<title>Characterization and stabilization of unstable cycles in a modulated class-B laser</title>

Abstract

AB&RACTWe have analyzed theordically diflrent chaos controlschemes fir a modulated class-B Ias including discoiitinuous and continuous delayed facks. The analysis is based on the detailed analytical and inirnaical studies of unstable manifolds evolution in phase space. A prescription foroptimal control is proposeci INTRODUC11ONThe problan ofcontrolling chaos has received naidi recent int&ea The basicconce* involved is that a chaotic attractor coains an infinite airnb of differecit unstthle orbits. So vari.is typ of regular behaviour from Intrinsically chaotic systans can be obtain ifan appropriate dynamical control technique is imp4nentecLSince the original method ofchaos control by Ott Grebogi, and Yorke (OGY) was presented [1] a zuimb of aoccessfiul applicatxxis oftbis method its variants for taming chaotic dynamics in ecperimantt including 1aans has been repoiteci. In particular, Roy al 1mw resolved the problem ofnnilthnode instability (so-called "grn" problem) in the Nd:YAG las& with intracavity frequency doubling on the KTP aystal[2], cBociix and coworkers have stabilized a chaotic output of selfpulsing fibu [3] and modulated lasers [4 and rofereies tharein]. At the same time, the OGY matbod itselfand its modifications have aome drawbacks which are essential for widespread applications in caI devices. The main ofthese are usually a prioii unknown parainaters of control, so the adjustments to 'correct values may become quite tedious the extremely high nipling frequencies in vy fast systans, and sometimes absence ofrobustness to noise.In this papa we analyze tloreticafly possibilities ofhnproving these techniques uãig as a paradigmatic ample the rate equations model for a modulated clasiB Iaan.At first we have consi&ed delayed impulsive feeack (DIF) technique proposed in Ref' [3]. The consideration vers the cases when cme or two ofthe following control parameters we acosenble : the losses ofcavity, the pump parameter and the enaj ofinjected pulse from a maste lasec. The key point hue is existence ofan oiznaI phase ofapplying impulsive cxrrection s. In partiailar, in a modulated las with control on cavity losses this phase conesponds to the vertiosi orientation of an unstable manifold of a saddle nT-periodic cycle to be stabilized and coineides with an optimal phase of amplification near a threshold of an instability winch, in its turn, can be readily identified theoretically and experimentally [5]. Moreover, there is also an optimal strength of the feedback kicks. It also can be expressed analytically through the parameters of characterizaticm of the unstable orbits. We present the theoretical results illustrating this optimal control technique applied to the model of a modulated CO2 laser with losses controlled by an additional spiking laser [5].We have also performed a linear stability analysis for a delayed continuous fbedback technique (DCF) introduced by Piragas , which has a number advantages in applications [6]. For example it is better suited for stablli7ing high-speed chaotic dynamics and robust to no...