Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 4, 39, 2001 SYNCHRONISATION EFFECTS AND CHAOS IN THE VAN DER POL-MATHIEU OSCILLATOR Jerzy Warmiński Department of Applied Mechanics, Technical University of Lublin e-mail: jwar@archimedes.pol.lublin.pl Analysis of a nonlinear oscillator system with one degree of freedom, which includes the van der Pol self-excitation term and parametric exci- tation of the Mathieu type is carried out in this paper. Interactions be- tween the parametric and self-excited vibrations for regular and chaotic motion are investigated. Synchronisation areas near the main parame- tric resonance and transition conditions from regular to chaotic motion are determined in this paper. It is also presented that a small external force causes qualitative andquantitative changes in themain parametric resonance and that the external harmonic force transits the system from chaos to regular motion. Key words: vibration, self- and parametric excitation, synchronisation, chaos 1. Introduction Self-excited vibrations can occur in many mechanical systems, for exam- ple in journal bearings lubricated with a thin oil film, flutter of a plane wing, shimmyofvehiclewheels, chatter vibrationsduringmachine cutting.Todescri- be self-excitation, in engineer practice, different types of models are applied. Self-excited vibrations ofmechanical systems are oftenmodelled byRayleigh’s function,whichdependsonly onvelocity of the system (−α+βx′2)x′, however morepopular is vanderPol’smodel (Hayashi, 1964;Tondl, 1978;Awrejcewicz, 1990; Kapitaniak and Steeb, 1991), which depends on the velocity and square of the generalised coordinate (−α+βx2)x′. In literature lots of papers are devoted to interactions of self-excited sys- tems with other types of excitation. The interactions between two different types of vibrations can lead to very interesting phenomena. One of the first 862 J.Warmiński very important works concerned with a self-excited system and external exci- tation, was presented in the monograph by Hayashi (1964). Analysis of the interactions of the van der Pol oscillator with a harmonic external force was carried out in that monograph. An analytical averaging method and analo- gue simulation results, for regular vibrations around harmonic, subharmonic and superharmonic resonance regions were there presented. Synchronisation phenomena depending on pulling in the self-excited vibration frequency by the external force and quasi-periodic limit cycles were observed on the special Hayashi plane. In the last years analysis of the van der Pol-Duffing oscilla- tor forced by external inertial and static forces was presented byAwrejcewicz (1990, 1996). Analysis of characteristic multipliers demonstrated that for sys- tem two types of transition from regular to chaoticmotionwere possible: hard – when one of the multipliers is +1, and soft – after a cascade of period do- ubling bifurcations. Analysis