World Journl of Applied Physics 08; (: -50 http://www.sciencepublishinggroup.com/j/wjp doi: 0.68/j.wjp.0800. ISSN: 67-5990 (Print; ISSN: 67-6008 (Online Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion Hervé Lucs Koudhoun, *, Yélomè Judicël Fernndo Kpomhou, Jen Aknde, Dmien Kêgnidé Kolwolé Adjï Deprtment of Physics, University of Abomey-Clvi, Abomey-Clvi, Benin Deprtment of Industril nd Technicl Sciences, University of Abomey, Abomey, Benin Emil ddress: * Corresponding uthor To cite this rticle: Hervé Lucs Koudhoun, Yélomè Judicël Fernndo Kpomhou, Jen Aknde, Dmien Kêgnidé Kolwolé Adjï. Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion. World Journl of Applied Physics. Vol., No., 08, pp. -50. doi: 0.68/j.wjp.0800. Received: June 9, 08; Accepted: July, 08; Published: August 6, 08 Abstrct: In this pper, chotic dynmics of cubic-quintic-septic Duffing oscilltor subjected to periodic excittion is investigted. The multiple scles method is used to determine the vrious resonnce sttes of the model. It is found tht the considered model posses thirteen resonnce sttes whose seven re thoroughly studied. The stedy-stte solutions nd theirs stbilities re determined. The frequency-mplitude curves show tht the considered system presents mixed behvior, limit cycles, hysteresis, jump nd bifurction phenomen. It is lso noticed tht these phenomen re strongly influenced by quintic-septic nonlinerity nd excittion mplitude. Bifurction structures displyed by the model for ech considered type of resonnt sttes re investigted numericlly using the fourth-order Runge-Kutt lgorithm. As results, the quintic-septic nonlinerity, liner dissiption nd excittion mplitude cn be used to control the chotic behvior of the system. Keywords: Extended Duffing Oscilltor, Resonnce Sttes, Stbility, Limit Cycles, Bifurction nd Jump Phenomen, Periodic nd Qusi-periodic Oscilltions, Chos. Introduction In recent decdes, the studies on chotic dynmics of nonliner dissiptive systems hve received considerble ttention of mny investigtors in order to understnd the complex dynmicl behvior of these systems. These studies re of gret importnce becuse chos hs lrge prcticl pplictions in sciences but lso in engineering nd in technology. For instnce chos hs been used in physics for refining the understnding of plnetry orbits nd lso for forecsting the intensity of solr ctivity []. In medicine, biology nd physiology, the chotic behvior hs been used to detect nd explin some physiologicl phenomen [-]. In meteorology, chos cn be used to indicte situtions of better or worse predictbility. In nnotechnology, the width of the conductnce correltion function is determined by the verge lifetime of trnsient chos. Moreover in this discipline, chos hs been used to mke the chnnels of few hundred microns in cross-section [5]. In communiction, chos hs been used to trnsmit informtion [6]. In engineering, chotic dynmics ws used for controlling the temperture of kerosene fn heters nd the direction of ir flow of ir conditioners to provide useful fluctutions tht re expected to be comfortble to humns [7-8], etc. In view of the existence nd importnce of chos in vrious brnches of sciences nd engineering, quntittive or qulittive nlysis of chotic systems is become chllenge for investigtors. There re some numericl tools such s bifurction digrm, phse portrit, Poincré section mp, Lypunov exponent digrm nd so on tht re used to mesure, detect, predict or quntify the chotic behvior of dynmicl systems. Mny investigtors in the study of chotic dynmics hve used Lypunov exponent s good indictor of chos. According to this tool, positive vlue of Lypunov exponent is
World Journl of Applied Physics 08; (: -50 5 quntittive mesure of chos [9]. Frnkly speking, mong the three fundmentl forced oscilltors such s Vn der Pol, Ryleigh nd Duffing oscilltors, this ltter with ϕ potentil hving hrdening or softening nonlinerity u" + µ u' + ( bu u = f cos( Ω τ ( the prime ('represents differentition with respect to time, hs been intensively used in context of vrious physicl nd engineering problems due to it rich vriety of nonliner dynmicl behvior such s hysteresis, bifurction, regulr nd chotic behviors nd its potentil pplictions [0-7]. Recently gin it hs been shown tht Duffing oscilltor with 6 ϕ potentil exhibits for pproprite prmeter choices wide rich vriety of dynmicl behvior tht the dynmicl system described by eqution ( cnnot disply [8-9]. Furthermore it hs been lso shown tht the quintic term of 6 ϕ Duffing oscilltor is responsible to produce the hrdening-softening or softening- hrdening behvior so-clled mixed behvior. The uthors of this pper hve underlined tht this behvior highly unstble, cn be dvntgeous or disdvntgeous ccording to the type of pplictions [0,, ]. In