Nonlinear dynamics of a solid-state laser with injection

We analyze the dynamics of a solid-state laser driven by an injected sinusoidal field. For this type of laser, the cavity round-trip time is much shorter than its fluorescence time, yielding a dimensionless ratio of time scales $\sigma \ll 1$. Analytical criteria are derived for the existence, stability, and bifurcations of phase-locked states. We find three distinct unlocking mechanisms. First, if the dimensionless detuning $\Delta$ and injection strength $k$ are small in the sense that $k = O(\Delta) \ll \sigma^{1/2}$, unlocking occurs by a saddle-node infinite-period bifurcation. This is the classic unlocking mechanism governed by the Adler equation: after unlocking occurs, the phases of the drive and the laser drift apart monotonically. The second mechanism occurs if the detuning and the drive strength are large: $k =O(\Delta) \gg \sigma^{1/2}$. In this regime, unlocking is caused instead by a supercritical Hopf bifurcation, leading first to phase trapping and only then to phase drift as the drive is decreased. The third and most interesting mechanism occurs in the distinguished intermediate regime $k, \Delta = O(\sigma^{1/2})$. Here the system exhibits complicated, but nonchaotic, behavior. Furthermore, as the drive decreases below the unlocking threshold, numerical simulations predict a novel self-similar sequence of bifurcations whose details are not yet understood.Comment: 29 pages in revtex + 8 figs in eps. To appear in Phys. Rev. E (scheduled tentatively for the issue of 1 Oct 98

Optimal Subharmonic Entrainment

For many natural and engineered systems, a central function or design goal is the synchronization of one or more rhythmic or oscillating processes to an external forcing signal, which may be periodic on a different time-scale from the actuated process. Such subharmonic synchrony, which is dynamically established when N control cycles occur for every M cycles of a forced oscillator, is referred to as N:M entrainment. In many applications, entrainment must be established in an optimal manner, for example by minimizing control energy or the transient time to phase locking. We present a theory for deriving inputs that establish subharmonic N:M entrainment of general nonlinear oscillators, or of collections of rhythmic dynamical units, while optimizing such objectives. Ordinary differential equation models of oscillating systems are reduced to phase variable representations, each of which consists of a natural frequency and phase response curve. Formal averaging and the calculus of variations are then applied to such reduced models in order to derive optimal subharmonic entrainment waveforms. The optimal entrainment of a canonical model for a spiking neuron is used to illustrate this approach, which is readily extended to arbitrary oscillating systems

Instabilities and subharmonic resonances of subsonic heated round jets, volume 2

When a jet is perturbed by a periodic excitation of suitable frequency, a large-scale coherent structure develops and grows in amplitude as it propagates downstream. The structure eventually rolls up into vortices at some downstream location. The wavy flow associated with the roll-up of a coherent structure is approximated by a parallel mean flow and a small, spatially periodic, axisymmetric wave whose phase velocity and mode shape are given by classical (primary) stability theory. The periodic wave acts as a parametric excitation in the differential equations governing the secondary instability of a subharmonic disturbance. The (resonant) conditions for which the periodic flow can strongly destabilize a subharmonic disturbance are derived. When the resonant conditions are met, the periodic wave plays a catalytic role to enhance the growth rate of the subharmonic. The stability characteristics of the subharmonic disturbance, as a function of jet Mach number, jet heating, mode number and the amplitude of the periodic wave, are studied via a secondary instability analysis using two independent but complementary methods: (1) method of multiple scales, and (2) normal mode analysis. It is found that the growth rates of the subharmonic waves with azimuthal numbers beta = 0 and beta = 1 are enhanced strongly, but comparably, when the amplitude of the periodic wave is increased. Furthermore, compressibility at subsonic Mach numbers has a moderate stabilizing influence on the subharmonic instability modes. Heating suppresses moderately the subharmonic growth rate of an axisymmetric mode, and it reduces more significantly the corresponding growth rate for the first spinning mode. Calculations also indicate that while the presence of a finite-amplitude periodic wave enhances the growth rates of subharmonic instability modes, it minimally distorts the mode shapes of the subharmonic waves

A study of poststenotic shear layer instabilities

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The Bulletin: Sidney Kimmel Medical College at Thomas Jefferson University, Volume 65, Issue 4, Fall 2016

This issue includes: Creating the Ideal Physician-One Note at a Time: The Dean\u27s concert series is more than a respite in the middle of a busy day at SKMC Tiny Bubbles, Big Future: New uses for microbubble-filled ultrasound contrast agents could reduce the need for highly invasive medical tests. Dean\u27s Column Findings: New pathway to treat heart failure #Giving Tuesday: A Message from Elizabeth Dale On Campus Clara Callahan, MD: A keen eye for aspiring physicians Kate Sugarman, MD \u2788: A career guided by social justice Nick Benvenuto: Finding Science and Art in Both Winemaking and Medicine Class Notes In Memoriam Time Capsule By the Number