of self-and external excited system with many degree-of-freedom, and their transition to chaotic and hyperchaotic motions werepresentedbyKapitaniakandSteeb (1991). Litak etal. (1999) investigated the interactions in Froude’s pendulum forced externally as well. By applying themethod of multiple scale of time, regular vibrations near the fundamental external resonance were determined. To find the critical value of the external force, which transits the system to chaos, Melnikov’s criterion was applied. Examples self-excited systems forced externally can be found in the works by Guckenheimer and Holmes (1997), Steeb andKunick (1987). Parametric and self-excited vibrations belong to another class of very im- portant dynamical systems. The main difference, when compared with exter- nally forced systems, is that their vibration generally results fromperiodically changing stiffness or mass moment of inertia. For example one can observe such vibrations in shafts whose bending stiffness changes periodically during rotation, crank shaftswithvariablemassmoment of inertia of thewhole crank- piston system, in a wheel of a car with periodically changing radial rigidity of the tyre or in toothed gear systems where the meshing siffeness changes periodically. Hill’s equation or, in a particular case, Mathieu’s equation often approximate this kind of vibration. Parametric and self-excited vibrations of the van der Pol-Mathieu oscil- lator for different nonlinearities were thoroughly analysed by Tondl (1978). For this class of oscillators, the synchronisation phenomenon appeared near parametric resonances regions. The amplitude of the synchronised vibrations was determined near the parametric resonances and a newmethod to describe quasi-periodic limit cycles was applied. The results were verified by an ana- logue simulation. Parametric and self-excited systems were classified byYano Synchronisation effects and chaos... 863 (1989) in three groups, depending on type of nonlinear terms. By applying the harmonic balance method regular motions near the main and fundamental parametric resonances for each class were compared. Rayleigh-Mathieu’s one- degree-of-freedom oscillators with the influence of an external harmonic force or inertial excitation were investigated by Szabelski andWarmiński (1995a,b). The amplitudes of regular vibration and a new effect of additional solutions in the main parametric resonance were presented as well. The review of the literature concludes that the transition from regular motion to chaos was intensively analysed in externally forced self-excited sys- tems. Nevertheless, the results cannot be applied directly to parametric and self-excited systems. Themain goal of this paper is to present new results for a simple one-degree of freedom oscillator, which includes three different ty- pes of excitation, i.e., self-excitation, parametric excitation and forced by an additional external harmonic force. 