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1 RESERCH IN LOL DYNMICS O DUIN-TYPE OSCILLTOR Y METHOD O COMPLETE IURCTION NLYSIS Ris Smirnov, Jurij Ivnov, Vldimir Nikishin Rig Technicl University, Ltvi rj@df.rtu.lv, ijm@df.rtu.lv, nikis@df.rtu.lv strct. Numerous reserches of the Duffing-type oscilltor hve reveled ig vriety of dynmics of this nonliner system. Nonliner effects, which result from chnges of initil conditions nd prmeters, cn e mnifested in vrious, sometimes seemingly unexpected, wys. The tsk of studying nonliner effects hs importnt prcticl vlue for engineering dynmicl systems working y the Duffing-type oscilltor. or the reserch of nonliner effects we usully use the method of hrmonic lnce, nlog simultion or the method of direct scnning. Such techniques re little limited regrding the ility to predict ll dynmicl chrcteristics, including ifurctions tht led to occurrence of unstle, suhrmonic nd chotic solutions. The given work is devoted to the study of complicted dynmics of the Duffing-type oscilltor nd the resons of its occurrence t the chnge of the system prmeters on the sis of the system pproch y the method of complete ifurction nlysis. The results re received y direct numericl simultion with the use of softwre NLO nd SPRIN. ifurction digrms t the reserch of dynmics of the Duffing-type oscilltor re constructed y mens of the prmeter continution pproch of stedy nd unstle periodic solution. The exmple of the Duffing-type oscilltor illustrtes high efficiency of the method of complete ifurction nlysis for providing n explntion of the resons of the irth of regulr nd chotic solutions, their coexistence nd influence on trnsient processes. t the systemtic pproch to certin prmeters, t which the complete ifurction nlysis specifies the preconditions of the irth of unexpected nonliner effects, in the investigted system n dditionl nlysis of vrious periodic nd chotic ttrctors, forecsting of rre ttrctors nd rre phenomen, studying of scenrios of their irth is crried out. Keywords: nonliner effect, dynmicl system, ifurction digrm, rre ttrctor, chos. Introduction n importnt tsk for ll nonliner dynmicl systems is the study of the nonliner effects tht llow predicting nd explining complicted ehviour of such systems. Modern reserches of nonliner phenomen in the nture nd engineering re relted to the study of typicl structures in the ehviour of nonliner dynmicl systems. This work is devoted to the reserch of regulr nd chotic dynmics of the Duffing-type oscilltor nd systemtic reveling of nonliner effects. The work illustrtes the dvntge of the method of complete ifurction groups for the reserch of nonliner effects in comprison to the pplied methods: hrmonic lnce nd direct scnning. Dynmicl model nd ifurction nlysis for regulr nd chotic dynmics reserch The im of the given work is to reserch in the dynmics of the Duffing-type oscilltor (), study regulr nd chotic solutions, nonliner effects [-]. The uthors pply up-to-dte numericl nd nlyticl methods of the nlysis of nonliner dynmicl systems. irstly, these re the points mpping method nd the prmeter continution method. It is offered to pply systemtic pproch to the reserch of periodic nd chotic forced oscilltions in nonliner systems: serch of periodic regimes, contour mpping, constructing of regime ttrction domins, construction of ifurction digrms. In this pper, periodic nd non-periodic stedy or trnsition oscilltions of the influence of the frequency of externl hrmonic forcing re oserved in computer-simulted solutions of the secondorder nonliner ordinry differentil eqution of the Duffing-type oscilltor (): ( t) & x& x& x x h cos, () where x displcement; nd coefficients of the Duffing-type restoring force; dissiption coefficient of the liner dissiptive force; (t) hcos(t) hrmonic driving force with the period Т π ; h mplitude; w frequency. DOI: 0.66/ERDev07.6.N