Optimal Control and Synchronization of Dynamic Ensemble Systems

Ensemble control involves the manipulation of an uncountably infinite collection of structurally identical or similar dynamical systems, which are indexed by a parameter set, by applying a common control without using feedback. This subject is motivated by compelling problems in quantum control, sensorless robotic manipulation, and neural engineering, which involve ensembles of linear, bilinear, or nonlinear oscillating systems, for which analytical control laws are infeasible or absent. The focus of this dissertation is on novel analytical paradigms and constructive control design methods for practical ensemble control problems. The first result is a computational method %based on the singular value decomposition (SVD) for the synthesis of minimum-norm ensemble controls for time-varying linear systems. This method is extended to iterative techniques to accommodate bounds on the control amplitude, and to synthesize ensemble controls for bilinear systems. Example ensemble systems include harmonic oscillators, quantum transport, and quantum spin transfers on the Bloch system. To move towards the control of complex ensembles of nonlinear oscillators, which occur in neuroscience, circadian biology, electrochemistry, and many other fields, ideas from synchronization engineering are incorporated. The focus is placed on the phenomenon of entrainment, which refers to the dynamic synchronization of an oscillating system to a periodic input. Phase coordinate transformation, formal averaging, and the calculus of variations are used to derive minimum energy and minimum mean time controls that entrain ensembles of non-interacting oscillators to a harmonic or subharmonic target frequency. In addition, a novel technique for taking advantage of nonlinearity and heterogeneity to establish desired dynamical structures in collections of inhomogeneous rhythmic systems is derived

Some observations of the vibrations of slender rotating shafts

March 1982Includes bibliographical references (page 29)The linear theory of a slender, initially bowed, rotating shaft is reviewed for both free and forced vibrations, and found to compare well with a simple experiment on such a shaft. The shaft behavior passing through the critical speed is described in detail, and the maximum bowed-out static deflection of the shaft was found dependent on the external damping and the initial bowing. The amplitude of the oscillatory deflections of the shaft due to gravity loads increased somewhat near the critical speed, but these increases were small compared to the large static deflection of the shaft. During rapid passage through the critical speed, low frequency whirling modes were excited transiently. At higher rotation speeds, the second critical speed was observed, and also the first mode was excited subharmonically and appeared as a backward whirl mode relative to the rotating shaft

Study of the resonant behaviour of bubbles embbeded in gelatin

Many diseases present abnormally high pressures at different points of the circulatory system, such as inside the heart, the portal vein or the pulmonary artery. For this reason, physicians need to know the pressure at these specific points either to help diagnosis or to monitor the evolution of a patient’s condition. Nowadays, these pressure measurements are performed invasively, for instance by navigating a catheter with a pressure sensor at its tip to the point of interest. As any invasive procedure, acquiring these measurements presents a number of shortcomings that medical doctors would like to avoid. Consequently, providing physicians with a non-invasive pressure measurement technique would represent a great advance in the diagnosis and/or treatment of these patients. Ultrasound Contrast Agents (UCAs), microbubbles injected into the blood stream to aid ultrasonic imaging, offer the possibility of obtaining the blood pressure at localized points of the circulatory system in non-invasive ways. Indeed, it can be shown mathematically how these microbubbles oscillate at a characteristic frequency (resonance frequency) when insonated with a pressure pulse with the appropriate features. More interestingly, it can also be shown that this resonant frequency depends on the ambient pressure at the bubble’s location. With these ideas in mind, in this thesis we have conducted an experimental and numerical investigation aimed at measuring how the resonance frequency of bubbles immersed in gelatin depends on the ambient pressure. Since the focus of this work is on the Physics, rather than on implementing a practical technique, we have worked with millimetric bubbles (commercial UCAs are micrometric) in order to overcome a number of experimental problems associated with using very small bubbles. Furthermore, fixing the bubbles in gelatin also allows for a prolonged observation and thus facilitates the experiments. But at the same time, gelatin constitutes a very realistic model of the rheological properties of the soft tissue that would surround bubbles in some real medical applications (think, for instance, of a bubble circulating through a narrow capillary surrounded by soft tissue). In our experiments, bubbles are insonated with chirps: pressure pulses that sweep a range of frequencies that contains the resonance one. Then, the radius vs. time evolution of the bubbles is obtained by applying digital image processing techniques to high-speed movies acquired synchronously with the acoustic insonation. Finally, the time evolutions of the bubble radii are processed using wavelets to extract the main frequency at which they oscillate. Although the experimental procedure designed and implemented in this thesis detects that bubbles oscillate at a well-defined frequency close to the expected resonance ones, our numerical simulations and the analysis of the pressure signals reveal that this frequency is actually arising from an improper behavior of the piezoelectric transducer used to generate the pulses. Finally, we point out future lines in which this work could be improved, most notably replacing the transducer by another one that performs better in the range of frequencies of interest. This solution is being implemented at the time of writing this dissertation.Ingeniería Mecánic

Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.Comment: 32 pages, 8 figures, 5 table