2. Model of the vibrating system Let us consider a one-degree-of-freedom oscillator, which consists of a non- linear spring with periodically changing stiffness and nonlinear damping de- scribed by the van der Pol model. Let us assume that an external harmonic force can additionally force the oscillator (Fig.1). Fig. 1. Physical model of the van der Pol-Mathieu oscillator The differential equation of motion of the oscillator has the form mx′′+f(x,x′)+(k−k0cos2νt)(x+k1x3)= { 0 Qcos(Ωt+ϕ) (2.1) 864 J.Warmiński where f(x,x′) denotes the van der Pol function f(x,x′)= (−c1+ ĉ1x2)x′ Introducing the dimensionless time τ = pt and a new dimensionless coor- dinate X = x/xst, where p = √ k/m is the natural frequency of the linear system and xst =mg/k means the static displacement, we get Ẍ+Fd(X,Ẋ)+(1−µcos2ϑτ)(X+γX3)= { 0 qcos(ωτ+ϕ) (2.2) In equation (2.2) the following notation is applied α= c1 mp β= c2p m x2st γ= k1x 2 st µ= k0 k q= Q mxst ϑ= ν p ω= Ω p The function Fd(X,Ẋ) is the dimensionless van der Pol damping function Fd(X,Ẋ)= (−α+βX2)Ẋ To find the analytical solutions we assumed that our system is weakly nonlinear and the parameters of the system are expressed by the formal small parameter ε Fd(X,Ẋ)= εF̃d(X,Ẋ) α= εα̃ β= εβ̃ γ= εγ̃ µ= εµ̃ q= εq̃ (2.3) Then, we can write the differential equation of motion in the form Ẍ+X = ε { F̃d(X,Ẋ)+ µ̃cos2ϑτX− γ̃X3+ q̃cos(ωτ+ϕ) } + (2.4) +ε2γ̃µ̃X3cos2ϑτ For ε equal to zero equation (2.4) describes vibrations of the linear oscillator with the natural frequency p equal to 1. For a nonlinear system (ε 6= 0) we have to take into account the first and second order nonlinear terms. Synchronisation effects and chaos... 865 3. Perturbation analysis Analytical examinations of the considered parametric and self-excited sys- tem were carried out by applying multiple scale of time method (Nayfeh, 1981). All analytical calculations were made in Mathematica 3.0 package. At the beginning we define different time scales Tn = ε nτ n=0,1,2, ... where T0 is a fast scale of time and T1,T2, ... are slower scales of time. Then, the time derivatives are expressed by the formulae d dτ =D0+εD1+ ... d2 dτ2 =D20 +2εD0D1+ ... where Dmn = ∂ m/∂Tn means the mth order partial derivative for nth order scale of time. In analytical calculations ε is still used as the small parameter. Solution to (2.4) is expressed in the form X(τ,ε) =X0(T0,T1,T2)+εX1(T0,T1,T2)+ε 2X2(T0,T1,T2)+ ... (3.1) Let us consider the solutions near the parametric resonances. Then the para- metric frequency ϑ has to fulfil the condition mϑ≈np (3.2) where m and n are natural numbers and p is the natural frequency of the system. In our case, after introducing the dimensionless time, p=1, we can write around the parametric resonances m2 n2 ϑ2 =1+εσ1 (3.3) Taking into account (3.3) we have Ẍ+ m2 n2 ϑ2X = ε [ F̃d(X,Ẋ)+ µ̃cos2ϑτX− γ̃X3+ (3.4) +σ1X+ q̃cos(ωτ +ϕ) ] +ε2γ̃µ̃X3cos2ϑτ Putting solution (3.1) into (3.4) and grouping the terms with the same power of εwe obtain a system of consecutive perturbation equations: 866 J.Warmiński —equation of the ε0 order D20X0+ m2 n2 ϑ2X0 =0 (3.5) — equation of the ε1 order D20X1+ m2 n2 ϑ2X1 =σ1X0−2D0D1X0+ F̃d0− γ̃X30 + (3.6) +µ̃X0cos2ϑτ + q̃cos(ωτ+ϕ) — equation of the ε2 order D20X2+ m2 n2 ϑ2X2 =σ1X1−D21X0−2D0D1X1−2D0D2X0+ (3.7) +F̃d1−3γ̃X20X1+ µ̃X1cos2ϑτ + γ̃µ̃X 3 0 cos2ϑτ where F̃d0 = α̃D0X0− β̃D0X0X20 F̃d1 = α̃(D0X1+D1X0)− β̃(X20 D0X1−X 2 0 D1X0+2X0X1D0X0) Further analysis will be carried out for the main parametric resonance around the frequency ϑ≈ 1. Influence of the external force will be considered in a particular case when the external force frequency ω = ϑ and the phase ϕ=0. According to (3.3) for m=1, n=1we can write ϑ2 =1+εσ1 (3.8) Then solution to equation (3.5) is expressed in the exponential form X0(T0,T1,T2)=A(T1,T2)exp(iϑT0)+A(T1,T2)exp(−iϑT0) (3.9) where A(T1,T2) is the complex conjugate function to A(T1,T2). Putting solution (3.9) into (3.6) we obtain D20X1+ϑ 2X1 = [ iϑA(α̃− β̃AA)−2iϑD1A+σ1A−3γ̃A2A+ (3.10) + 1 2 µ̃A+ q̃cosϑτ ] exp(iϑT0)+ [ −(iβ̃ϑ+ γ̃)A3+ 1 2 µ̃A ] exp(3iϑT0)+ c.c. Synchronisation effects and chaos... 867 where c.c. means complex conjugate terms. The condition of secular terms elimination leads to the relation iϑA(α̃− β̃AA)−2iϑD1A+σ1A−3γ̃A2A+ 1 2 µ̃A+ 1 2 q̃=0 (3.11) The particular solution to equation (3.10) has the form X1 = 1 8ϑ2 A ( iβ̃ϑA2+ γ̃A2− µ̃ 2 )exp(3iϑT0) (3.12) Introducing solution (3.12) into equation (3.7) we get D20X2+ϑ 2X2 = =−   D21A+2iϑD2A+(−α̃+2β̃AA)D1A+ β̃A 2D1A− − 3 2 γ̃µ̃AA 2 ( 1 8ϑ2 +1 ) + ( − 1 8 β̃2+ 1 ϑ iβ̃γ̃+ 3γ̃2 8ϑ2 ) A3A 2− − 1 2 γ̃µ̃A3 ( 1 8ϑ2 +1 ) + µ̃2 32ϑ2 A   exp(iϑT0)− (3.13) −   ( − 3 8 iµ̃+ 9 4ϑ iγ̃A2− 5 4 β̃A2 ) D1A+ + 1 4 α̃A3 (3 2 β̃− 3 ϑ iγ̃+ 3 2ϑ iµ̃ ) + 1 16ϑ2 µ̃σ1A− − 1 8ϑ A3 ( 7iβ̃µ̃+ 1 ϑ γ̃σ1 ) − 3 4 A4A ( β̃2+ 1 ϑ2 γ̃2 ) + + 3 ϑ iβ̃γ̃A4A− 3 2 γ̃µ̃A2A ( 1 4ϑ2 +1 )   exp(3iϑT0)− −    − 5 24 iα̃µ̃A+ 1 32ϑ2 µ̃2A− 1 2 γ̃µ̃A3 ( 1 2ϑ2 +1 ) + + ( 3 8ϑ2 γ̃2− 5 8 β̃2+ 2 ϑ iβ̃γ̃ ) A5− 25 24ϑ iβ̃µ̃A2A   exp(5iϑT0)+ c.c. From elimination of secular terms in the solution to equation (3.13) we have D21A+2iϑD2A+(−α̃+2β̃AA)D1A+ β̃A 2D1A− 3 2 γ̃µ̃AA 2 ( 1 8ϑ2 +1 ) + (3.14) + ( − 1 8 β̃2+ 1 ϑ iβ̃γ̃+ 3γ̃2 8ϑ2 ) A3A 2− 1 2 γ̃µ̃A3 ( 1 8ϑ2 +1 ) + µ̃2 32ϑ2 A=0 868 J.Warmiński Applying the reconstitutionmethod (Sanchez andNayfeh, 1990) we trans- form equations (3.11) and (3.14) into one first order differential equation 2iϑȦ+ε ( −iα̃ϑA+iβ̃ϑA2A−σ1A− 1 2 µ̃A+3γ̃A2A)+ +ε2    1 4 A ( −α̃2+ 3 8ϑ2 µ̃2− 1 ϑ2 σ21 ) + 3 2ϑ A2A ( iα̃γ̃+ 1 ϑ γ̃1σ1 ) − −A3A2 (7 8 β̃2+ 1 ϑ iβ̃γ̃+ 15 8ϑ2 γ̃2 ) − 1 2 γ̃µ̃A3 ( 7 8ϑ2 +1 ) + + 1 16 iβ̃µ̃A3+ 3 2 γ̃µ̃AA 2 ( 1 8ϑ2 +1 ) − 3 16ϑ iβ̃µ̃AA 2    + (3.15) +ε2q̃   − 1 8ϑ2 σ1+ 1 8ϑ α̃+ 1 16ϑ2 µ̃+ 3 4ϑ2 γ̃ ( AA− 1 2 A2 ) + + 1 4ϑ β̃ (1 2 A2−AA )  =0 The variable A is expressed as the exponential function A= 1 2 aexp(iΦ)+ 1 2 aexp(−iΦ) (3.16) where a and Φ are vibration amplitude and