillations are found to scale linearly with freeplay magnitude. The parameter plane of forward velocity and loading force is then further subdivided into areas of different types of dynamics. With the inclusion of freeplay one observes the appearance of low-frequency and high- frequency shimmy oscillations, bistable behavior, and stationary solutions of nonzero yaw. Considering the desirable case in which no shimmy occurs, the set of allowable freeplay profiles that satisfy a conservative stability criteria is defined. [ABSTRACT FROM AUTHOR]

We report on the first experimental observation of spontaneous mirror symmetry breaking (SSB) in coherently driven-dissipative coupled optical cavities. SSB is observed as the breaking of the spatial or mirror Z2 symmetry between two symmetrically pumped and evanescently coupled photonic crystal nanocavities, and manifests itself as random intensity localization in one of the two cavities. We show that, in a system featuring repulsive boson interactions (U>0), the observation of a pure pitchfork bifurcation requires negative photon hopping energies (J<0), which we have realized in our photonic crystal molecule. SSB is observed over a wide range of the two-dimensional parameter space of driving intensity and detuning, where we also find a region that exhibits bistable symmetric behavior. Our results pave the way for the experimental study of limit cycles and deterministic chaos arising from SSB, as well as the study of nonclassical photon correlations close to SSB transitions. [ABSTRACT FROM AUTHOR]