The stationary and highly non-stationary resonant dynamics of the harmonically forced pendulum are described in the framework of a semi-inverse procedure combined with the Limiting Phase Trajectory concept. This procedure, implying only existence of slow time
scale, permits one to avoid any restriction on the oscillation amplitudes. The main results relating to the dynamical bifurcation thresholds are represented in a closed form. The small parameter defining the separation of the time scales is naturally identified in the ana-
lytical procedure. Considering the pendulum frequency as the control parameter we reveal two qualitative tran-
sitions. One of them corresponding to stationary instability with formation of two additional stationary states, the other, associated with the most intense energy drawing from the source, at which the amplitude of pendulum oscillations abruptly grows. Analytical
predictions of both bifurcations are verified by numerical integration of original equation. It is also shown that
occurrence of chaotic domains may be strongly connected with the second transition

Leonid Manevitch

ROMEO, Francesco

Valeri Smirnov

English

arXiv

Archivio della ricerca- Università di Roma La Sapienza

CYBERNETICS AND PHYSICS, VOL. 5, NO. 3, 2016 , 91–95NON-STATIONARY RESONANCE DYNAMICS OFTHE HARMONICALLY FORCED PENDULUMLeonid I. ManevitchPolymer and Composite Materials DepartmentN. N. Semenov Institute of Chemical Physics RASRussialeonidmanevitch3@gmail.comValeri V. SmirnovPolymer and Composite Materials DepartmentN. N. Semenov Institute of Chemical Physics RASRussiavvs@polymer.chph.ras.ruFrancesco RomeoDepartment of Structural and Geotechnical EngineeringSapienza University of RomeItalyfrancesco.romeo@uniroma1.itAbstractThe stationary and highly non-stationary resonant dy-namics of the harmonically forced pendulum are de-scribed in the framework of a semi-inverse procedurecombined with the Limiting Phase Trajectory concept.This procedure, implying only existence of slow timescale, permits one to avoid any restriction on the os-cillation amplitudes. The main results relating to thedynamical bifurcation thresholds are represented in aclosed form. The small parameter defining the separa-tion of the time scales is naturally identified in the ana-lytical procedure. Considering the pendulum frequencyas the control parameter we reveal two qualitative tran-sitions. One of them corresponding to stationary in-stability with formation of two additional stationarystates, the other, associated with the most intense en-ergy drawing from the source, at which the amplitudeof pendulum oscillations abruptly grows. Analyticalpredictions of both bifurcations are verified by numeri-cal integration of original equation. It is also shown thatoccurrence of chaotic domains may be strongly con-nected with the second transition.Key wordsNonlinear dynamics, forced pendulum, limiting phasetrajectory1 IntroductionThe harmonically forced pendulum is one of the basicmodels of nonlinear dynamics; as such, it has numer-ous applications in different fields of physics and me-chanics [Sagdeev, Usikov, Zaslavsky 1988; Chirikov,Zaslavsky, 1972; Arnold, Kozlov, Neishtadt, 2006;Neishtadt, Vasiliev, 2005]. There are two main di-rections in the study of this model. It can be han-dled as an application of general mathematical pertur-bation theory in which the integrable conservative sys-tem is the generating model [Bogoliubov, Mitropol-ski, 1961; Hale, 1963; Nayfeh, 2004; Chirikov, Za-slavsky, 1972; Neishtadt, 1975]. In this framework,the obtained results hold mainly in quasi-linear ap-proach and capture only the basic features of nonlin-ear forced oscillators: finiteness of resonance ampli-tude and possibility of its abrupt change due to insta-bility of stationary states. The second direction, whichwas predominantly developed by physicists and me-chanicians, deals with analytical description of the sta-tionary states (still in the quasi-linear approximation),their stability analysis and merely numerical study ofnon-stationary dynamics [Nayfeh, 2004; Manevitch,Manevitch, Manevitch, 2005]. Recently it was shownthat the