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Recent Advances in Electrical Engineeringsuch dependences for the collinear and transverseperturbations are shown in Fig. 1. The parameters aand b of the linear relationshipN = aνω0+ b, (18)accumulated for all the data sets, are plotted in Fig.2against the relative perturbation amplitude є .One can notice two remarkable features in theseplots. First, the parameters a and b, in their entirety,look like the mirror images of each other, with thehorizontal reflection plane located somewherebetween 0 and 50. Second, their absolute valuesappear to oscillate and exponentially decay with є,eventually converging at the mirror plane ordinate.On the basis of these considerations, the four setsof the coefficients a and b are fitted using a functiona1exp ( − a2є) ⎡⎣cos ( a3є)+ a4⎤⎦+ a5where a 1 , a 2 , a 3 , a 4 , a 5 are the fit constants.The fit equations for the separate groups of thesecoefficients are shown in Fig.2 beside each fit curve.Assuming the true mirror arrangement of a and b,the fit equations for the coefficients related to thecollinear and transverse perturbations are( acol− 14.5) =−( bcol−14.5)(19)= 41.3exp( − 14.8є) ⎡⎣cos( 92.3є)+ 2.9⎤⎦and( atr− 18.0) =−( btr−18.0)(20)= 36.7exp( − 13.1є ) ⎡⎣cos( 92.3є)+ 1.1 ⎤⎦.Furthermore, the similarity of the panels of Fig. 2suggests that the time evolution of the SLW in bothcases is likely to be described by a singlerelationship. The fitting procedure incorporating allthe four groups of the coefficients results in( atot− 17.5) =−( btot−17.5)(21)= 39.2exp( − 14.1є ) ⎡⎣cos( 92.0є)+ 2.0 ⎤⎦.If (19)-(20) or (21) are still correct in the regionof weak fields, then the characteristic number ofperturbation cycles is about 20-40 for νω 0 = 1 andinfinitely small amplitude. In the area of highrelative amplitudes, the characteristic number ofperturbation cycles calculated from (19)-(21) turnsout to be zero. As a matter of fact, this physicallymeaningless result implies that the number of fieldcycles saturating the SLW in this region is,expectedly, as low as about unity.Because of the paucity of the data used, it is yetunclear whether the obtained equations (19)-(21)can be applied to average the influence of all initialphases. The broad dispersion of the coefficientvalues (Fig. 2) suggests that the field phase plays animportant role in the stochastic layer development.Fig.2 Coefficients of (18) vs. the relativeperturbation amplitude for the collinear (upperpanel) and transverse (lower panel)perturbations. Blue circles and fit lines: a; greensquares and fit lines: b. The fit equations aregiven for each group of the coefficients.4 ConclusionA phenomenological approach to finding the timeevolution of the stochastic layer of a perturbedpendulum dipole has been proposed in this work. Asa result of this approach, the complete equation (13)and its simplified version (16) have been derived.These equations in principle allow predicting theexponential decrease of the stochastic layerbroadening rate, which has been observed in thesimulations.The area of moderately high perturbationamplitudes and intermediate frequencies studied inthis work is of special importance, since no rigoroustheoretical basis was proposed to account for thenonlinear pendulum dynamics in this region.In our simulations, the characteristic stochasticlayer saturation time, expressed as the characteristicISBN: 978-960-474-318-6 90
Recent Advances in Electrical Engineeringnumber of perturbation cycles, has been found tovary linearly with the relative perturbationfrequency. The coefficients of this linear relationconverge in an oscillatory manner to a constantvalue with the relative perturbation amplitudeincrease. In the zero-field limit, the characteristicnumber of perturbation cycles has proven to attainsome tens for the main resonance frequency. In thestrong-field limit, the fit equations have predicted anear-zero number of cycles for this frequency,which is quite reasonable, taking into account thestrong scattering and uncertainty of the fittedcoefficients.The found results are interesting in thedevelopments of enhanced technologies ofmicrowave heating and chemistry, and incontrolling molecules by EM field [13].References:[1] B.V. Chirikov, A Universal Instability ofMany-Dimensional Oscillator Systems, Phys.Rep., Vol.52, No.5, 1979, pp. 263-379.[2] G.M. Zaslavsky, Physics of Chaos in HamiltonSystems, Imperial College Press, 1998.[3] S.V. Kapranov, G.A. Kouzaev, Stochasticity inNonlinear Pendulum Motion of Dipoles inElectric Field, In: Recent Advances in SystemsEngineering and Applied Mathematics,Selected papers from WSEAS Conference inIstanbul, Turkey, May 27-30, 2008, WSEASPress, 2008, pp. 107-111.[4] S.V. Kapranov, G.A. Kouzaev, StochasticDynamics of Electric Dipole in ExternalElectric Fields: A Perturbed NonlinearPendulum Approach, Physica D, Vol.252,No.1, 2013, pp. 1-21.[5] A.C.J. Luo, R.P.S. Han, The dynamics ofStochastic and Resonant Layers in aPeriodically Driven Pendulum, Chaos, SolitonsFractals, Vol.11, No.14, 2000, pp. 2349-2359.[6] V.V. Vecheslavov, Chaotic Layer of aPendulum Under Low- and Medium-FrequencyPerturbations, Tech. Phys., Vol.49, No.5, 2004,pp. 1-5.[7] I.I. Shevchenko, The Width of a Chaotic Layer,Phys. Lett. A, Vol.372, No.6, 2008, pp. 808-816.[8] S.M. Soskin, R. Mannella, Maximal Width ofthe Separatrix Chaotic Layer, Phys. Rev. E,Vol.80, No.6, 2009, 066212.[9] A.C.J. Luo, K. Gu, R.P.S. Han, Resonant-Separatrix Webs in Stochastic Layers of theTwin-Well Duffing Oscillator, NonlinearDynam., Vol.19, No.1, 1999, pp. 37-48.[10] A.C.J. Luo, R.P.S. Han, Dynamics ofStochastic Layers in Nonlinear HamiltonianSystems, Int. J. Nonlinear Sci. Numer. Simul.,Vol.1, No.2, 2000, pp. 119-132.[11] S.M. Soskin, R. Mannella, O.M. Yevtushenko,Matching of Separatrix Map and ResonantDynamics, with Application to Global ChaosOnset Between Separatrices, Phys. Rev. E,Vol.77, No.3, 2008, 036221.[12] Ya. Goltser, A. Domoshnitsky, AboutReducing Integro-Differential Equations withInfinite Limits of Integration to Systems ofOrdinary Differential Equations, Adv. Diff.Equ., 2013:187, 2013.[13] G.A. Kouzaev, Applications of AdvancedElectromagnetics. Components and Systems.Springer-Verlag, 2013.ISBN: 978-960-474-318-6 91

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