Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

Some approximate solutions of dynamic problems in the linear theory of thin elastic shells

Some aspects of wave propagation in thin elastic shells are considered. The governing equations are derived by a method which makes their relationship to the exact equations of linear elasticity quite clear. Finite wave propagation speeds are ensured by the inclusion of the appropriate physical effects.

The problem of a constant pressure front moving with constant velocity along a semi-infinite circular cylindrical shell is studied. The behavior of the solution immediately under the leading wave is found, as well as the short time solution behind the characteristic wavefronts. The main long time disturbance is found to travel with the velocity of very long longitudinal waves in a bar and an expression for this part of the solution is given.

When a constant moment is applied to the lip of an open spherical shell, there is an interesting effect due to the focusing of the waves. This phenomenon is studied and an expression is derived for the wavefront behavior for the first passage of the leading wave and its first reflection.

For the two problems mentioned, the method used involves reducing the governing partial differential equations to ordinary differential equations by means of a Laplace transform in time. The information sought is then extracted by doing the appropriate asymptotic expansion with the Laplace variable as parameter.

Analysis of temporal structure in spike trains of visual cortical area MT

The temporal structure of neuronal spike trains in the visual cortex can provide detailed information about the stimulus and about the neuronal implementation of visual processing. Spike trains recorded from the macaque motion area MT in previous studies (Newsome et al., 1989a; Britten et al., 1992; Zohary et al., 1994) are analyzed here in the context of the dynamic random dot stimulus which was used to evoke them. If the stimulus is incoherent, the spike trains can be highly modulated and precisely locked in time to the stimulus. In contrast, the coherent motion stimulus creates little or no temporal modulation and allows us to study patterns in the spike train that may be intrinsic to the cortical circuitry in area MT. Long gaps in the spike train evoked by the preferred direction motion stimulus are found, and they appear to be symmetrical to bursts in the response to the anti-preferred direction of motion. A novel cross-correlation technique is used to establish that the gaps are correlated between pairs of neurons. Temporal modulation is also found in psychophysical experiments using a modified stimulus. A model is made that can account for the temporal modulation in terms of the computational theory of biological image motion processing. A frequency domain analysis of the stimulus reveals that it contains a repeated power spectrum that may account for psychophysical and electrophysiological observations.

Some neurons tend to fire bursts of action potentials while others avoid burst firing. Using numerical and analytical models of spike trains as Poisson processes with the addition of refractory periods and bursting, we are able to account for peaks in the power spectrum near 40 Hz without assuming the existence of an underlying oscillatory signal. A preliminary examination of the local field potential reveals that stimulus-locked oscillation appears briefly at the beginning of the trial.

Finding needles in the haystack : a search for AM CVn systems using the Palomar Transient Factory

The AM CVn systems are a rare class of ultra-compact astrophysical binaries. With orbital periods of under an hour and as short as five minutes, they are among the closest known binary star systems and their evolution has direct relevance to the type Ia supernova rate and the white dwarf binary population. However, their faint and rare nature has made population studies of these systems difficult and several studies have found conflicting results.

I undertook a survey for AM CVn systems using the Palomar Transient Factory (PTF) astrophysical synoptic survey by exploiting the "outbursts" these systems undergo. Such events result in an increase in luminosity by a factor of up to two-hundred and are detectable in time-domain photometric data of AM CVn systems. My search resulted in the discovery of eight new systems, over 20% of the current known population. More importantly, this search was done in a systematic fashion, which allows for a population study properly accounting for biases.

Apart from the discovery of new systems, I used the time-domain data from the PTF and other synoptic surveys to better understand the long-term behavior of these systems. This analysis of the photometric behavior of the majority of known AM CVn systems has shown changes in their behavior at longer time scales than have previously been observed. This has allowed me to find relationships between the outburst properties of an individual system and its orbital period.

Even more importantly, the systematically selected sample together with these properties have allowed me to conduct a population study of the AM CVn systems. I have shown that the latest published estimates of the AM CVn system population, a factor of fifty below theoretical estimates, are consistent with the sample of systems presented here. This is particularly noteworthy since my population study is most sensitive to a different orbital period regime than earlier surveys. This confirmation of the population density will allow the AM CVn systems population to be used in the study of other areas of astrophysics.

Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

The instability of fluids with time dependent heating

The stability of a fluid having a non-uniform temperature stratification is examined analytically for the response of infinitesimal disturbances. The growth rates of disturbances have been established for a semi-infinite fluid for Rayleigh numbers of 10³, 10⁴, and 10⁵ and for Prandtl numbers of 7.0 and 0.7.

The critical Rayleigh number for a semi-infinite fluid, based on the effective fluid depth, is found to be 32, while it is shown that for a finite fluid layer the critical Rayleigh number depends on the rate of heating. The minimum critical Rayleigh number, based on the depth of a fluid layer, is found to be 1340.

The stability of a finite fluid layer is examined for two special forms of heating. The first is constant flux heating, while in the second, the temperature of the lower surface is increased uniformly in time. In both cases, it is shown that for moderate rates of heating the critical Rayleigh number is reduced, over the value for very slow heating, while for very rapid heating the critical Rayleigh number is greatly increased. These results agree with published experimental observations.

The question of steady, non-cellular convection is given qualitative consideration. It is concluded that, although the motion may originate from infinitesimal disturbances during non-uniform heating, the final flow field is intrinsically non-linear.