concept of Limiting Phase Trajectories (LPTs),introduced in [Manevitch, 2007], allows to find an ef-ficient analytical description of highly non-stationaryresonance dynamics in which the oscillator (pendulum)draws the maximum possible (at given conditions) en-ergy from the source (periodic field). In particular, in[Manevitch, Musienko, 2009], the non-stationary dy-namics of a nonlinear forced oscillator was studiedin the quasi-linear approximation. The goal of thiswork is to remove the restrictions on the amplitude ofthe forced pendulum’s oscillations extending the re-sults obtained in [Manevitch, Romeo, 2015] for twoweakly coupled conservative pendula. Towards thisgoal, a semi-inverse approach in combination with theLPT concept is adopted and the analytical descriptionof qualitative transitions in both stationary and highly92 CYBERNETICS AND PHYSICS, VOL. 5, NO. 3non-stationary dynamics is derived. Furthermore, theconditions of chaotization for different oscillation am-plitudes are clarified.The paper is organized as follows. At first, the gov-erning equations and the semi-inverse asymptotic ap-proach are introduced and the stationary regimes ofoscillations are studied. Then, in the following sec-tion, the non-stationary (global) dynamical transitionsare discussed and identified. The numerical validationcarried out by means of Poincare´ sections is eventuallydescribed followed by the concluding remarks.2 Governing Equation and Asymptotic ApproachWe discuss the undamped dynamics of a pendulumexcited by a harmonic external field, undergoing uni-directional motion. Corresponding equation of motionisd2qdt2+ sin q = f sin [(! + s) t]; (1)where q is the pendulum angular coordinate, f and !are the harmonic forcing amplitude and frequency, re-spectively, and s is the detuning parameter. By intro-ducing the complex amplitude of the pendulum oscil-lations as =1p2(1p!dqdt+ ip!q)q = ip2!(    ); dqdt=r!2( +  );(2)equation (1) can be rewritten as followsid dt+!2( +  )+12!1Xk=01(2k + 1)!12!k(    )2k+1=f2p2!ei(!+s)t   e i(!+s)t(3)Let us search a solution to equation (3) in the form = 'ei!t, where ' is a slowly varying function oftime t. Substituting into equation (3) and assuming thedetuning parameter s to be small enough, one can con-sider the function ' as a new variable depending onlyon slow time  = s t.Then, multiplying equation (3) by e i!t and integrat-ing with respect to the “fast” time t, the condition pro-viding elimination of resonance (secular) terms is ob-tained as:is@'@  !2'+1p2!J1 r2!j'j!'j'j =fp2!ei ;(4)Figure 1. (Color online) Frequency-amplitude curves for the pen-dulum stationary oscillations. Black, red and blue curves show thebackbone natural and the forced frequencies, respectively. Detuningparameter s = 0:05 and force amplitude f = 0:1. Solidcurves correspond to equation (7), while the dashed one shows theexact backbone value.where J1 is the Bessel function of the first kind.It is easy to check that the function ' =pXei isthe solution of equation (4) if frequency ! satisfies therelations  !2+1p2X!J1 r2X!!  fp2X!= 0 (5)Taking into account the relationship X = !2 Q2 be-tween oscillation amplitude Q and the amplitude of itscomplex representationpX (see definition (2) of thefunction  ), one can rewrite equation (5) as follows!2 + 2s !   1Q(2J1(Q)  f) = 0: (6)This equation provides with a very simple expressionfor the pendulum stationary oscillations frequency as afunction of their amplitude:! =  s+r1Q(2J1 (Q)  f) + s2 (7)Fig. 1 shows the frequency-amplitude dependence forboth the forced and unforced oscillations. It is worthnoticing that the high-frequency branch of the forcedoscillations is described by negative values of both de-tuning s and force amplitude f . It can also be seenthat the limit fs ! 0; f ! 0g leads to the backbonefrequency of the natural vibrations of the pendulum!0 =p2J1(Q)=Q. The latter approximates the ex-act value of natural frequency of pendulum oscillationsin a wide range of amplitudes, up to 9=10 (see Fig.1).CYBERNETICS AND PHYSICS, VOL. 5, NO. 3 93Equation (7) allows us to estimate the amplitudeat which the bifurcation of stationary states occurs.One can see from Fig. 1 that this bifurcation takesplace at the extreme of low-frequency branch of theforced oscillations. Therefore, by solving the equationd!