Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals

Frequency locking by external forcing in systems with rotational symmetry

We study locking of the modulation frequency of a relative periodic orbit in a general $S^1$-equivariant system of ordinary differential equations under an external forcing of modulated wave type. Our main result describes the shape of the locking region in the three-dimensional space of the forcing parameters: intensity, wave frequency, and modulation frequency. The difference of the wave frequencies of the relative periodic orbit and the forcing is assumed to be large and differences of modulation frequencies to be small. The intensity of the forcing is small in the generic case and can be large in the degenerate case, when the first order averaging vanishes. Applications are external electrical and/or optical forcing of selfpulsating states of lasers.Comment: 5 figure

Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered

Shear-induced bifurcations and chaos in models of three coupled lasers

Copyright © 2011 Society for Industrial and Applied MathematicsWe study nonlinear dynamics in a linear array of three coupled laser oscillators with rotational $\mathbb{S}^1$ and reflectional $\mathbb{Z}_2$ symmetry. The focus is on a coupled-laser model with dependence on three parameters: laser coupling strength, $\kappa$, laser frequency detuning, $\Delta$, and degree of coupling between the amplitude and phase of the laser, $\alpha$, also known as shear or nonisochronicity. Numerical bifurcation analysis is used in conjunction with Lyapunov exponent calculations to study the different aspects of the system dynamics. First, the shape and extent of regions with stable phase locking in the $(\kappa,\Delta)$ plane change drastically with $\alpha$. We explain these changes in terms of codimension-two and -three bifurcations of (relative) equilibria. Furthermore, we identify locking-unlocking transitions due to global homoclinic and heteroclinic bifurcations and the associated infinite cascades of local bifurcations. Second, vast regions of deterministic chaos emerge in the $(\kappa,\Delta)$ plane for nonzero $\alpha$. We give an intuitive explanation of this effect in terms of $\alpha$-induced stretch-and-fold action that creates horseshoes and discuss chaotic attractors with different topologies. Similar analysis of a more accurate composite-cavity mode model reveals good agreement with the coupled-laser model on the level of local and global bifurcations as well as chaotic dynamics, provided that coupling between lasers is not too strong. The results give new insight into modeling approaches and methodologies for studying nonlinear behavior of laser arrays

Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand side is small. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand side. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic.Comment: 42 pages. The main result is generalized to cover more general first order quasilinear hyperbolic systems; improved presentation, a correction in Example in Subsection 3.

A Journey Through the Dynamical World of Coupled Laser Oscillators

The focus of this thesis is the dynamical behaviour of linear arrays of laser oscillators with nearest-neighbour coupling. In particular, we study how laser dynamics are influenced by laser-coupling strength, $\kappa$, the natural frequencies of the uncoupled lasers, $\tilde{\Omega}_j$, and the coupling between the magnitude and phase of each lasers electric field, $\alpha$. Equivariant bifurcation analysis, combined with Lyapunov exponent calculations, is used to study different aspects of the laser dynamics. Firstly, codimension-one and -two bifurcations of relative equilibria determine the laser coupling conditions required to achieve stable phase locking. Furthermore, we find that global bifurcations and their associated infinite cascades of local bifurcations are responsible for interesting locking-unlocking transitions. Secondly, for large $\alpha$, vast regions of the parameter space are found to support chaotic dynamics. We explain this phenomenon through simulations of $\alpha$-induced stretching-and-folding of the phase space that is responsible for the creation of horseshoes. A comparison between the results of a simple {\it coupled-laser model} and a more accurate {\it composite-cavity mode model} reveals a good agreement, which further supports the use of the simpler model to study coupling-induced instabilities in laser arrays. Finally, synchronisation properties of the laser array are studied. Laser coupling conditions are derived that guarantee the existence of synchronised solutions where all the lasers emit light with the same frequency and intensity. Analytical stability conditions are obtained for two special cases of such laser synchronisation: (i) where all the lasers oscillate in-phase with each other and (ii) where each laser oscillates in anti-phase with its direct neighbours. Transitions from complete synchronisation (where all the lasers synchronise) to optical turbulence (where no lasers synchronise and each laser is chaotic in time) are studied and explained through symmetry breaking bifurcations. Lastly, the effect of increasing the number of lasers in the array is discussed in relation to persistent optical turbulence