Search results for "bifurcation"
showing 10 items of 204 documents
One- and two-photon lasers with injected signal in a high-Q fabry-Pérot cavity
2000
Explicit models are derived for good cavity one- and two-photon lasers with an injected signal in a Fabry-Perot cavity. The steady solutions and their stability properties are obtained analytically and compared with the corresponding ring cavity model ones. Only quantitative differences between both types of cavities are found. In particular we show that (i) the Fabry-Perot cavity reduces significantly the domain of self-pulsing with respect to the ring cavity, and for the two-photon laser case (ii) larger output can be extracted from a Fabry-Perot cavity than from a ring cavity under certain conditions, something impossible in free-running lasers. We conclude that ring cavity models are se…
Cavity solitons in nondegenerate optical parametric oscillation
2000
Abstract We find analytically cavity solitons in nondegenerate optical parametric oscillators. These solitons are exact localised solutions of a pair of coupled parametrically driven Ginzburg–Landau equations describing the system for large pump detuning. We predict the existence of a Hopf bifurcation of the soliton resulting in a periodically pulsing localised structure. We give numerical evidence of the analytical results and address the problem of cavity soliton interaction.
Two-photon laser dynamics.
1995
Degenerate as well as nondegenerate three-level two-photon laser (TPL) models are derived. In the limit of equal cavity losses for both fields, it is shown that the nondegenerate model reduces to the degenerate one. We also demonstrate the isomorphism existing between our degenerate TPL model and that of a dressed-state TPL. All these models contain ac-Stark and population-induced shifts at difference from effective Hamiltonian models. The influence of the parameters that control these shifts on the nonlinear dynamics of a TPL is investigated. In particular, the stability of the periodic orbits that arise at the Hopf bifurcation of the system and the extension of the self-pulsing domains of…
Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms
1996
AbstractWe study a generic, real analytic unfolding of a planar diffeomorphism having a fixed point with unipotent linear part. In the analogue for vector fields an open parameter domain is known to exist, with a unique limit cycle. This domain is bounded by curves corresponding to a Hopf bifurcation and to a homoclinic connection. In the present case of analytic diffeomorphisms, a similar domain is shown to exist, with a normally hyperbolic invariant circle. It follows that all the ‘interesting’ dynamics, concerning the destruction of the invariant circle and the transition to trivial dynamics by the creation and death of homoclinic points, takes place in an exponentially small part of the…
Progress in Modelling Coherently Pumped Far-Infrared Laser Dynamics
1990
Coherently pumped lasers (CPL) operating in the far-infrared spectral region shown a wealth of instabilities1, including a behavior remarkably similar2,3 with the predictions of the paradigmatic Lorenz-Haken model of a single-mode homogeneously broadened laser4,5. The qualitative agreement; between theory and experiments2,3 was rather surprising, for the model4,5 refers to a two-level system whereas the CPL operate on a three-level scheme, where the pumping and lasing transitions share a common upper level. Dupertuis et al.6 have identified conditions for the mathematical reduction of the CPL equations to the Lorenz-Haken equations4, but these conditions were not all fulfilled in the experi…
Turing Instability and Pattern Formation for the Lengyel–Epstein System with Nonlinear Diffusion
2014
In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel---Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we c…
How to Get a Model in Pedestrian Dynamics to Produce Stop and Go Waves
2016
Stop and go waves in granular flow can often be described mathematically by a dynamical system with a Hopf bifurcation. We show that a certain class of microscopic, ordinary differential equation-based models in crowd dynamics fulfil certain conditions of Hopf bifurcations. The class is based on the Gradient Navigation Model. An interesting phenomenon arises: the number of pedestrians in the system must be greater than nine for a bifurcation—and hence for stop and go waves to be possible at all, independent of the density. Below this number, no parameter setting will cause the system to exhibit stable stop and go behaviour. The result is also interesting for car traffic, where similar model…
Entangled states and coherent interaction in resonant media
2014
The entanglement features of some solid state materials, as well as of particular systems of interacting atoms and fields are analyzed. A detailed investigation of the rich phase structure of low dimensional spin models, describing the natural mineral azurite and copper based coordination compounds, has revealed regimes with the most robust entanglement behavior. Using the dynamical system approach, the phase structure of some classical models on hierarchical (recursive) lattices has been also studied and, for the first time, the transition between chaotic and periodic regimes by means of tangent bifurcation has been detected.A detailed description of entanglement properties of three atoms …
Nonlinear Analysis of Phase-Locked Loop (PLL): Global Stability Analysis, Hidden Oscillations and Simulation Problems
2013
In the middle of last century the problem of analyzing hidden oscillations arose in automatic control. In 1956 M. Kapranov considered a two-dimensional dynamical model of phase locked-loop (PLL) and investigated its qualitative behavior. In these investigations Kapranov assumed that oscillations in PLL systems can be self-excited oscillations only. However, in 1961, N. Gubar’ revealed a gap in Kapranov’s work and showed analytically the possibility of the existence of another type of oscillations, called later by the authors hidden oscillations, in a phase-locked loop model: from a computational point of view the system considered was globally stable (all the trajectories tend to equilibria…
Cycles in continuous and discrete dynamical systems : computations, computer-assisted proofs, and computer experiments
2009
The present work is devoted to calculation of periodic solutions and bifurcation research in quadratic systems, Lienard system, and non-unimodal one-dimensional discrete maps using modern computational capabilities and symbolic computing packages.In the first chapter the problem of Academician A.N. Kolmogorov on localization and modeling of cycles of quadratic systems is considered. For the investigation of small limit cycles (so-called local 16th Hilbert’s problem) the method of calculation of Lyapunov quantities (or Poincaré-Lyapunov constants) is used. To calculate symbolic expressions for the Lyapunov quantities the Lyapunov method to the case of non-analytical systems was generalized. …