[PDF]    http://dx.doi.org/10.3952/lithjphys.48109

Open access article / Atviros prieigos straipsnis

Lith. J. Phys. 48, 5–16 (2008)


ANALYTICAL TREATMENT OF THE DELAYED FEEDBACK CONTROLLED LORENZ SYSTEM CLOSE TO A SUBCRITICAL HOPF BIFURCATION
V. Pyragas and K. Pyragas
Semiconductor Physics Institute, A. Goštauto 11, LT-01108 Vilnius, Lithuania
E-mail: viktpy@pfi.lt, pyragas@kes0.pfi.lt

Received 23 October 2007; revised 27 November 2007; accepted 22 February 2008

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modification based on an unstable delayed feedback controller. The analytical approach employs the centre manifold theory, the near identity transformation, and averaging. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.
Keywords: chaos, dynamical systems, delayed feedback control, Lorenz system, subcritical Hopf bifurcation, centre manifold theory, near identity transformation, averaging
PACS: 05.45.Gg, 02.30.Yy, 02.30.Ks


UŽDELSTO GRĮŽTAMOJO RYŠIO VALDOMOS LORENCO SISTEMOS ANALIZINIS TYRIMAS SUBKRIZINĖS HOPFO BIFURKACIJOS APLINKOJE
V. Pyragas, K. Pyragas
Puslaidininkių fizikos institutas, Vilnius, Lietuva

Išplėtojame analizinę teoriją uždelsto grįžtamojo ryšio valdikliui (UGRV), valdančiam Lorenco sistemą, esančią arti subkrizinės Hopfo bifurkacijos taško. Šios bifurkacijos metu atsirandančių periodinių orbitų topologija riboja UGRV, ir jų neįmanoma stabilizuoti įprastiniu UGRV metodu. Topologiniam ribojimui apeiti naudojame nestabilų valdiklį. Analiziniai tyrimai grindžiami centrinės daugdaros teorija, beveik tapačia transformacija bei vidurkinimo metodu. Analiziškai gauname charakteringas lygtis, kurių šaknys yra valdomos orbitos Flokė rodikliai. Taip pat gauname paprastas išraiškas, nusakančias valdiklio grįžtamojo ryšio stiprio stabilumo slenkstinę bei optimalią vertę. Analizinius rezultatus patvirtina išeitinės valdomos Lorenco sistemos skaitinis integravimas.


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