the light of ll the bove, forced dmped Duffing oscilltor remins up to now very importnt nonliner model to exmine in the study of chotic dynmics in nonliner dissiptive systems. Therefore we consider in this work the following eqution: α u + µ u + u + u u = F cos( Ωτ The prticulrity of this model is tht it contins one prmeter controlling the quintic nd septic nonlinerities simultneously. This model cn rise in the modeling of clssicl prticle in double-well potentil []. It ws lso used to model the dynmic behvior of crgo system []. It is interesting to point out tht this model without dmping hs been obtined in the modeling of the purely elstic structures whose the tension ws relted to the norml strin by power lw [5]. The eqution ( is n extended form of cubic dmped Duffing oscilltor defined by (. From this eqution, when α = 0, the softening Duffing oscilltor with ( ϕ potentil describes by eqution ( is obtined with = nd b =. As foregoing mentioned, vrious studies hve been dedicted to the softening Duffing oscilltor. The interest in this system lies in the vriety of physicl phenomen tht it models, such s the rolling motion of ship [6-7] nd the fct tht it is isomorphic with other systems of importnce in physics nd engineering [8]. The present study is motivted by the lrge vriety of dynmicl behvior of softening Duffing oscilltor [] nd lso by the fct tht the chotic dynmics of high-order Duffing oscilltor hve not been intensively studied up to now [8]. Therefore, in this work the following question is necessry to be sked in order to understnd the chotic behvior of dynmicl system governed by (: is tht the prmeter α controlling the quintic nd septic nonlinerities simultneously cn ffect the dynmics of softening Duffing oscilltor eqution? In order to nswer to this eqution, the uthors of this pper suppose tht α hs significnt effect on the dynmics of the softening Duffing oscilltor with ϕ potentil. To verify this ssumption, we determine the vrious resonnt sttes of ( by mens of the multiple scles method s well s the influence of the significnt prmeters on the ech stedy-stte solution (Section. We fterwrd nlyze its bifurction nd trnsition to chos using numericl simultions (Section. Finlly we present the conclusion of this work (Section.. Resonnce Sttes In this section, we use the multiple scles method [9] for investigting the vrious resonnce sttes of ( s well s the effects of the significnt prmeters on the ech stedy-stte response. To pply this method, we introduce into eqution ( the smll perturbtion prmeter ε such s 0 ε <. Thus, eqution ( cn be written s follows α 5 u" + u = ε µ u + u u u + F cos( Ωτ µ = εµ nd α = εα. Now, we seek n uniform first-order solution of eqution (in the form: 0 0 0 ( u( τ, ε = u ( T, T + εu ( T, T + 0( ε ( T o = τ nd T = ετ represent the fst nd slow time scles respectively. In terms of the new time scles, the time derivtives become d D0 εd 0( ε τ = + + & d d dτ 0 εd0d ε = D + + 0( (5 Where Dn =, n = 0, represents differentil opertor. T.. Primry Resonnt Stte n In this prt, we put tht F = εf. The reltionship between both nturl nd externl frequencies is given by ω = + εσ, σ is the detuning prmeter. Inserting equtions ( nd (5 into ( nd equting the coefficients of like power of zero-order 0 ε nd first-order 0 D0u0 u0 ε on both sides, we obtin: ε : + = 0 (6 ε : D0u + u = D0Du 0 µ Du 0 0 + u0 α 5 u0 u0 F cos( τ + Ω The solution of eqution (6 cn be expressed in the complex form: it (, ( ( 0 0 (7 0 i u T T = A T e + A T e (8
6 Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion A is the complex conjugte of A. Substituting the solution (8 into (7yields to α 5 Du 0 + u = ida iµ A + A A 0A A A A e F e iω + + NST + cc. NST denotes the terms does not produce seculr terms nd cc. designtes the complex conjugte. Introducing the primry resonnce condition given bove into (9nd eliminting seculr terms, we obtin the following eqution: α 5 F ia i A + A A A A A A e σ + = i T µ 0 0 i (9 (0 the prime ( denotes the derivtives with respect to T. Writing A in the following polr form: ( ( i T A = T e θ ( T ( nd θ ( T re mplitude nd phse response respectively. Inserting eqution ( into (0 nd seprting rel nd imginry prts, we obtin the modultion equtions: F µ = sinϕ ( α 5 5 6 F ϕ σ = + cosϕ 6 6 56 + ( ϕ = σt θ, The stedy-stte response cn be obtined by putting = ϕ = 0 in equtions ( nd (. Thus, the stedy-stte mplitude nd phse re governing by the following equtions: F µ sin ϕ = ( F α 5 5 6 cosϕ = σ + 6 6 56 (5 with 0. For non-trivil solution, eliminting of ϕ from equtions ( nd (5, yields to the following eqution: α 5 5 6 F µ σ = + 6 6 56 ± (6 Eqution (6 represents the frequency-response eqution of the primry resonnce cse. The pek mplitude of the system under study t the