Search results for "Names"
showing 10 items of 6843 documents
Hopf bifurcation at infinity for planar vector fields
2007
We study, from a new point of view, families of planar vector fields without singularities $ \{ X_{\mu}$  :  $-\varepsilon < \mu < \varepsilon\} $ defined on the complement of an open ball centered at the origin such that, at $\mu=0$, infinity changes from repellor to attractor, or vice versa. We also study a sort of local stability of some $C^1$ planar vector fields around infinity.
FLUCTUATION-INDUCED LOCAL OSCILLATIONS AND FRACTAL PATTERNS IN THE LATTICE LIMIT CYCLE MODEL
2003
The fractal properties of the Lattice Limit Cycle model are explored when the process is realized on a 2-dimensional square lattice support via Monte Carlo Simulations. It is shown that the structure of the steady state presents inhomogeneous fluctuations in the form of domains of identical particles. The various domains compete with one another via their borders which have self-similar, fractal structure. The fractality is more prominent, (fractal dimensions df < 2), when the parameter values are near the critical point where the Hopf bifurcation occurs. As the distance from the Hopf bifurcation increases in the parameter space the system becomes more homogeneous and the fractal dimens…
Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System
2012
In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control.
Class-B two-photon Fabry–Pérot laser
1998
Abstract We study the stationary operation and stability properties of a class-B two-photon Fabry–Perot laser. We show that, differently from the one-photon laser, the intensity emitted by the two-photon laser is larger in a Fabry–Perot than in a ring cavity. The lasing solution loses stability through a subcritical Hopf bifurcation, as it occurs in the unidirectional ring laser. The stability domain in the parameter space is larger in the Fabry–Perot than in the ring cavity configuration.
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…