Conditions for Synchronization of One Oscillation System

Самойленко, А. М.

doi:10.1007/s11253-005-0250-3

Cited by 7 publications

(7 citation statements)

References 13 publications

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confidence: 73%

“…The present paper extends previous work on this topic for particular types of the vector field f and the forces g [23,20,19]. In particular, the considered special cases in [23,20,19] are doubly degenerate in the sense that not only the averaged forcing (1.9) vanishes but also the second averaging term turns to zero. Remark that in [18] related results are described for the case that both differences of modulation and wave frequencies are small, and [17] concerns the case when the internal state as well as the external forcing are not modulated.…”

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confidence: 73%

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Frequency Locking by External Forcing in Systems with Rotational Symmetry

Самойленко¹,

Тkachenko²,

Yanchuk³

2012

SIAM J. Appl. Dyn. Syst.

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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.

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supporting

confidence: 73%

supporting

confidence: 73%

Frequency Locking by External Forcing in Systems with Rotational Symmetry

Самойленко¹,

Тkachenko²,

Yanchuk³

2012

SIAM J. Appl. Dyn. Syst.

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“…The reduction of the differential equations considered above to equations with a constant matrix of coefficients in their separated linear part is essential for the subsequent investigation of these equations. A confirmation of this statement can be found, e.g., in [4,8], where, however, the problem of this reduction was only partially solved.…”

Section: 2mentioning

confidence: 62%

Some problems of the linear theory of systems of ordinary differential equations

Самойленко¹

2011

Ukr Math J

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We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet-Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude-phase coordinates in a neighborhood of a periodic trajectory of a dynamical system.

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“…Our aim is to prove the existence of stable solutions, e.g., of a globally asymptotically stable equilibrium position or of a periodic solution of system (5)- (7). System (5)-(7) has the following specific features:…”

Section: Reduced Systemmentioning

confidence: 99%

“…The problems of synchronization of coupled lasers are of especial interest [2][3][4][5][6][7][8][9][10][11][12]. These problems are encountered, in particular, in connection with the necessity to increase the power of radiation by using laser lattices [13] and in the construction of optical systems aimed at the transmission of protected data.…”

Section: Introductionmentioning

confidence: 99%

Amplitude synchronization in a system of two coupled semiconductor lasers

Yanchuk

Schneider

Lykova³

2008

Ukr Math J

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We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.

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Conditions for Synchronization of One Oscillation System

Cited by 7 publications

References 13 publications

Frequency Locking by External Forcing in Systems with Rotational Symmetry

Frequency Locking by External Forcing in Systems with Rotational Symmetry

Some problems of the linear theory of systems of ordinary differential equations

Amplitude synchronization in a system of two coupled semiconductor lasers

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