Relaxation oscillations in a class of delay-differential equations.

We study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics. Under certain circumstances the model exhibits self-sustained oscillations, with periods which can be significantly longer than the basic cell cycle time. We show that the long periods in the oscillations occur when the cell generation rate is small, and we provide an asymptotic analysis of the model in this case. This analysis bears a close resemblance to the analysis of relaxation oscillators (such as the Van der Pol oscillator), except that in our case the slow manifold is infinite dimensional. Despite this, a fairly complete analysis of the problem is possible

Canard explosion in delayed equations with multiple timescales

We analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of `classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold.Comment: arXiv admin note: substantial text overlap with arXiv:1404.584

Bifurcation analysis of a semiconductor laser with filtered optical feedback

We study the dynamics and bifurcations of a semiconductor laser with delayed filtered optical feedback, where a part of the output of the laser reenters after spectral filtering. This type of coherent optical feedback is more challenging than the case of conventional optical feedback from a simple mirror, but it provides additional control over the output of the semiconductor laser by means of choosing the filter detuning and the filter width. This laser system can be modeled by a system of delay differential equations with a single fixed delay, which is due to the travel time of the light outside the laser. In this paper we present a bifurcation analysis of the filtered feedback laser. We first consider the basic continuous wave states, known as the external filtered modes (EFMs), and determine their stability regions in the parameter plane of feedback strength versus feedback phase. The EFMs are born in saddle-node bifurcations and become unstable in Hopf bifurcations. We show that for small filter detuning there is a single region of stable EFMs, which splits up into two separate regions when the filter is detuned. We then concentrate on the periodic orbits that emanate from Hopf bifurcations. Depending on the feedback strength and the feedback phase, two types of oscillations can be found. First, there are undamped relaxation oscillations, which are typical for semiconductor laser systems. Second, there are oscillations with a period related to the delay time, which have the remarkable property that the laser frequency oscillates while the laser intensity is almost constant. These frequency oscillations are only possible due to the interaction of the laser with the filter. We determine the stability regions in the parameter plane of feedback strength versus feedback phase of the different types of oscillations. In particular, we find that stable frequency oscillations are dominant for nonzero values of the filter detuning. © 2007 Society for Industrial and Applied Mathematics

Nonlinear Spin Dynamics in Ferromagnets with Electron-Nuclear Coupling

Nonlinear spin motion in ferromagnets is considered with nonlinearity due to three factors: (i) the sample is prepared in a strongly nonequilibrium state, so that evolution equations cannot be linearized as would be admissible for spin motion not too far from equilibrium, (ii) the system considered consists of interacting electron and nuclear spins coupled with each other via hyperfine forces, and (iii) the sample is inserted into a coil of a resonant electric circuit producing a resonator feedback field. Due to these nonlinearities, coherent motion of spins can develop, resulting in their ultrafast relaxation. A complete analysis of mechanisms triggering such a coherent motion is presented. This type of ultrafast coherent relaxation can be used for studying intrinsic properties of magnetic materials.Comment: 1 file, LaTex, 23 page