The general problem of stability of microwave and RF devices and circuits in the presence of multi-frequency or broadband signals is addressed. The problem is formulated in terms of the system of nonlinear non-autonomous differential equations for generalized coordinates in the phase space that can be attributed to, e.g., amplitudes of electric currents in circuits with lumped elements or mode amplitudes of the electromagnetic field in the distributed systems. An approach to the stability analysis is proposed that allows predicting the appearance of various kinds of instabilities including the deterministically chaotic motion. The new criterion based on the analysis of Lyapunov exponents is discussed that establishes the relation between maximal stable amplitude of oscillations and the levels of nonlinearity and damping in the system. The examples of one- and two-mode oscillators have been considered in detail.Razmotren je općeniti problem stabilnosti mikrovalnih i radiofrekvencijskih aktivnih elemenata i sklopova u prisutnosti višefrekvencijskih ili širokopojasnih signala. Problem je formuliran kao sustav nelinearnih neautonomnih diferencijalnih jednadžbi u poopćenim koordinatama faznog prostora koje se mogu pridružiti npr. amplitudama električnih struja u sklopovima od elemenata s koncentriranim parametrima ili amplitudama modova elektromagnetskog polja u sustavima s raspodijeljenim parametrima. Predložena je analiza stabilnosti koja omogućava predviđanje različitih vrsta nestabilnosti uključujući determinističko kaotično gibanje. Razmotren je novi kriterij koji se osniva na analizi Ljapunovljevih eksponenata. Taj kriterij uspostavlja vezu između najveće stabilne amplitude oscilacija te razina nelinearnosti i prigušenja u sustavu. Podrobno su analizirani primjeri jednomodnog i dvomodnog oscilatora

Vladimir B. Ryabov

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ISSN 0005–1144ATKAAF 45(1–2), 19–22 (2004)Vladimir B. RyabovStability of Microwave and Electronic Devices: an Approachfrom Dynamical Systems TheoryUDK 621.37.01IFAC IA 5.8.0;4.1.4Original scientific paperThe general problem of stability of microwave and RF devices and circuits in the presence of multi-fre-quency or broadband signals is addressed. The problem is formulated in terms of the system of nonlinearnon-autonomous differential equations for generalized coordinates in the phase space that can be attributedto, e.g., amplitudes of electric currents in circuits with lumped elements or mode amplitudes of the electro-magnetic field in the distributed systems. An approach to the stability analysis is proposed that allows pre-dicting the appearance of various kinds of instabilities including the deterministically chaotic motion. The newcriterion based on the analysis of Lyapunov exponents is discussed that establishes the relation between maxi-mal stable amplitude of oscillations and the levels of nonlinearity and damping in the system. The exam-ples of one- and two-mode oscillators have been considered in detail.Key words: chaos, Lyapunov exponents, microwave circuits, oscillators, instability1 INTRODUCTIONThe analysis of the structure of electromagneticfield in electronic devices or complex transforma-tion of broadband signals in RF circuits can bereduced in many cases to the study of interacti-on between several coupled oscillators. Such clas-sical problems of circuit theory as generation ofoscillations, frequency conversion, parametric am-plification, etc. can be efficiently formulated andsolved in terms of temporal variation of amplitu-des and phases of generalized modes. From themathematical viewpoint, such a description requi-res solving a set of deterministic differential equa-tions for the time evolution of the generalized co-ordinates controlled by the set of external and in-ternal parameters (physical properties of the sys-tem). It should be noted, that the equations de-scribing the dynamics of the system are typicallynonlinear, therefore, the solutions cannot be ex-pressed in elementary or special functions and so-me approximate or numerical methods have to beused.The issue of stability plays a key role in thetheory of nonlinear oscillations, due to its impor-tance in applications in the development and uti-lization of concrete devices and systems. Until ve-ry recently, the stability analysis in nonlinear dy-namical systems was based on the assumption thatonly periodic and quasiperiodic motion can exist inthe stationary regime. Under such an assumption,the final goal of predicting the stability of thegiven system at the given values of controls canbe reached by the Floquet-type analysis [1, 2] oflinearized equations in the vicinity of a stationaryperiodic or quasiperiodic oscillation. Nowadays, withthe discovery of the phenomenon of deterministicchaos, it became clear that the traditional met-hods of stability analysis have to be revised withtaking into account the possibility for the chaoticmotion