2 Let us show the ppliction of the method of hrmonic lnce [] for the reserch of the Duffingtype oscilltor. t clcultion y the method of hrmonic lnce on the first hrmonics the solution of eqution () is set in the form of (). Prmeters nd re unknown vlues. Solution () is plced in eqution () nd the mplitudes re equted t the sine nd cosine of the first hrmonics. fter the trnsformtions we receive system of two nonliner lgeric equtions (). The system of equtions () is solved numericlly t zero initil pproch in softwre MTL. System () is reduced to one cuic eqution () for the mplitude of the sinus components of the solution. We find the root of cuic eqution () nd determine the mplitude of the cosine component of the solution sed on expression (5). Cuic eqution () is recorded in the cnonicl form (6). or the cnonicl form (6) the discriminnt of the eqution is (7). t positive discriminnt (7) cuic eqution () hs one rel root; t negtive discriminnt the cuic eqution will hve three rel roots (ig..). The system descried y eqution () possesses one or three possile regimes ccording to the quntity of rel roots of cuic eqution (). In cse of coexistence of three regimes of the sic group there is, so-clled in the electricl engineering, the trigger or hysteresis effect when there is discontinuous chnge of the regime t the chnge (increse nd reduction) of the prmeters (see ig. ). 6 ( h) ( t) cos( t) x sin. () ) ( ) h ) ( ) ) 0. () h ( ) h 0 ) 6 s 6 z. () h. (5) r s z 0. (6) 8 r p q D, p 7 ) h ( h) ( ). h, ( ( ) ), s r, q r r s z 7. (7) When clculting with the help of the hrmonic lnce method on the first nd third hrmonics the solution of eqution () is set in the form of (8). Prmeters,,, re unknown vlues. Solution (8) is plced in eqution () nd the mplitudes re equted t the sine nd cosine of the first nd third hrmonics. fter the trnsformtions we receive system of four nonliner lgeric equtions (). The system of equtions () is solved numericlly t zero initil pproch in softwre MTL. ( t) cos( t) sin( t) cos( t) x sin (8) 0

3 0 0 0 h () ig.. ifurction digrms of the Duffing-type oscilltor of sic periodic regimes t the chnge of the frequency of externl hrmonic forcing pplying the hrmonic lnce method on the first hrmonics (left fig.) nd pplying the method of complete ifurction nlysis of the prmeter continution pproch of stle (drk gry) nd unstle (light gry) periodic solutions with the sic ifurction group Т (right fig.). System prmeters:,, 0.08, h 5, w vr The method of hrmonic lnce (HM) is the pproched method of definition of periodic regimes in nonliner oscilltory systems, sed on the theory tht t presence of nonlinerity the estlished oscilltions re frequently close to the hrmonic. The HM is lso suitle for the study of oscilltions of utonomous systems. The HM in mny cses cn give successful results in terms of qulity nd often in quntity not only t smll nonlinerities, ut lso for essentilly nonliner systems. High efficiency HM in the reserch of the sic regime cn e explined y filtering properties of mny nonliner systems, which llows pproximting the solution y the sum of hrmonics. or the reserch of su-hrmonic nd chotic solutions the pplicility of the method is strongly limited y the complexity of trnsformtions (see ig. nd ig. ). or the disply of regulr nd chotic sttionry regimes (ttrctors) in ifurction digrms the scnning method ( method of direct scnning) is usully used [5]. This method llows defining quntittive vlues of phse coordintes of ttrctors tht correspond to the results of the integrtion fter the completion of the trnsition process t the movement on the prmeter of one or severl initil conditions elonging to the re of ttrction of the sttionry regime (see ig. ). To receive complete ifurction imge we need to crry out whole set of dditionl reserches t fixed vlues of prmeters in order to define the quntity of ttrctors, which crries occsionl rther thn systemtic chrcter. The ide of the RTU professor M. Zkrzhevsky of continution of stedy nd unstle solutions of one ifurction group efore the construction of complete ifurction imge llows crrying out complete ifurction nlysis nd systemticlly find nd explin nonliner effects. Under the supervision of M. Zkrzhevsky the Spring nd NLO softwre hve een developed for the reserch of dynmicl systems [6-8]. T w x