Studies on photoblastic germination in lettuce seed

It was shown, with the aid of osmotic inhibition of germination, that the action of the far-red-absorbing form of phytochrome (Pf) in promoting germination can be completed even if the seed is held under conditions where germination is not possible. An effect of the continuing action of Pf beyond the point of complete germination promotion was demonstrated by enhancement of germination rate after removal of the osmotically active solute.

Previous reports that the rate of growth in water of seeds freed from the expansion-restricting endosperm is independent of the state of phytochrome were confirmed. However, a marked, phytochrome-mediated enhancement of the growth potential of such seeds was demonstrated through restricting water uptake by incubation in an osmoticum.

An experimental system, utilizing the appearance of a geotropic curvature in the radicle of the excised axial portion of the seed, was developed for more detailed studies of the phytochrome-enhanced growth potential. It was possible to demonstrate the light effect in water as well as in osmotica; this apparently is not possible with de-endospermed entire seeds. As in intact seeds, the effect of the continuing action of Pf is to enhance the rate of the response. Secretion of a chemical inhibitor of growth by the endosperm as a possible mechanism of induction of light sensitivity has been ruled out.

The phytochrome-dependent rate of appearance of geotropic curvature in osmotica is paralleled in time by a similar dependence of the rate of early extension growth of the embryonic axis. Only the first small increment of growth is a differentially responsive to red (R) and far-red (F); the rate of later increase in length is independent of the light regime.

It was shown that the high concentrations of gibberellic acid required for germination promotion in the intact seed are due at least in part to a diffusion barrier in the endosperm, and that the occasional reports in the literature of the ineffectiveness of kinetin are probably due to the same phenomenon. It was shown that gibberellin, like red light, enhances the growth potential of the axis, but kinetin does not. The difference in rates of response obtained after R-irradiation or gibberellin treatment, together with other results reported in the literature, strongly suggests that gibberellic acid and red light promote germination by different means. The idea that kinetin promotes germination by yet another mechanism, probably operating in the cotyledons, was supported through two different experimental approaches.

The phenomenon of temperature-dependent dark germination was examined in detail, using a wide range of both temperatures and incubation times. With the aid of the half-seed system, it was demonstrated that the promotive effect of low temperature on germination could not be due to a low optimum temperature for early growth of the radicle, since the rate of that process increased with increasing temperature, up to the highest temperature used.

It was shown that phytochrome does not function at high temperatures. This fact is of considerable importance in interpreting the phenomenon of thermodormancy, since in the literature only a small part of the effect of high temperature has been ascribed to an effect on phytochrome, and at that, only to an acceleration of dark reversion of Pf to the red-absorbing form of phytochrome (Pr). Partial denaturation of phytochrome may also make some contribution.

It was shown that the germination-promoting effect of low temperature depends on the presence of Pf, and concluded that low temperatures act by delaying or preventing transformation of Pf. Support for the assumption that Pf, not Pr, is the active form of phytochrome in lettuce seeds was drawn from the same evidence.

Attempts to stimulate germination by repeated irradiation with F over relatively prolonged incubation times resulted in failure, as have similar attempts reported in the literature. However, an enhancement of growth potential in the half-seed system by the maintenance of a small amount of Pf over long periods at ordinary temperatures by repeated irradiation with F was demonstrated.

It was observed that cold storage of the dry seed prevents or delays loss of dark dormancy during post-harvest storage. No change in the response of the half-seed in osmoticum to R and F was observed in seeds that has lost dark dormancy; that is, no internal change took place to measurably increase the growth potential of the embryonic axis. This suggests that the endosperm is the seat of changes responsible for after-ripening of photoblastic lettuce seed.

Conditioned responses in the reticular formation

Unit activity was recorded from the midbrain and pons of 40 freely moving rats in an appetitive classical conditioning situation. Responses to auditory stimuli were observed from 100 units before and during a conditioning procedure in which presentation of food occurred 1 sec after the onset of the auditory stimulus. Conditioned unit responses (i.e., spike rate accelerations or decelerations) were considered to be positive when 1) no similar responses appeared prior to conditioning, and 2) latencies were equal to or less than those of sensory responses derived from the inferior colliculus. Such short latency conditioned unit responses were recorded from 11 probes located in the mid-lateral pert of the ventral region of the brain stem. This region was differentiated from paramedian, far lateral and dorsal parts of the brain stem reticular formation. Conditioned unit responses of considerably longer latencies were recorded from 76 probe located in these other regions. Among the longer latency responses interesting differences appeared in experiments conducted after the first conditioning series was completed. With additional training, units in the "reticular activating system" of midbrain and pons tended to yield stabilized responses in the early portion of the CS-US interval closely related in time to the orientation responses evoked by the CS. In contrast, the responses of units in the limbic midbrain tended to stabilize in the later part of the CS-US interval closely related in time to preparatory responses tied to the US. During extinction when the auditory stimulus was no longer followed by presentation of food, many of the responses were reduced to their pre-conditioning levels. However, there was a tendency for units which had displayed short latency responses on the first conditioning day to be more resistant to extinction than units which had displayed longer latency conditioned responses. The data were interpreted as indicating a local correlate of learning in the reticular formation of midbrain end pons and a separation of the midbrain system into at least two areas: 1) the classical "reticular activating system" related to orienting reactions, and 2) the limbic midbrain areas related to drives and rewards. Because the ventral and mid-lateral area with very short latency conditioned responses was not clearly tied to either of these; it was considered as possibly representing a third division.