to appear. On the other hand, the burst ofactivity in the field of nonlinear dynamics spurredby this discovery in the middle of 60-s, has led tothe invention of new powerful methods of analy-sis that can be efficiently applied for predictingthe raise of instabilities in dynamical systems ofvirtually any physical origin, including various elec-tronic devices and circuits [3].In this paper we apply a recently proposed met-hod of stability analysis [4] based on the notionof Lyapunov exponents for predicting the generalkind of instability in oscillatory systems. From aslightly different perspective, the method establishesthe threshold of stability in terms of the amplitu-de of oscillations that guarantees the stable opera-tion of the devices or circuits described by simi-lar sets of differential equations. Lyapunov cha-racteristic exponents (LCE) provide a quantitativemeasure of stretching and contracting deforma-tions of an infinitesimally small phase space sphe-re in the vicinity of an arbitrary trajectory in adynamical system. So defined, they also characte-rize the divergence (convergence) rates of two ini-tially close trajectories residing on an attractor andserve as indicators of the stability of motion. To-tal number of Lyapunov exponents that a systempossesses is equal to the dimension of the phasespace, or, in other words, the number of indepen-dent variables necessary to fully characterize theAUTOMATIKA 45(2004) 1–2, 19–22 19holds all the time, the system is asymptoticallystable, i.e. all the perturbations are exponentiallyshrinking with time and, hence, unstable motionsare precluded. From the inequality (6), togetherwith Equations (2), (3), it appears possible to ob-tain the relation between the control parametersand phase space coordinates which guaranteesthat the system is »safe« in the sense that, if thetrajectory never leaves the region with negativevalues of µ1, no instability appears. The goal isreached by analyzing the structure of the functi-on P1(ϕ1(t), ϕ2(t), ..., ϕn−1(t)), together with soluti-ons of Equation (2), which define the dynamicsof angles ϕm through Equations (5). It should behowever noted that a straightforward calculationof the function P1 does not always lead to theexplicit equation for the border of the asymptoticstability area in the phase space. This happens dueto the presence of both the expanding and con-tracting directions around a typical trajectory thatis a consequence of the affine character of thephase flow in the vicinity of a generic stable fixedpoint. Fortunately, the particular form of the func-tion P1 depends on the choice of coordinates, andin many cases it turns out possible to obtain theborders of the asymptotic stability area by introdu-cing a linear change of coordinates diagonalizingthe linear part of the flow F(x, t) in the vicinity ofan arbitrary point in the phase space. This kindof transformation is known to be a standard toolin the analysis of differential equations [7].3. NON AUTONOMOUS PASSIVE NONLINEAROSCILLATORAs an example of a particular system governedby the equations of the type (1), we take the sin-gle nonlinear oscillator (motion in a potential well with a potential U(x) defined by )(7)that has been used as a basic model in manyproblems of mechanics, electronics, optics, electro-magnetic field theory, etc. By introducing the variables x1 = x′, x2 = d x/dt thesystem (7) is transformed to the standard form(8)and the variational equations (3) in the vicinityof an arbitrary trajectory x*(t) for this systemlook likedddd12222 0 1 1( ) ( )xxtxx x N x f ttδ ω ε== − − − +dddd2202( ) ( )xxx N x f tttδ ω ε+ + + =dd( )( )U xN xx≡(9)where y1, 2 are the components of the perturbationvector y,Then, after the coordinate transform recasting thelinear part of Equation (8) to the canonical (dia-gonal) form, the explicit expression for the growthrate µ1 follows directly from its definition and theequation for the norm of ||y|| = ρ in the polar co-ordinates:Under the assumption that ϕ can take any valuein the interval [0;2π] we obtain the explicit formu-las for the border of the asymptotic stability area V1 < εV(x) < V2 (10)whereInequalities (10) thus define the limits of varia-tion for the function V(x) and, hence, for the coor-dinate x of the nonlinear oscillator (7), ensuringthe asymptotic stability of motion. 