4 The method of complete ifurction groups nd the complex of lgorithms developed on its sis nd the softwre llow finding qulittively new previously unfmilir regulr nd chotic regimes, rre ttrctors (R) nd rre phenomen, new ifurction groups for typicl nd widely pplied clssicl nonliner dynmicl models nd studying the interction of vrious ifurction groups [8-]. The completeness of ifurction group is defined, if we find for it ll stedy nd unstle regimes nd lso connected with this ifurction group in spce of prmeters of dissiptive nonliner dynmicl systems. t vrition of prmeters in nonliner oscilltory systems qulittive chnges in ehviour of the system tke plce, which leds to the loss of stility of the periodic solution, the cscde ifurctions of period douling, to the formtion of res with infinite numer of unstle regimes UPI (Unstle Periodic Infinitium) nd to the irth of chotic ttrctors [0]. In nonliner dynmicl systems the irth of chotic oscilltions is connected with the sic ifurction group Т, suhrmonic ifurction groups nt nd their coexistence []. E T E T E E E 5 ig.. ifurction digrm of the Duffing-type oscilltor pplying the method of complete ifurction nlysis of the prmeter continution pproch of stedy nd unstle periodic solution with the sic ifurction group Т nd suhrmonic ifurction group T (left fig.). System prmeters:,, 0.08, h 5, w vr. Nonliner multistility phenomenon of the coexistence of ttrctors: t w 5 the right figure displys ll the domins of ttrction for stle periodic regimes E, E, E 5 nd coordintes for unstle periodic regimes E, E (see ig. ) E E E 5 E E E E E 5 ig.. Phse portrits nd phse trjectories of stle periodic regimes (ttrctors E, E, E 5 ) nd phse portrits of unstle periodic regimes E, E from the ifurction groups Т nd T (see ig. ). Initil conditions: E (x 5.0, v 6.758); E (x.7577, v 0.56); E (x -.6, v ); E (x -.75, v 5.86); E 5 (x , v 0.07). The nonliner driven Duffing-type oscilltor prmeters:,, 0.08, h 5, w 5

5 or disply of the chotic trnsient process, regulr nd chotic ttrctors in ifurction digrms the scnning method is used to determine the phse spce of the quntittive vlue. ig.. ifurction digrm (dependence of sttionry regulr periodic nd chotic solutions displcement x nd velocity v) of the Duffing-type oscilltor pplying the scnning method ( method of direct scnning) on the excittion force frequency w. System prmeters: -,, 0., h 7, w vr T T T 5T twins UPI 5T T twins R P5 T twins T R P5 5T twins UPI 5T ig. 5. Complete ifurction digrm of stle (drk gry) nd unstle (light gry) periodic solutions (dependence displcement x nd velocity v) nd chotic motions or ttrctors (UPI regions) of 7 ifurction groups (T, three T nd three 5T) of the driven Duffing-type oscilltor pplying the method of complete ifurction nlysis of the prmeter w (frequency of externl hrmonic forcing) continution. New nonliner phenomen hve een discovered: rre ttrctors (R) (see ig. 6). System prmeters: -,, 0., h 7, w vr w.5 w.666 w.0 ig. 6. Chotic ttrctors (w.5 nd w.0) from initil conditions (x -, v 0) during periods on the Poincré plne nd the phse portrit of suhrmonic solution - ttrctor P 5 of the Duffing-type oscilltor. t the vlue of the prmeter w.666, we cn oserve trnsient chos (chotic oscilltions) during 70 periods from initil conditions (x -, v 0). Then, periodic ttrctor P 5 (rre ttrctor R) is formed in the system. It is displyed in the middle imge. The nonliner driven Duffing-type oscilltor prmeters: -,, 0., h 7, w vr Efficient lgorithms of softwre Spring llow conducing reserches of regulr nd chotic solutions of ll types ccording to the method of complete ifurction groups. The exmples in ig. 5 of the ifurction nlysis for the Duffing-type oscilltor y mens of softwre Spring illustrte the dvntge nd systemticity of the reserch of dynmicl systems pplying the method of complete ifurction groups.