primry resonnce obtined from F (6 is given by p = which is inversely proportionl to µ µ. From this eqution, we cn sy tht incresing of dmping fctor µ cn decrese the vlue of p. Furthermore we p notice tht the coefficient of quintic- septic nonlinerity does not ffect the pek mplitude of the primry resonnce response. However the corresponding loction of the pek mplitude, depend on α, µ nd F ccording to the following reltionship: F α 5F σp = 5F µ 6 6µ Equtions ( nd (5 show tht is no trivil solution t = 0. To determine the stbility of the nontrivil solution, let, nd = + (7 0 ϕ = ϕ + ϕ (8 0 ϕ re slight vritions. Inserting equtions (7 nd (8into equtions ( nd ( nd cnceling nonliner terms, we obtin the eigenvlues of the corresponding Jcobin mtrix, which re the roots of: λ + µλ + P = 0 (9 µ 5α 7 9 5α 9 P = + ( ( σ + σ 6 6 6 + 6 6 6 According to the Routh-Wurwitz criteri for stbility [0], the stedy stte solution for the primry resonnce cse is symptoticlly stble if nd only if µ nd P re greter thn zero. The mplitude response curves of the primry resonnce cse showing the effects of α, µ ndf re plotted in Figures(-(d. In Figure ( we note tht when α increses nd tkes the vlue, the softening behvior displyed by α = 0 persists but the corresponding loction of the pek mplitude decreses. For high vlue of α the hrdening-softening behvior so-clled mixed behvior is obtined with wide decresing of loction of the pek mplitude. Therefore it ppers in the system hysteresis, jump nd bifurction phenomen. For negtive vlue of α, the hrdening behvior is dominnt nd softening-hrdening behvior is lso observed (Figure (b. From these observtions we cn conclude tht the prmeter α cptures more the nonlinerity of the system understudy nd is effectively responsible of mixed behvior. When the excittion mplitude F decreses, the mximum vlue of the mplitude decreses considerbly nd hysteresis, jump nd bifurction phenomen dispper in the system (Figure (c. In Figure (d we observe the inverse vritions illustrted in Figure(c. For certin vlues of the system prmeters, the model cn exhibit limit cycles whose topology is perturbed by the quintic-septic nonlinerity s illustrted in Figures (-(d. In these figures, we observe periodic limit cycles oscilltions when α is positive nd different shpes of qusi-periodic stte signture for negtive vlues of α... Other type of Resonnt Sttes In this subsection, we consider tht the periodic externl force is wide, tht is to sy ( F = ε F. In this condition other 0
World Journl of Applied Physics 08; (: -50 7 type of oscilltions cn be displyed by the model, nmely sub-hrmonic, super-hrmonic, sub-super-hrmonic nd super-sub-hrmonic oscilltory sttes. Applying the multiple time scles method, we obtin the following system: 0 0 0 0 ε : D u + u = F cos( Ω τ (0 : D0u + u = D0Du 0 Du 0 0 + u0 ε µ α u u 5 0 0 The generl solution of eqution (0 is: F Λ = i iω ( 0 0 ( u T, T = AT ( e + Λ e + cc. ( ( Ω. Substituting ( into ( yields to: α D u u ida i A A A A A A 5 + 60Λ A A + 0Λ A A A 0Λ A A 0 + = [ µ + + Λ (0 6 5A AΛ 70 AΛ ] e i α + [ iµ Ω + Λ AA + Λ (0Λ A A + 60Λ AA 5 + 0Λ 70ΛA A 5Λ A A 5 5 7 iω 0 AAΛ Λ ] e α 05 + [ ΛA (0Λ AA + 0Λ A ΛA A 5 0Λ AA 05 A Λ ] e 5 i( Ω i(ω α 05 6 + [ Λ A (0Λ AA + 0Λ A Λ A 0Λ A A 0 Λ AA ] e Λ α 5 + [ (0Λ AA + 5Λ + 05Λ A A 5 7 iω 05 Λ AA Λ ] e α 5 05 i( Ω [5ΛA ΛAA Λ A ] e α i(ω [0Λ A 05Λ AA 70 Λ A ] e α 05 5 i(ω [0Λ A Λ A 70 Λ AA ] e α 05 6 i(ω [5Λ A Λ AA Λ Ae ] α 5 7 7 5 Ω 7 6 [ Λ Λ Λ AAe ] + αλa e 8 5 i(ω 5 T 0 5 i(5ω + αλ A e + αλ A e 8 7 6 i(6ω T 5 0 i( Ω + αλ Ae + αλ A e 8 8 5 i(ω T α 0 7 7iΩ + αλ A e + Λ e + NST + cc. 8 8 5 i i( Ω 6 ( resonnces sttes such s: Sub-hrmonic Ω = + εσ ; Ω = 5 + εσ ; Ω = 7 + εσ b Super-hrmonic Ω = + εσ ; 5Ω = + εσ nd 7Ω = + εσ c Sub-super-hrmonic Ω = 6 + εσ ; Ω = + εσ nd Ω = 5 + εσ d Super-sub-hrmonic Ω = + εσ ; 5Ω = + εσ nd 6Ω = + εσ.... Sub-hrmonic Resonnt Sttes Considering in this prgrph Ω = + εσ, nd injecting this condition into ( nd setting seculr terms equl to 0, we obtin: NST is non seculr term nd c. c denotes the complex conjugte term. Evlution of ( shows tht the system presents three sub-hrmonics, three super-hrmonics, three sub-super-hrmonics nd three super-sub-hrmonics
8 Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion Figure. Frequency-response curves for primry resonnce showing the effects of: (, b α; (c F nd (d µ. Figure. Effects of the quintic-septic nonlinerity on the limit cycles oscilltions with the prmeters µ = 0.0, F = 0.00 nd Ω =. ( α = 0; (b α = 0.; (c α = 0.0 nd (d α = 0. ( α 5 6 ida iµ A + A A + Λ A 0A A + 60Λ A A + 0Λ A A A 0Λ A A 5A AΛ 70AΛ + α 05 5 i T A ( 0 AA 0 A A A 0 AA 05A Λ Λ + Λ Λ Λ Λ e σ = 0 (