=dQ = 0, the amplitude Q at which the bifurcationoccurs can be obtained from2QJ2(Q) = f (8)The expression (8) identifies the frequency of abruptincreasing of oscillation amplitude as the frequencyof external force increases along the low-amplitudebranch of equation (7). However, the abrupt decreas-ing of oscillation amplitude along the high-amplitudebranch can not be tracked due to the absence of any lim-itation on the oscillation amplitude in the initial equa-tion of motion. As a result, the system does not preventthe motion from the transition into rotating regimes.3 Dynamical TransitionsThe results of previous section relate to states cor-responding to fixed points in the system phase space.Thus, they reflect the stationary dynamics of the forcedpendulum (let us note that the term “stationary” doesnot always mean “stable”). However, non-stationaryregimes in the dynamics of pendulum exist, which arecharacterized by a large modulation of the oscillations.Such processes are accompanied by intensive energyexchange between pendulum and energy source.This non-stationary dynamics of the forced pendulumcan be tackled by introducing phase () and ampli-tude a =p!=2Q, such that the slowly varying func-tion ' can now be expressed as ' = aei(). Havingintroduced the phase shift  =    , the equation ofmotion (4) can be written assa _ + 1p2!J1q2!a   s+ !2  a (9)= f2p2!coss _a = f2p2!sin; (10)where the overdot indicates the derivative with respectto the slow time  . The corresponding integral of mo-tion readsH =s2"a2s+!2+af cosp2!+ J0 r2!a!  1#(11)There are three control parameters in the consideredsystem (11): detuning s, forcing f , and frequency !.They are coupled with the stationary oscillation ampli-tude Q by relation (7). So, by fixing detuning s andforce amplitude f and varying pendulum frequency !,the phase portrait of system (11) on the-Q plane canFigure 2. (Color online) Evolution of the -Q phase portrait forf = 0:06 and s = 0:1 for increasing frequency !. (a) Beforethe non-stationary transition, ! = 0:79; (b) at the non-stationarytransition, ! = 0:8106; (c) after the non-stationary transition,! = 0:82; (d) after the stationary transition, ! = 0:85.be considered in order to study the non-stationary pro-cesses (Figs. 2 (a-d)).Fig. 2(a) shows the typical phase portrait for the“low” frequency region. The three stationary pointsare associated with stable, i.e. the nonlinear normalmodes (NNM), and unstable branches in Fig. 1. Itis worth noticing that the attraction areas of the stablestates are bounded by two specific trajectories. The firstone is the dynamic heteroclinic separatrix (blue curve)passing through the unstable state and surrounding thelarge-amplitude stationary point. The second one stemsform zero amplitude initial conditions (without depen-dence of initial phase shift) and leads to move along theclosed trajectory bounding the attraction area of the sta-ble stationary points at  = 0 (red curve in Fig. 2(a)).Since the latter trajectory is the farthest from the small-amplitude stationary point, we refer to it as the Lim-iting Phase Trajectory (LPT). The principal differencebetween this trajectory and the separatrix being the fi-nite time of the corresponding period. The LPT corre-sponds to the most intensive energy taking off by thependulum from the energy source (at given initial con-ditions). The remaining trajectories require non-zeroinitial conditions.Increasing frequency !, the LPT grows until it coa-lesces with the dynamic separatrix (Fig. 2(b)), and anabrupt change of the LPT amplitude takes place. Thisqualitative transformation of the phase portrait repre-sents the non-stationary transition, at which the LPTsplits in two branches, surrounding the two stable stateswith phase shift  = 0 and  = . The latter turnsout to be significantly larger than the former. Thereforethe amplitudes of non-stationary oscillations increasesabruptly together with the energy flow from the source94 CYBERNETICS AND PHYSICS, VOL. 5, NO. 3Figure 3. (Color