6 Results nd discussion This work shows the dvntge of the method of complete ifurction groups in comprison to usully pplied methods of hrmonic lnce nd direct scnning, which do not llow complete predicting of mny nonliner effects, including ifurctions tht led to the occurrence of unstle, su-hrmonic nd chotic solutions. With the exmple of the Duffing-type oscilltor the uthors hve considered nd illustrted new possiilities in the reserch of the irth nd existence of nonliner effects, for exmple, such s multistility, chotic oscilltions nd rre ttrctors. These new possiilities hve ppered owing to the method of complete ifurction groups, which is the sis of the serch of ll sic periodic solutions nd their continution in the investigted prmeter, studying the topology of their ifurctions. Thus, ifurction digrms disply ll ifurction groups, which llow studying the irth of nonliner effects, nlyzing interction of ifurction groups t the chnge of prmeters. typicl element of ifurction group with chotic ttrctor is the res in spce of prmeters with infinite numer of unstle periodic regimes resulting from the cscde of ifurctions of period douling. The systemtic pproch to crrying out ifurction nlysis on the sis of the method of complete ifurction groups llows explining the origin of rre ttrctors - nonliner effects tht re hrd to e dignosed nd which exist in smll intervl of chngele prmeter. Conclusions. The exmple of the Duffing-type oscilltor hs een used to show the dvntge of crrying out ifurction nlysis with the ppliction of the method of complete ifurction groups.. The nlysis of coexistence of vrious ifurction groups hs enled the systemtic study of mechnisms of the irth of nonliner effects in the investigted dynmicl system.. It hs een shown tht the method of complete ifurction groups llows defining wy to chotic oscilltions nd rre ttrctors.. The results of the reserch of ttrctors, the irth of which is closely connected with unstle solutions, cn e used to forecst nd void ctstrophic situtions in electric mechnicl nd other nonliner dynmicl systems. References. Duffing., Erzwungene Schwingungen ei vernderlicher Eigenfrequenz und ihre technische edeutung,. Vieweg, runschweig, /, 8.. Kovcic I., rennn M.J. The Duffing Eqution: Nonliner Oscilltors nd their ehviour. John Wiley nd Sons, Chichester, 0, 0 p.. Ued Y. The Rod to Chos - II, eril Press Inc. - Snt Cruz, 00.. Virtions in Engineering: Reference ook. In 6 volumes. M.: Mchine Engineering, V.. Oscilltions of Nonliner Mechnicl Systems/Edited y I.I. lehmn. 7, p.5 (in Russin). 5. Lkshmnn M., Rjsekr S. Nonliner Dynmics. Integrility, Chos nd Ptterns. Springer- Verlg, erlin, Heidelerg, Zkrzhevsky M., Ivnov Yu., rolov V. NLO: Universl Softwre for lol nlysis of Nonliner Dynmics nd Chos. In Proceeding of the nd ENOC, Prgue 6. v.. pp Zkrzhevsky M. New concepts of nonliner dynmics: complete ifurction groups, protuernces, unstle periodic infinitiums nd rre ttrctors. Journl of Viroengineering, Volume 0, Issue, 008, pp Schukin I., Zkrzhevsky M., Ivnov Yu., Kugelevich V., Mlgin V., rolov V. ppliction of softwre SPRIN nd method of complete ifurction groups for the ifurction nlysis of nonliner dynmicl system. Journl of Viroengineering, Volume 0, Issue, 008, pp Smirnov R.S., Zkrzhevsky M.V., Schukin I.T., Yevstignejev V.Yu. The Influence of Nonliner Dissiption on the irth of Rre ttrctors in Nonliner Dynmics. Interntionl Symposium R on Rre ttrctors nd Rre Phenomen in Nonliner Dynmics, Rīg, 0, pp Smirnov R. The trnsition to chos nd criteri of chotic ehvior in dynmicl systems with liner nd nonliner dmping: DYVIS-0 - the Dynmics of Viroimpct (Strongly Nonliner) Systems, Russin cdemy of Sciences, Moscow- Klin, 0, pp Smirnov R., Zkrzhevsky M., Schukin I. lol nlysis of the Nonliner Duffing-vn der Pol Type Eqution y ifurction Theory nd Complete ifurction roups Method. Viroengineering Procedi, 0, Vol.6, Iss.,.-.lpp. ISSN 5-05.