World Journl of Applied Physics 08; (: -50 9 Using the polr form into ( nd seprting rel nd imginry prts, we obtin the following modultion equtions: nd nd = µ + Qsinθ (5 Q θ = σ Y + cosθ (6 α 5 5 Y = Λ 5 6 + Λ + Λ 6 5 6 05 5 6 Λ Λ 5Λ 56 8 5 5 05 6 05 05 5 Q α = Λ 8 Λ + Λ Λ Λ Λ 8 8 θ = σt β ( T. For stedy stte solution, = θ = 0, in (5 nd (6 we get: Qsinθ = µ (7 σ Q cosθ = + Y (8 Equtions (7 nd (8 show tht there re two possibilities: (trivil solution t = 0 nd (nontrivil solution t 0. Squring nd dding (7 nd (8 we get the frequency -response eqution for sub-hrmonic resonnce of order : Q µ σ = Y ± (9 The stbility nlysis of the trivil solution is equivlent to the nlysis of the liner solution of eqution ( by neglecting the non-liner terms. Thus, we obtin: ( ia iµ A + Λ A + 0Λ 70Λ = 0 (0 iσt To solve (0, we inject A = e ( BT ( + ibt ( B nd b re rel into eqution (0. Seprting rel nd imginry prts, we get: ( µ 5 B = B + σ α 7 b + Λ Λ Λ ( ( 5 µ b = σ α 7 B b + Λ Λ Λ ( The eigenvlues of the Jcobin mtrix stisfy the eqution: λ + µ λ + P = 0 ( ( µ 5 P = + σ α 7 + Λ Λ Λ. According to the Routh-Wurwitz criteri for stbility, the trivil stedy-stte solution for the sub-hrmonic resonnce of order is symptoticlly stble since 0 µ > nd P > 0. The stbility of the nontrivil stedy-stte sub-hrmonic resonnce of order is determined by the eigenvlues of the corresponding Jcobin mtrix, which re the roots of nd λ + P λ + P = ( 0 Q ( µ 0 = 0 Q ( 0 P Q ( 0 P = 0 µ ( Y( 0 σ 0 ( Y( 0 σ Y ( 0 Q ( 0 + + According to the Routh-Wurwitz criteri for stbility, the nontrivil stedy-stte solution for the sub-hrmonic resonnce of order / is symptoticlly stble if nd only if P > 0 nd P > 0. The frequency-response curves for sub-hrmonic resonnce of order/illustrting the effects of the prmeters α, F nd re represented in Figures (-(d. In Figure ( we observe tht when α increses the softening behvior observed in the cse α = 0 chnge into hrdening behvior, tht is to sy the response mplitude of the sub-hrmonic resonnce, which consist of two brnches move to the right. Then it ppers in the system jump nd bifurction phenomen. It is importnt to point out tht this incresing of α ffect the mximum vlue of the response mplitude nd the loction of minimum vlue of the response mplitude. For negtive vlue of α the softening behvior is dominnt but the loction of minimum vlue of the response mplitude increses while the mximum mplitude vlue decreses (Figure (b. From Figure (c we note tht jump nd bifurction phenomen exist since the hrdening behvior is obtined. Moreover the resonnce bndwidth nd the loction of minimum vlue of the response mplitude increse with incresing of F. However the mximum
0 Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion response mplitude decreses when the excittion mplitude F increses. When µ increses, the minimum vlue of the response mplitude highly increses but this vrition of µ hs no effect on the mximum vlue of the resonnce response (Figure (d. sub-hrmonic Figure. Frequency-response curves for / sub-hrmonic resonnce illustrting the effects of: (, b α;(cf nd (d µ. Considering the cse Ω = 5 + εσ, nd inserting this reltion into ( nd eliminting seculr terms, we get α ida iµ A + A A + Λ A (0A A 5 + 60Λ A A + 0Λ A A A 0Λ A A 6 5A AΛ 70 AΛ α 5 05 i T [5ΛA ΛAA Λ A ] e σ = 0 (5 ( Using the polr form ( i T A = T e β into eqution (5 nd seprting rel nd imginry prts, we obtin the following modultion equtions: = µ + Rsinθ (6 5R θ = σ 5Y + cosθ (7 α 5 6 R Λ Λ Λ = 6 6 nd θ = σt 5 β ( T Introducing the stedy-stte condition = θ = 0, into equtions (6 nd (7, we obtin: Rsinθ = µ (8 σ R cosθ = + Y (9 5
World Journl of Applied Physics 08; (: -50 Equtions (8 nd (9 show tht there re two possibilities: (trivil solution t = 0 nd (nontrivil solution t 0. The stbility of the trivil stedy-stte sub-hrmonic resonnce of order is the sme tht those exmined in the 5 nd R ( µ 0 5 = 0 R ( 0 P cse of the sub-hrmonic resonnce of order. Squring nd dding (8 nd (9 we get fter some mthemticl mnipultions the following frequency-response eqution of the sub-hrmonic resonnce of order 5 : R µ σ = 5 Y ± (0 The stbility of the nontrivil stedy-stte sub-hrmonic resonnce of order is exmined by the eigenvlues of the 5 corresponding Jcobin mtrix, which re the roots of λ + P λ + P = ( 5 6 0 R ( 0 P6 = 0 µ ( 5 Y( 0 σ 0 ( 5 Y( 0 σ Y ( 0 R( 0 + + 5 If the two inequlities P 5 > 0 nd P 6 > 0 re stisfied then the stedy-stte solution will be symptoticlly stble. Figures (-(c present the effects of α nd F on the sub-hrmonic resonnce solution of order. In Figure ( 5 we note tht the frequency-response curves of this cse of oscilltion is obtined for high positive vlues of detuning prmeter. In this figure we observe tht when α increses positively the response mplitude curves move to the right with lrge incresing of the resonnce frequency. When α tkes negtive vlue