4 TWO COUPLED OSCILLATORSTwo nonlinear oscillators with a diffusive coup-ling is a classical system in the circuits theory. Itconstitutes a natural generalization of a single-de-gree-of-freedom nonlinear oscillator to a morecomplex dynamical system, necessary for understan-ding the multi-mode interactions in the spatiallydistributed devices, like, e.g., optical wave-guidesor electronic tubes. In the simplest case of identi-2 21,2 0142V δ ω δ= −∓*12 201 2( ) ( ( ))cos(2 ) ,2 4t V x tεµ δ ϕω δ  = − − − dd *1*1 ( )( ( )) .x x tNV x tx==dddd122 *20 1 2 1(( ( ))yytyy y V x t ytω δ ε== − − −AUTOMATIKA 45(2004) 1–2, 19–22 21V. B. Ryabov Stability of Microwave and Electronic Devices:...Fig. 1 Areas of asymptotic stability (shaded) for the case ofcubic potential (Duffing oscillator)cal oscillators with cubic nonlinearity (N(x) = βx3)and linear coupling, the system is described by thefollowing equationswhere κ stands for the coefficient of coupling bet-ween the oscillators. The direct application of theapproach given above combined with the canoni-cal linear coordinate transform results in the follo-wing equation for the largest local expansion rate:d dddd ddd22 31 10 1 1 2 1222 32 20 2 2 1 22( )( )x xx x x f tttx xx x x f tttω β δ κω β δ κ+ + + + =+ + + + =change of harmonic to bifrequency excitation in anequation of class (7) results in considerable lowe-ring of the instability onset in the intensity of theexternal force. A natural question stems from thesefindings: what is the lowest possible level of exci-tation that can result in a destabilization of thesystem? As we have demonstrated with severalexamples of nonlinear oscillators, the analysis ofasymptotic stability in terms of LCE allows answe-ring this question and estimating the maximal sta-ble amplitude of motion, and thus provides a ne-cessary condition for any bifurcation to occur. Wewould like to stress that the method we propose isindependent from the type of external force anddimensionality of the dynamical system, therefore,it yields a fundamental limit for all types of insta-bilities, including chaotic motion, to appear in abroad class of nonlinear circuits and systems.REFERENCES[1] C. Hayashi, Nonlinear Oscillations in Physical Systems.McGraw-Hill, New York, 1964.[2] G. Floquet, Sur les Equations Differentielles Lineaires aCoefficients Periodiques. Ann. Ecole Norm. Super., vol.2–12, pp. 47–88, 1883.[3] V. B. Ryabov, D. M. Vavriv, Chaotic Instabilities in Single-and Multi-Mode Electronic Devices, in Low Temperatureand General Plasmas. M. Miloslavljevic, Z. Petrovic (eds.),Nova Science, Belgrade, 1996.[4] V. B. Ryabov, Using Lyapunov Exponents to Predict theOnset of Chaos in Nonlinear Oscillators. Phys. Rev. E,vol. 66, no. 1, pp. 016214-1–016214-17, 2002.[5] J.-P. Eckmann, D. Ruelle, Ergodic Theory of Chaos andStrange Attractors. Rev. Mod. Phys., vol. 57, no. 3, pp.617–656, 1985.[6] V. I. Oseledec, A multiplicative Ergodic Theorem: Liapu-nov Characteristic Numbers for Dynamical Systems. Trans.Moscow Math. Soc. vol. 19, pp. 197–231, 1968.[7] S. Wiggins, Introduction to Applied Nonlinear DynamicalSystems and Chaos. Springer, New York, 1990.[8] A. D. Grishchenko, D. M. Vavriv, Dynamics of a Pendu-lum with a Quasiperiodic Perturbation. Tech. Phys., vol.42, pp. 1115–1120, 1997.22 AUTOMATIKA 45(2004) 1–2, 19–22Stability of Microwave and Electronic Devices:... V. B. Ryabov2 211 21 23 1 12 ( ) sin 2 cos sin 2 sin21 1sin cos sin 2 cos sin sin 2t UVβµ δ φ θ ϕ θω ωφ ϕ θ φ ϕ θω ω  = − − + +   + +  where φ, ϕ, θ are the angles in Equation (5). Then, the condition of absolute stability (6) can be formu-lated for this system asTherefore, the stability of the system of two cou-pled oscillators as a whole is defined by the am-plitude of oscillations of each of its subsystems.5 CONCLUSIONIt has been recently recognized that in manyoscillatory systems the threshold of instability maystrongly depend on the frequency content of theexternal signal. As it was shown, e.g., in [8], the2 2 1 21 21 2, .3 ( )x xδω ωβ ω ω<+22 2 2 2 2 21 2 1 2 1,2 0; ; ,4U x x V x xδω ω κ= + = − = − ∓AUTHOR’S ADDRESSVladimir B. Ryabov, ProfessorComplex Systems Department Future University Hakodate116-2 Kameda Nakano-cho, 041-8655 Hakodate cityHokkaido, JapanPhone: +81 - 0138-346129Fax: +81 - 0138-34-6301E-mail: riabov@ fun.ac.jp Received: 2004–10–23Stabilnost mikrovalnih i elektroni~kih aktivnih elemenata: Analiza primjenom teorije dinami~kih sustava.Razmotren je op}eniti problem stabilnosti mikrovalnih i radiofrekvencijskih aktivnih elemenata i sklopova u pri-sutnosti vi{efrekvencijskih ili {irokopojasnih signala. Problem je formuliran kao sustav nelinearnih neautonom-nih diferencijalnih jednad`bi u poop}enim koordinatama faznog prostora koje se mogu pridru`iti npr. amplitu-dama elektri~nih struja u sklopovima od elemenata s koncentriranim parametrima ili amplitudama modova elek-tromagnetskog polja u sustavima s raspodijeljenim parametrima. Predlo`ena je analiza stabilnosti koja omogu}avapredvi|anje razli~itih vrsta nestabilnosti uklju~uju}i deterministi~ko kaoti~no gibanje. Razmotren je novi kriterijkoji se osniva na analizi Ljapunovljevih eksponenata. Taj kriterij uspostavlja vezu izme|u najve}e stabilne ampli-tude oscilacija te razina nelinearnosti i prigu{enja u sustavu. Podrobno su analizirani primjeri jednomodnog idvomodnog oscilatora.Klju~ne rije~i: kaos, Ljapunovljevi eksponenti, mikrovalni sklopovi, oscilatori, nestabilnost

Stabilnost mikrovalnih i elektroničkih aktivnih elemenata: Analiza primjenom teorije dinamičkih sustava

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