online) Dynamical transitions thresholds on the(f -!) plane for s = 0:1. First (blue) and second (red) thresholds,analytical (solid), numerical (dashed), and points corresponding tothe phase portraits shown in Fig. 2.to the pendulum. Further increase of the frequency !leads to weak decreasing of LPT and separatrix, up tothe annihilation of the latter occurring at the station-ary transition whose frequency is determined throughequation (8).In order to estimate the threshold of the abrupt changeof oscillation amplitude (i.e. the non-stationary tran-sition), it is worth noticing that the value of Hamilto-nian (11) turns out to be equal to zero at this bifurcationpoint. So, by solving the equationH(Q;!)j=0 = 0 (12)jointly with equation (6) with respect to ! and Q, atfixed values f and s, one can calculate the thresholdfrequency as a function of detuning s and forcing f .Fig. 3 shows the threshold values of ! at detunings = 0:1 as a function of forcing f (solid red curve).The bifurcation value of ! that leads to the annihila-tion of unstable stationary point, given by (8), is alsorepresented in Fig. 3 by solid blue curve; the dashedcurves correspond to the same thresholds obtained nu-merically.4 Poincare´ sectionsIn this section a numerical validation of the dynamicregimes exhibited by the forced pendulum is proposedby resorting to Poincare´ sections obtained from di-rect integration of the starting equation of motion (1).Besides confirming the two dynamical transitions dis-cussed in the preceding section, the Poincare´ sectionsallow one to identify the onset of nonregular pendu-lum response and its connection with the LPTs and thedynamic separatrix. Having fixed the forcing ampli-tude and shift, the dynamic regimes evolution is thedescribed for varying forcing frequency values. Fors = 0:1, in Figs. 4 and 5 Poincare´ sections are shownfor f = 0:06 and f = 0:08, respectively. More indetail, in Fig. 4a the scenario at ! = 0:85 correspond-ing to point D in Fig. 3, whose phase plane is shownin Fig. 2d, is depicted; it is characterized by the pres-ence of one stationary point (NNM at  = ), andthe LPT (red curve) encircles it. Next, in Fig. 4 b, thePoincare´ sections at ! = 0:82, corresponding to pointC in Fig. 3, whose phase plane is shown in Fig. 2 c, isshown; in this case, having crossed the second (station-ary) transition occurring at ! = 0:825, the newbornNNM at  = 0 and unstable hyperbolic point can beseen together with the LPT (red curve) encircling theNNM at  =  and the heteroclinic separatrix (bluecurve). As the value of ! is lowered to 0:8047, thefirst (non-stationary) transition, corresponding to pointB in Fig. 3 and phase-plane in Fig. 3b, occurs; the LPT(red curve) and the heteroclinic separatrix coalesce im-plying the most intense energy exchange between thesource of excitation and the pendulum. At last, afterthe second (stationary) transition the LPT (red curve)localization is clearly seen in Fig. 4 d in which, for! = 0:79, consistently with the phase plane shown inFig. 2a, the LPT amplitude undergoes a significant re-duction entailing a weak energy exchange with the har-monic forcing. For lower values of ! the regular mo-tion region, so far characterizing the whole phase plane,splits into two regions surrounding the two stationarypoints, separated by a chaotic sea (see Figs. 4e and 4f).The Poincare´ sections reported in Fig. 5, correspondingto the case with f = 0:08, show the analogous qualita-tive evolution of the previous case with f = 0:06. Themain difference lies on the onset of separatrix chaosthat for f = 0:08 occurs for higher values of !, in be-tween the two dynamic transitions (see Fig. 5b). Asshown in Fig. 5c, at the non-stationary transition suchchaotization involves the LPT as well. Afterwards, as !decreases, a trend similar to the case for f = 0:06 canbe observed. It is seen that for relatively weak forcingamplitude (Fig. 3) one can observe a regular behav-ior which corresponds exactly to analytical predictionswhile increase of forcing amplitude leads to manifesta-tion of chaotic behavior (Fig. 4, 5). Two scenarios ofchaotization are possible. In the first scenario, this