Computing unstable manifolds of periodic orbits in delay differential equations

Krauskopf, Bernd; Green, Kirk

doi:10.1016/s0021-9991(03)00050-0

Cited by 32 publications

(64 citation statements)

References 24 publications

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“…However, this is changing with the introduction of advanced tools allowing detailed bifurcation studies of DDEs. These consist of (a) the publicly available Matlab continuation package DDE-BIFTOOL for numerical bifurcation analysis (see [8] and section 4 below) and (b) an algorithm, using DDE-BIFTOOL to obtain the necessary starting data, to compute unstable manifolds of saddle periodic orbits in a suitable Poincaré section [25]. (This algorithm was used in [19] to identify the break-up of a torus in the PCF laser and its subsequent disappearance in a crisis bifurcation.…”

mentioning

confidence: 99%

A Two-Parameter Study of the Locking Region of a Semiconductor Laser Subject to Phase-Conjugate Feedback

Green¹,

Krauskopf²,

Samaey³

2003

SIAM J. Appl. Dyn. Syst.

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Abstract. We present a detailed bifurcation analysis of a single-mode semiconductor laser subject to phaseconjugate feedback, a system described by a delay differential equation. Codimension-one bifurcation curves of equilibria and periodic orbits and curves of certain connecting orbits are presented near the laser's locking region in the two-dimensional parameter plane of feedback strength and pump current. We identify several codimension-two bifurcations, including a double-Hopf point, Belyakov points, and a T-point bifurcation, and we show how they organize the dynamics. This study is the first example of a two-parameter bifurcation study, including bifurcations of periodic and connecting orbits, of a delay system. It was made possible by new numerical continuation tools, implemented in the package DDE-BIFTOOL, and showcases their usefulness for the study of delay systems arising in applications.Key words. semiconductor lasers, phase-conjugate feedback, delay differential equations, two-parameter continuation, heteroclinic orbits, T-point bifurcation AMS subject classifications. 37N20, 34K18, 37G10, 37G20PII. S11111111024165751. Introduction. The majority of lasers in application today are semiconductor lasers. They can be found, for example, in CD-players, laser printers, and optical communication networks. Semiconductor lasers are so-called Class B lasers, in which the polarization of the electric field can be adiabatically eliminated. As a consequence, they can be described well by three-dimensional rate equations, one for the complex electric field E(t) and one for the population inversion N (t) (the number of excited states that can produce a single photon). It turns out that the phase φ(t) of the electric field follows the two equations for the optical intensity P (t) = |E(t)| 2 and the inversion N (t). Therefore, a solitary semiconductor laser is essentially a two-dimensional dynamical system that cannot exhibit chaotic dynamics. The only observable dynamical behavior is a damped periodic exchange between the electric field and the inversion. These oscillations are referred to as relaxation oscillations in the laser literature (not to be confused with relaxation oscillations in slow-fast systems); see, for example, [26,41] for an introduction to the theory of semiconductor lasers.

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confidence: 99%

A Two-Parameter Study of the Locking Region of a Semiconductor Laser Subject to Phase-Conjugate Feedback

Green¹,

Krauskopf²,

Samaey³

2003

SIAM J. Appl. Dyn. Syst.

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“…In section 5.2 we use the Arneodo system [16] with an artificial delay to show that our methods can also be used to compute manifolds accumulating on limit cycles. Our last example (in section 5.3) shows the usefulness of these new methods for analyzing the dynamics of phase-conjugate feedback (PCF) laser systems previously studied by Green, Krauskopf and collaborators [17,8,7,18]. We conclude by discussing the limitations of our approach and list several topics for future research in section 6.…”

Section: Introductionmentioning

confidence: 92%

“…Our implementation uses ǫ = 0.01, δ min = ∆/2, and δ max = 2∆. We note that the above algorithm exploits a combination of ideas in [1,28] with those in the work of Green and Krauskopf on approximating one-dimensional unstable manifolds of periodic orbits of DDEs [7]. : Algorithm for computing geodesic curves.…”

Section: Computation Of Manifolds Of Ddesmentioning

confidence: 99%

“…This system exhibits "stable periodic operation interspersed with "bubbles" of chaotic dynamics"; it has been previously studied numerically by Krauskopf, Green and co-authors in [17,8,7,18]. The system is posed in (E x , E y , N) and possesses Z 2 symmetry corresponding to rotation by π in the complex E plane.…”

Section: Pcf Laser Feedbackmentioning

confidence: 99%

“…Several numerical methods for approximating higher-dimensional invariant manifolds of ODEs have been developed over the years [1,2,3,4,5]; a recent overview and comparison of these can be found in [6]. For delay differential equations (DDEs), an algorithm for computing one-dimensional invariant manifolds (in the Poincaré map) of periodic orbits has been introduced by Krauskopf and Green in [7,8]. Very little, however, has been done so far to approximate higher dimensional invariant manifolds of DDEs, though some relevant theoretical convergence results can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Methods for Approximating Invariant Manifolds of Delayed Systems

Sahai¹,

Vladimirsky²

2009

SIAM J. Appl. Dyn. Syst.

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In this paper we develop methods for computing k−dimensional invariant manifolds of delayed systems for k ≥ 2. For small delays, we consider methods for approximating delay differential equations (DDEs) with ordinary differential equations (ODEs). Once these approximations are made, any existing method for computing invariant manifolds of ODEs can then be used directly. We derive bounds on errors incurred by the most natural of these approximations. For large delays, we extend to DDEs the method originally introduced by Krauskopf and Osinga [1] for invariant manifolds of ODEs. We test the convergence of the resulting algorithm numerically and further illustrate our approach by computing two-dimensional unstable manifolds of equilibria in the context of phase-conjugate feedback lasers.

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Bifurcation Analysis of Lasers with Delay

Krauskopf

2005

Unlocking Dynamical Diversity

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Computing unstable manifolds of periodic orbits in delay differential equations

Cited by 32 publications

References 24 publications

A Two-Parameter Study of the Locking Region of a Semiconductor Laser Subject to Phase-Conjugate Feedback

A Two-Parameter Study of the Locking Region of a Semiconductor Laser Subject to Phase-Conjugate Feedback

Numerical Methods for Approximating Invariant Manifolds of Delayed Systems

Bifurcation Analysis of Lasers with Delay

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