the obtined vritions re contrry to those shown in Figure (b. From Figure (c we note the sme vritions s illustrted in Figure ( but with high vlue of excittion mplitude.... Super-hrmonic Resonnt Sttes Using the super-hrmonic resonnce reltion Ω = + εσ into eqution (, the condition for eliminting seculr terms in the problem is given by: ( α 5 ida iµ A + A A + Λ A 0A A + 60Λ A A + 0Λ A A A 0Λ A A 5Λ A A 70AΛ Λ α ( 0 5 05 05 7 i T + Λ AA + Λ + Λ A A Λ AA Λ e σ = 0 6 ( ( Using the polr form ( i T A = T e β into eqution ( nd seprting rel nd imginry prts, we obtin the following modultion equtions: Where = µ + M sinθ ( M θ = σ Y + cosθ ( Λ α 5 05 05 5 7 M = 5 5 Λ + Λ + Λ Λ Λ 6 frequency-response eqution for super-hrmonic resonnce of order. M µ σ = Y ± (7 From equtions (5 nd (6 we notice tht there only exists non-trivil solution t 0. The stbility of the stedy-stte super-hrmonic resonnce of order is determined by the eigenvlues of the corresponding Jcobin mtrix, which re the roots of λ + P λ + P = (8 7 8 0 nd θ = σt β ( T. Using the stedy-stte solution, = θ = 0 in equtions ( nd (, we get: M sinθ = µ (5 M cosθ = σ + Y (6 After few lgebric opertions we obtin the nd M ( µ 0 7 = 0 M ( 0 P M ( 0 P8 = 0 µ + ( Y ( 0 σ + 0 ( Y ( 0 σ Y ( 0 M ( 0
Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion Figure. Frequency-response curves for /5 sub-hrmonic resonnce exhibiting the effects of: (, b α nd (cf. According to the Routh-Wurwitz criteri for stbility, the stedy-stte super-hrmonic resonnce of order is symptoticlly stble if nd only if P 7 > 0 ndp 8 > 0. Figure 5. Frequency-response curves for super-hrmonic resonnce of order showing the effects of: (, b αnd (c F. Figures 5(-(c show the effects of α nd F on the super-hrmonic resonnce response of order. In Figure 5(, we notice tht when α tkes the vlue the response mplitude consists of two curves move to the right. Therefore
World Journl of Applied Physics 08; (: -50 hrdening behvior is obtined nd jump nd bifurction phenomen exist. However the hysteresis phenomenon disppers in the system. Furthermore incresing of α reduced considerbly the mximum vlue of the response mplitude nd when α = 5, the resonnce curves touch t one loction. At this loction, the corresponding mplitude is criticl. For negtive vlue of α softening behvior becomes dominnt with decresing of the response mplitude. The resonnce curves touch t different loctions nd become open like those of the undmped oscilltor (Figure 5(b. From Figure 5(c we note tht incresing of F decreses the mximum vlue of the response mplitude but the resonnce bndwidth increses. Moreover we observe tht when F = the lower brnch of the super-hrmonic resonnce of order disppers on the considered intervl for detuning prmeter σ. Another cse of oscilltion 5Ω = + εσ is treted in this prt. Injecting this condition into ( nd eliminting the resulting seculr we get: Λ 7 i T ( α 5 6 α 5 7 0 60 0 0 5 70 5 σ id A iµ A + A A + Λ A A A + Λ A A + Λ A A A Λ A A Λ A A AΛ Λ Λ AA e = 0 (9 Inserting the polr form of the A into (9 nd seprting rel nd imginry prts, we obtin the following modultion equtions: = µ + S sinθ (50 M θ = σ Y + cosθ (5 α 7 S = Λ Λ Λ 5 5 7 nd θ = σt β ( T Introducing the stedy-stte solution, = θ = 0, into equtions (50 nd (5, we get: S sinθ = µ (5 S cosθ = σ + Y (5 Eliminting of θ from this eqution systems yields to fter few lgebric mnipultions the following frequency-response eqution for super-hrmonic resonnce of order 5: S µ σ = Y ± (5 The stbility of the stedy-stte 5:super-hrmonic resonnce response is determined by the eigenvlues of the corresponding Jcobin mtrix, which re the roots of λ + P λ + P = (55 9 0 0 S ( µ 0 9 = 0 S ( 0 P S ( P0 = 0 µ + Y ( 0 σ + 0( Y ( 0 σ Y ( 0 S( 0 0 And ( According to the Routh-Wurwitz criteri for stbility, the non-trivil stedy-stte solution of the super-hrmonic resonnce of order 5 is symptoticlly stble if nd only if P > ndp 0 > 0. 9 0 The frequency-response curves for super-hrmonic resonnce of order5 exhibiting the effects of α nd F is presented in Figures 6(-(c. In this cse of oscilltion, the response mplitude is obtined for lrge vlue of detuning prmeter. From Figures 6( nd (c, the response mplitude consists of two curves which move to the right nd hve hrdening behvior. Therefore jump nd bifurction phenomen exist in the system. For negtive vlue of α resonnce curves move to the left pointing out tht the system displys the softening behvior (Figure 6 (b.... Sub-super- nd Super-sub-hrmonic Resonnt Sttes In this prgrph we consider the cse of sub-super-hrmonic resonnce Ω nd therefore set Ω = + εσ. Eliminting seculr terms from (, we obtin the following eqution: α ida iµ A + A A + Λ A 0A A + 60Λ A A + 0Λ A 5 6 A A 0Λ A A 5Λ A A 70AΛ α ( 0 05 70 i T Λ A Λ AA Λ A e σ = 0 ( (56 Injecting the polr form of thea into eqution (56 nd seprting rel nd imginry prts, we obtin the following modultion equtions: = µ + Lsinθ (57 L θ = σ Y + cosθ (58 5αΛ L = 7 6 Λ 8 nd θ = σt β ( T For stedy stte solution, = θ = 0, in equtions (57 nd (58, we obtin:
Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion Lsinθ = µ (59 Lcosθ = σ + Y (60 Equtions (59 nd (60 show tht there re two possibilities: (trivil solution t = 0 nd (nontrivil solution t 0. Squring nd dding (59 nd (60 we get the following frequency-response eqution fter few mthemticl mnipultions: L µ σ = Y ± (6 The stbility of the trivil sub-super-hrmonic resonnce of order is the sme tht those obtined in the cse of the sub-hrmonic resonnce of order. However the stbility of the nontrivil stedy-stte response is nlyzed by the eigenvlues of the corresponding Jcobin mtrix, which re roots of λ + P λ + P = (6 0 5 L ( µ 0 = 0 L( 0 P ( L ( 0 P = 0 µ + ( Y ( 0 σ + 0 ( Y ( 0 σ Y ( 0 L( 0 According to the Routh-Wurwitz criteri for stbility, the stedy-stte solution of the sub-super-hrmonic resonnce of order is symptoticlly stble if nd only if P > 0 nd P > 0. The frequency-response curves for sub-super-hrmonic resonnce of order displying the effects of α nd F re represented in Figures 7(-(c. In Figure 7( we observe tht the response mplitude is obtined for high positive vlues of detuning prmeter. This response composed of two brnches move to the right. Then hrdening behvior nd bifurction phenomenon exist in the system. Furthermore when α increses the resonnce bndwidth nd the resonnce frequency increse. When α tkes negtive vlue the softening behvior is dominnt in the system (Figure 7(b. In Figure 7(c when F increses, we notice the sme vritions illustrted in Figure 7(. Figure 6. Frequency-response curves for super-hrmonic resonnce of order 5 showing the effects of: (, b αnd (cf.
World Journl of Applied Physics 08; (: -50 5 α ida iµ A + A A + Λ A 0A A + 60Λ A A + 0Λ A 5 6 A A 0Λ A A 5Λ A A 70AΛ α ( 05 5 6 i T Λ A Λ AA Λ A e σ = 0 ( (6 Inserting the polr form of the A into (6 nd seprting rel nd imginry prts, we obtin the following modultion equtions:, = µ + K sinθ (6 K θ = σ Y + cosθ (65 5 05 6 K α = Λ Λ Λ ndθ = σt 6 β ( T For stedy-stte solution, = θ = 0, in (6 nd (65 we obtin: K sinθ = µ (66 K cosθ = σ + Y (67 Equtions (66 nd (67 show tht there re two possibilities: (trivil solution t = 0 nd (nontrivil solution t 0. Squring nd dding (66 nd (67, we get fter few mthemticl opertions the following frequency-response eqution: K µ σ = Y ± (68 The stbility nlysis of the trivil solution is equivlent to the nlysis of the liner solution of (6 by neglecting the non-liner terms. Thus we get: α µ + Λ Λ Λ + Λ Λ = 6 6 i T ia i A A 0 A 70 A (5 A A e σ 0 (69 Figure 7. Frequency-response curves for / sub-super-hrmonic resonnce exhibiting the effects of: (, b α nd (c F. At present we consider the cse of super-sub-hrmonic resonnce Ω nd therefore set Ω = + εσ. Eliminting seculr terms from (, we obtin the following eqution: i σt To solve (69 one lets A e ( BT ( ibt ( = + B nd b re rel nd imginry prts, nd we get fter inserting A into (69, the following liner equtions system: B µ B α = + σ (5 9 b + Λ Λ Λ b α = σ (5 9 B µ b + Λ Λ Λ (70 (7 The chrcteristic eqution of the corresponding Jcobin
6 Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion mtrix is given by P λ + µλ + P = 0 (7 µ α = + + Λ Λ (5 9 Λ σ According to the Routh-Wurwitz criteri for stbility, the trivil solution of the super-sub-hrmonic resonnce of order is symptoticlly stble since P > 0 nd µ is here positive. The stbility of the nontrivil stedy-stte super-sub-hrmonic resonnce of order given by (68 is determined by the eigenvlues of the chrcteristic eqution, which re the roots of: λ + P λ + P = (7 5 0 µ K ( 0 P = 5 0 K( 0 nd K ( 0 P5 = 0 µ + ( Y( 0 σ + 0 ( Y ( 0 σ Y ( 0 K( 0 The nontrivil stedy-stte solution for the super-sub-hrmonic resonnce of order is symptoticlly stble if nd only if P > 0 ndp 5 > 0. Figures 8(-(c represent the effects of α nd F on the super-sub-hrmonic resonnce of order. In this oscilltion cse, the response mplitude is obtined for high positive vlues. In Figure 8( nd (c, the response mplitude consists of two brnches which bend to the right nd hve hrdening nonlinerity. Therefore it ppers in the system jump nd bifurction phenomen. However the hysteresis phenomenon does not exist. In these figures, when α nd F increse, the resonnce bndwidth increses. For negtive vlue of α, the response mplitude displys softening behvior (see Figure 8(b. Figure 8. Frequency-response curves for super-sub-hrmonic resonnce of order illustrting the effects of: (, b αnd (cf.. Bifurction nd Trnsition to Chos Since the forced dmping Duffing oscilltor is