phase, respectively. Substituting (3.16) into (3.15) and then separating the real and imaginary parts we get two first order modulation equations ȧ= 1 2 εα̃a− 3 16ϑ2 ε2α̃γ̃a3− 1 8 εβ̃a3+ 1 32ϑ2 ε2β̃γ̃a5+ 1 64ϑ2 ε2β̃µ̃a3cos2Φ+ + 1 4ϑ εµ̃a ( −1+ 5 16ϑ3 εγ̃a2− 1 2 εγ̃a2 ) sin2Φ+ +εq̃ 1 8ϑ [1 ϑ ε ( −α̃+ 1 4 β̃a2 ) cosΦ− ( 5− 1 ϑ2 − 9 4ϑ2 εγ̃a2+ 1 ϑ2 εµ̃ ) sinΦ ] (3.17) aΦ̇= 1 8ϑ ε2α̃a(−α̃+ β̃a2)− a 8ϑ [ (−1+ϑ2) ( 5− 1 ϑ2 ) − 3 8ϑ2 ε2µ̃2 ] + +a3 ( 3 8ϑ εγ̃+ 3 16ϑ3 εγ̃(−1+ϑ2) ) − 1 256ϑ ε2a5 ( 7β̃2+ 15 ϑ2 γ̃2 ) − − 1 32ϑ2 εβ̃ sin2Φ− 1 4ϑ εµ̃a [ 1+ ( 1 8ϑ2 +1 ) εγ̃a2 ] cos2Φ+ +εq̃ 1 8ϑ [( −5+ 1 ϑ2 + 3 8ϑ3 εγ̃a2+ 1 ϑ2 εµ̃ ) cosΦ+ 1 ϑ ε ( −α̃2+ β̃a2 ) sinΦ ] Synchronisation effects and chaos... 869 Basing on equation (3.1) an approximate solution has the form X = acos(ϑτ+Φ)+ε [ 1 32ϑ2 γ̃a3cos(3ϑτ +3Φ)− (3.18) − 1 16ϑ2 µ̃acos(3ϑτ +Φ)− 1 32ϑ β̃a3 sin(3ϑτ+3Φ) ] For the steady state ȧ=0, Φ̇=0, and then from (3.17), we obtain bifurcation equations 1 2 εα̃a− 3 16ϑ2 ε2α̃γ̃a3− 1 8 εβ̃a3+ 1 32ϑ2 ε2β̃γ̃a5+ 1 64ϑ2 ε2β̃µ̃a3cos2Φ+ + 1 4ϑ εµ̃a ( −1+ 5 16ϑ3 εγ̃a2− 1 2 εγ̃a2 ) sin2Φ= = εq̃ 1 8ϑ [ − 1 ϑ ε ( −α̃+ 1 4 β̃a2 ) cosΦ+ ( 5− 1 ϑ2 − 9 4ϑ2 εγ̃a2+ 1 ϑ2 εµ̃ ) sinΦ ] (3.19) 1 8ϑ ε2α̃a(−α̃+ β̃a2)− a 8ϑ [ (−1+ϑ2) ( 5− 1 ϑ2 ) − 3 8ϑ2 ε2µ̃2 ] + +a3 [ 3 8ϑ εγ̃+ 3 16ϑ3 εγ̃(−1+ϑ2) ] − 1 256ϑ ε2a5 ( 7β̃2+ 15 ϑ2 γ̃2 ) − − 1 32ϑ2 εβ̃ sin2Φ− 1 4ϑ εµ̃a [ 1+ ( 1 8ϑ2 +1 ) εγ̃a2 ] cos2Φ= = εq̃ 1 8ϑ [( 5− 1 ϑ2 − 3 8ϑ3 εγ̃a2− 1 ϑ2 εµ̃ ) cosΦ− 1 ϑ ε ( −α̃2+ β̃a2 ) sinΦ ] Equations (3.19) enable us to determine the amplitude a and phase Φ for the steady state in the second order approximation. For parametrically and self-excited systems without external forces it is necessary to introduce q = 0 and then the right-hand sides of the bifurcation equations become zero. Then the bifurcation points of the trivial into nontrivial solution for a system without the external force can be determined by putting a = 0 and rearranging equations (3.19) 1 4 −3ϑ2+ 23 2 ϑ4−15ϑ6+ 25 4 ϑ8+ +ε2 (1 2 α̃2ϑ2−3α̃2ϑ4+ 13 2 α̃2ϑ6− 3 16 µ̃2+ 9 8 ϑ2µ̃2− 31 16 ϑ4µ̃2 ) + (3.20) +ε4 (1 4 α̃4ϑ4− 3 16 α̃2ϑ2µ̃2+ 9 256 µ̃4 ) =0 870 J.Warmiński Onecannotice that thebifurcationpoints found from(3.20) donotdepend on the parameter β. Considering only the first order perturbation equation (3.11) we find a resonance curve in the direct form (1−ϑ2)2+ε2 ( α̃2ϑ2− µ̃2 4 ) + [ − 1 2 ε2α̃β̃ϑ2+ 3 2 εγ̃(1−ϑ2) ] a2+ (3.21) +ε2 ( 9 16 γ̃2+ 1 16 β̃2ϑ2 ) a4 =0 and hence we get the amplitude and the phase of stable vibrations a= √√√√√√ − ( −1 2 ε2α̃β̃ϑ2+ 3 2 εγ̃(1−ϑ2) ) ∓ √ d1 2ε2 ( 9 16 γ̃2+ 1 16 β̃2ϑ2 ) (3.22) tanΦ= εϑ(4α̃− β̃a2) 4−4ϑ2+3γ̃εa2 where d1 = [ − 1 2 ε2α̃β̃ϑ2+ 3 2 εγ̃(1−ϑ2) ]2 − −4ε2 ( 9 16 γ̃2+ 1 16 β̃2ϑ2 )[ (1−ϑ2)2+ε2 ( α̃2ϑ2− µ̃2 4 )] For a=0 from (3.21) we have 1−2ϑ2+ϑ4+ε2 ( α̃2ϑ2− µ2 4 ) =0 (3.23) and then the bifurcation points of the trivial into non-trivial solution in the first order approximation are ϑ∗1,2 = √ 2−ε2α̃2∓ε √ −4α̃2+ε2α̃4+ µ̃2 2 (3.24) The points described by (3.24) can appear if −4α̃2 + ε2α̃4 + µ̃2 > 0. It means that the shape of the resonance curve depends on the self-excitation α and the parametric excitation µ. Assuming that ε is a small parameter we can neglect the term ε2α̃4 as a small value, and then determine the condition µ α ­ 2 (3.25) Synchronisation effects and chaos... 