phe-nomenon appears after both dynamical transitions (Fig.4e) and may be identified as breaking of regular sepa-ratrix. In the second, realized around f = 0:08 (Fig.5b, c, d, e and f), chaos manifestation occurs after thestationary transition (Fig. 5b) and becomes especiallyclear at the non-stationary transition (Fig. 5c).5 ConclusionBy adopting a self-consistent semi-inverse procedureand the LPT concept the fundamental solutions for bothstationary and highly non-stationary dynamics werederived for the harmonically forced pendulum. In par-ticular, the analytical representation of the dynamicalpendulum states was found without any restrictionson the oscillations amplitude. Two qualitative transi-tions (in parametric space) were revealed. One of themcorresponds to the birth of a pair of additional, onestable and one unstable, stationary states and hetero-CYBERNETICS AND PHYSICS, VOL. 5, NO. 3 95Figure 4. (Color online) Evolution of the Poincare´ sections forf = 0:06 and s = 0:1 for decreasing frequency!. (a) Before thestationary transition, ! = 0:85; (b) after the stationary transition,! = 0:82; (c) at the non-stationary transition, ! = 0:8047; (d)after the non-stationary transition, ! = 0:79; (e) ! = 0:7; (f)! = 0:6.clinic separatrix. Manifestation of the other transitionis connected with coalescence of dynamic separatrixand LPT entailing an abrupt growth of oscillation am-plitude. As a result, at this purely non-stationary tran-sition, the conditions of maximum energy that can bedrawn from the source are realized. Furthermore, wehave shown that chaotization of dynamic trajectories isclearly manifested in the vicinity of the non-stationarytransition. The described highly non-stationary reso-nance dynamics of the forced pendulum lends itself tobe exploited in many applications involving manipula-tion of mechanical energy transfer.AcknowledgmentsAuthors are grateful the Russia Basic Research Foun-dation /grant n. 16-02-00400/ and the Department ofChemistry and Material Science of RAS /Programm #3/ for the financial support.ReferencesArnol’d, V. I., Kozlov, V. V., Neishtadt A. I. (2006).Mathematical Aspects of Classical and Celestial Me-chanics, 3rd ed., Encyclopaedia Math. Sci., v. 3,Springer, Berlin.Bogoliubov, N., Mitropolsky, Y. A. (1961). AsymptoticMethods in the Theory of Non-Linear Oscillations,Gordon and Breach.Chirikov, B.V., Zaslavsky, G.M. (1972). Stochastic in-Figure 5. (Color online) Evolution of the Poincare´ sections forf = 0:08 and s = 0:1 for decreasing frequency !. (a) Beforethe stationary transition, ! = 0:85; (b) after the stationary transi-tion, ! = 0:8; (c) at the non-stationary transition, ! = 0:7834;(d) after the non-stationary transition, ! = 0:75; (e) ! = 0:7; (f)! = 0:6.stability of nonlinear oscillations, Sov. Phys. Uspekhi,Vol. 14, pp. 549.Hale, J. K. (1963). Oscillations in Nonlinear Systems,McGraw-Hill, New York.Manevitch, A. I., Manevitch, L. I. (2005). The Mechan-ics of Nonlinear Systems with Internal Resonances,Imperial College Press, London.Manevitch, L. I. (2007). New approach to beatingphenomenon in coupled nonlinear oscillatory chains,Arch Appl Mech, Vol. 77, pp. 301.Manevitch, Leonid I., Musienko, Andrey I. (2009)Limiting phase trajectories and energy exchange be-tween anharmonic oscillator and external force, Non-linear Dyn., Vol. 58, pp. 633642Manevitch, L. I. , Romeo, F. (2015). Non-stationaryresonance dynamics of weakly coupled pendula, Eu-rophys. Let., Vol.112, pp. 3005.Nayfeh, A. H., Mook, D. T. (2004). Nonlinear oscilla-tions, Wiley-VCH Verlag GmbH and Co.Neishtadt, A. I. (1975). Passage through a Separatrix ina Resonance Problem with a Slowly-Varying Param-eter, J. Appl. Math. Mech., Vol. 39(4), pp. 594.Neishtadt, A. I., Vasiliev, A.A. (2005). Capture intoResonance in Dynamics of a Classical HydrogenAtom in an Oscillating Electric Field, Phys. Rev. E,Vol. 71, pp. 056623.Sagdeev, R. Z., Usikov, D. A., Zaslavsky, G. M. (1988).Nonlinear Physics: From the Pendulum to Turbu-lence and Chaos, Harwood Acad. Publ., New York.

Non-stationary resonance dynamics of the harmonically forced pendulum

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