subjected to quintic-septic nonlinerity, then the complex phenomen must rise in the model described by ( for different resonnt sttes. Therefore the im of this section is to investigte the conditions under which the complex phenomen rise in the model becuse they re of interest in mny physics nd engineering problems. For this purpose the fourth-order Runge-Kutt lgorithm is used to solve numericlly nd drw the bifurction digrm nd its corresponding lrgest Lypunov exponent of the (. The lrgest Lypunov exponent tht is used s good mesure of chrcteriztion of chos in the system is defined s follows: ( du + dvu ln Ly = lim (7 t t Where du nd dvu re the vritions of u nd u respectively. The time periodic of the periodic stroboscopic π bifurction digrm used to mp the trnsition is T =. Ω
World Journl of Applied Physics 08; (: -50 7 For set of prmeters α = 0.5, µ = 0.5 nd Ω =, the bifurction digrm nd its corresponding Lypunov exponent re represented in Figure 9. From these figures we cn notice tht the existence of periodic, qusiperiodic nd chotic motions. In order to hve n ide on the system behvior predicted by these digrms, vrious phse portrits re plotted for severl different vlues of the excittion mplitude F using the prmeters of Figure 9 (see Figures0 nd. Figure 9. Bifurction digrm nd its corresponding Lypunov of n extended Duffing oscilltor versus F with α = 0.5, µ = 0.5; Ω =. Figure 0. Periodic orbits of n extended Duffing oscilltor with the prmeters of Figure 9. (: Orbit T, F = ; (b: Orbit T, F =.75; (c:orbit T, F = 0 nd (d: Orbit T, F = 5. Figure. Chotic orbits of n extended Duffing oscilltor with the prmeters of Figure 9. (: F = 5.5 (b: F = 8.0;(c: F = 6.5 nd (d: F =.5.
8 Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion The influence of the quintic-septic nonlinerity nd liner dmping coefficients on the bifurction sequences re investigted nd reported in Figures. From these figures it is found tht decresing of the dissiption coefficient µ ccentutes the chotic motions while incresing of quintic-septic coefficient α reduces the choticity of the system but increses the regions of the periodic oscilltions. In order to detect the resonnt sttes the chotic motions re dominnt, the bifurction digrms showing the vritions of u versus Ω for different vlues of the excittion mplitude F re presented in Figures. We cn note tht the number of bifurctions depend on the vlue of F. Furthermore the chotic motions re lso influenced by the excittion mplitude nd resonnt frequency vlues. To illustrte such sitution, the phse portrits re represented for different vlues of the excittion mplitudef (see Figures. We observe tht for F = 8.0 the system under study hs the possibility to disply chotic motions for the primry resonnce nd super-sub-hrmonic resonnce of order. Moreover when F =.5, the system presents in ddition chotic behvior t the sub-super-hrmonic resonnce of order. Figure. Bifurction digrms of n extended Duffing oscilltor versus Ω with the prmeters of Figure 9. (: F = 8.0 nd (b: F = 9.5.. Conclusion Figure. Effect of α = 0.0 (left µ = 0.(right on the Bifurction digrm nd its corresponding Lypunov with the prmeters of Figure 9. In this work we hve presented the study of chotic dynmics of cubic-quintic-setpic Duffing oscilltor under periodic excittion. The prticulrity of this study is linked t fct tht one only prmeter is here used to control the quintic-septic nonlinerity. The multiple scles method pplied to this extended Duffing eqution hs generted thirteen resonnce sttes which of seven re thoroughly studied. The modultion equtions of ech considered resonnce oscilltion cse re derived. The stedy-sttes solutions nd theirs stbilities re determined. The frequency-mplitude curves obtined show tht the model displys mixed behvior, limit cycles, hysteresis, bifurction nd jump phenomen. It is found tht these phenomen re considerbly ffected by quintic-septic nonlinerity coefficient nd excittion mplitude. The numericl simultions directly performed on the model through bifurction nd Lypunov exponent digrms hve reveled tht periodic, multi-periodic oscilltions nd chotic motions tke plce in the system. These motions predicted by these digrms re confirmed by vrious phse portrits. The effects of liner dissiption µ, quintic-septic nonlinerity α nd externl mplitude influence strongly the chotic behvior of
