[PDF]    http://dx.doi.org/10.3952/lithjphys.45603

Open access article / Atviros prieigos straipsnis

Lith. J. Phys. 45, 457–461 (2005)


AN ANALYTICAL TREATMENT OF THE DELAYED FEEDBACK CONTROL ALGORITHM AT A SUBCRITICAL HOPF BIFURCATION
Vikt. Pyragas and K. Pyragas
Semiconductor Physics Institute, A. Goštauto 11, LT-01108 Vilnius, Lithuania
E-mail: viktpy@pfi.lt, pyragas@kes0.pfi.lt

Received 21 October 2005

We develop an analytical approach for the delayed feedback control algorithm applied to a dynamical system close to a subcritical Hopf bifurcation. A simple nonlinear electronic circuit is considered as a prototypical model of the subcritical Hopf bifurcation. The periodic orbit arising at this bifurcation is torsion free and cannot be controlled by the conventional delayed feedback algorithm. We show the necessity of employing an unstable degree of freedom in a feedback loop as well as a nonlinear coupling between the controlled system and controller. Close to the bifurcation point the system is weakly nonlinear and the problem is treated analytically using the method of averaging.
Keywords: chaos, dynamical systems, delayed feedback, subcritical Hopf bifurcation, analytical theory
PACS: 05.45.Gg, 02.30.Yy, 02.30.Ks


ANALIZINIS VALDYMO ALGORITMO VĖLUOJANČIU GRĮŽTAMUOJU RYŠIU TYRIMAS ARTI SUBKRITINĖS HOPFO BIFURKACIJOS
Vikt. Pyragas, K. Pyragas
Puslaidininkių fizikos institutas, Vilnius, Lietuva

Nagrinėjame vėluojančio grįžtamojo ryšio metodo taikymą nestabiliai periodinei orbitai, atsirandančiai dinaminėje sistemoje arti subkritinės Hopfo bifurkacijos. Bifurkacijos taško aplinkoje sistema yra silpnai netiesinė ir uždavinys sprendžiamas analiziškai, naudojant suvidurkinimo metodą. Aptariame būtinybę įjungti papildomąjį nestabilųjį laisvės laipsnį į grįžtamojo ryšio grandinę bei panaudoti netiesinį ryšį tarp valdomos sistemos ir valdiklio. Analiziniam tyrimo būdui pademonstruoti buvo išnagrinėta paprasta netiesinė elektroninė grandinė, kuri modeliuoja subkritinę Hopfo bifurkaciją. Išplėtota analizinė teorija tinka bet kuriai dinaminei sistemai, kai ji yra arti subkritinės Hopfo bifurkacijos.


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