World Journl of Applied Physics 08; (: -50 9 the system. Furthermore it is lso found tht the cpcity of the system hs developed more chotic motions depend on the vlues of excittion mplitude F nd resonnt frequency Ω. Finlly we cn ffirm tht the crried on objective in this work is ttined since the quintic-septic nonlinerity ffects the dynmic of softening dmping Duffing oscilltor. University Press, New York. [6] Hyes, Scott., Grebogi, Celso., Ott, Edwrd nd Mrk, Andre. (99, Experimentl Control of Chos for Communiction, Physicl review letters, 7(, 78-78. [7] Aihr, Kzuyuki. (00. Chos Engineering nd Its Appliction to Prllel Distributed Processing With Chotic Neurl Networks using energy blnce method, Proceeding of the IEEE, 90(5, 99-90. [8] Aihr, Kzuyuki. (0. Chos nd its pplictions, Procedi IUTAM, 5, 99-0. [9] Khn Ayub nd Kumr, Snjy. (08. Study of chos in chotic stellite systems, Prmn-J. Phys, 90(, -9. [0] Mllik, A. K. (00. Response of A Hrd Duffing Oscilltor to Hrmonic Excittion-An Overview, indin institute of technology, khrgpur 70, 8-0, -5. [] Luo, Albert C. J., nd Hung Jinzhe. (0. Period- Motions to Chos in Softening Duffing Oscilltor, Interntionl Journl of Bifurction nd Chos, (, 000-000-6 [] Ghndchi-Tehrni Mrym., Wilmshurst, Lwrence I., nd Stephen, J. Elliote. (05. Bifurction control of Duffing oscilltor using pole plcement, Journl of Vibrtion nd Control, (, 88-85. [] Kovcic, Ivnnd Brennn, Michel J. (0. The Duffing Eqution Nonliner Oscilltors nd their Behviour, John Wiley & Sons, Ltd. [] Lou, Jing-jun., He, Qi-wei., nd Zhu Shi-jin. (00. Chos In The Softening Duffing System Under Multi-Frequency Periodic Forces, Applied Mthemtics nd Mechnics, 5(, -7. [5] Berger, J. E., nd Nunes, G. Jr. (997. A mechnicl Duffing oscilltor for the undergrdute lbortory, Americn Journl of Physics, 65(9, 8-86. [6] Nyfeh, A. H. nd D. T. Mook. (995. Nonliner Oscilltions, John Wiley & Sons, New York. Figure. Phse Portrits of n extended Duffing oscilltor for super-subnd sub-super-hrmonic resonnce sttes with the prmeters of Figure 9. (: F = 8.0, Ω = nd (b: F =.5, Ω =. References [] Monwnou, A. V., Miwdinou, C. H., Aïnmon, C., nd Chbi Orou, J. B. (08. Hysteresis, Qusiperiodicity nd Choticity in Nonliner Dissiptive Hybrid Oscilltor, Interntionl Journl of Bsic nd Applied Sciences, 7(, 7. [] Higgins, John P. (00. Nonliner Systems in Medicine, Yle Journl of biology nd medicine, 75(5-6, 7-60. [] Moon, Frncis C. (00. Chotic Vibrtions An Introduction for Applied Scientists nd Engineers, John Wiley. [] Weiss, Jmes N., Grfinkel, Alin., Spno, Mrk L., nd Ditto, Willim L. (99. Chos nd Chos Control in Biology, The journl of Clinicl Investigtion, 9(, 55-60. [5] Tmstel, nd Gruiz, Mrton (006. Chotic Dynmics n Introduction Bsed on Clssicl Mechnics, Cmbridge [7] Donso, Guillermo nd Celso L. Lder. (0. Nonliner dynmics of mgneticlly driven Duffing-type spring-mgnet oscilltor in the sttic mgnetic field of coil, Europen Journl of Physics, (6, 7-86. [8] Chu, Vivien. (0. Cubic-Quintic Duffing Oscilltors, www.its.cltech.edu/mson/reserch/duf.pdf, -9. [9] Oyesny, Moses O., nd Nwmb, J. I. (0. Stbility nlysis of dmped cubic-quintic Duffing oscilltor, WorldJournl of Mechnics, (, -57. [0] Kcem, N., Bguet, S., Dufour R., nd Hentz, S. (0. Stbility control of nonliner micromchnicl resontors under simultneous primry nd superhrmonic resonnces, Applied Physics Letters, 98(9, 9507-9507-. [] Kcem, N., nd Hentz, S. (009 Bifurction topology tuning of mixed behvior in nonliner micromechnicl resontors, Applied PhysicsLetters, 95(8, 80-80-. [] Elshurf, Amro M., Khirllh, Kreem, Twfik, Hni H., Ahmed, Emir., Ahmed K. S. Abdel Aziz nd Sedky, Sherif. M. (0. Nonliner Dynmics of Spring Softening nd Hrdening in Folded-MEMS Comb Drive Resontors, Journl of Microelectromechnicl Systems, 0(, 9-958.
50 Hervé Lucs Koudhoun et l.: Chotic Dynmics of n Extended Duffing Oscilltor Under Periodic Excittion [] Oyesny, M. O., nd Nwmb, J. I. (0. Duffing Oscilltor with Heptic Nonlinerity under single Periodic Forcing, Int. J. of Mechnics nd Applictions, (, 5-. [] Lesge, J. C., Liu, M. C. (008. On the investigtion of restrined crgo system modeled s Duffing oscilltor of vrious orders, Proceeding of Erly Creer Technicl Conference, ASME, Mimi, Florid, USA. [5] Koudhoun, L. H., Kpomhou, Y. J. F., Adjï, D. K. K., Aknde, J., Rth, B. Mllick, P. nd Monsi, D. M. (06. Periodic Solutions for nonliner oscilltions in Elstic Structures vi Energy Blnce Method, vixr 60v, -. [6] Delenu, Dumitru. (06. Trnsient nd Stedy-Stte Responses for the Ship Rolling Motion with Multiple scles Lindstedt Poincré Method, Mirce Cel Btrn Nvl Acdemy Scientific Bulletin, XIX(, 08-5. [7] Luo, Albert C. J. (00. Dynmicl Systems: Discontinuity, Stochsticity nd Time-Dely, Springer-Verlg, New York. [8] Mrinc, V. nd Herisnu, N. (005. Forced Duffing Oscilltor with Slight Viscous Dmping nd Hrdening Non-Linerity, Fct Universittis, Series: Mechnics, Automtic Control nd Robotics, (7, 5-55. [9] Nyfeh, Ali H. (00. Perturbtion Methods, WILEY-VCH. [0] Hyshi, Chihiro. (986. Nonliner Oscilltions in Physicl Systems, Princeton University Press, Willim Sheet, Princeton, New Jersey 0850. Mc Grw-Hill, Inc.