871 whichmust be satisfied in order to get the bifurcation points. This is in agre- ement with the result obtained by Szabelski and Warmiński (1995a,b) for Rayleigh’s model. In the opposite case, if condition (3.25) is not satisfied, the resonance curve has a closed formand lies above the ϑ axis. Analysing (3.22)1 we can define the conditionswhen the parametric resonance does not take pla- ce. Such situations can appear if it is not possible to have a real solution for the vibration amplitude, i.e. when d1 is smaller than zero. 4. Regular vibration, numerical examples Exemplary calculations were made by using analytical dependencies and numerical simulation. Numerical data were based on the papers by Tondl (1978), Yano (1989), Szabelski and Warmiński (1995a) which concerned self- and parametrically excited systems. The basic data are following α=0.01 β=0.05 γ=0.1 µ=0.2 First, we examine vibration of the system without the external force (q = 0) and near the main parametric resonance region. Fig.2 presents a resonant curve obtained from the perturbation analysis (index AR) and nu- merical simulation (RKG).Near themainparametric resonanceweobserve the synchronisation phenomenon. In this region the parametric vibrations pull in the self-excited ones and the system vibrates only with the single frequency ϑ equal to the half of the parametric excitation frequency 2ϑ. The analy- tical solution is plotted by the solid or dashed line for stable and unstable solutions, respectively. Outside the synchronisation area the system vibrates quasi-periodically.Then the responseof the system includes strong influenceof self-excitation andwe observe vibrationwithmodulated amplitude. This kind of vibration consists of two signals,whose frequency ratio is nota rational num- ber. On the phase plane such motion is represented by stable quasi-periodic limit cycles. Double points outside the resonance region denote extreme values of the modulated amplitude. The amplitude and phase modulation is determined by equations (3.17). We can observe behaviour of the system for increasing parametric frequency ϑ from ϑ=0.8 up to ϑ=1.5. At the beginning (ϑ≈ 0.8) the systemvibrates quasi-periodically, then, around ϑ∗ =0.9489, the first nontrivial steady but unstable solution appears (the first bifurcation point of the trivial into nontrivial solution). This unstable region (unstable focus) is 872 J.Warmiński Fig. 2. Synchronisation area around themain parametric resonance, AR – analytical results, RKG – numerical simulation Fig. 3. Bifurcation values of µ and α parameters Synchronisation effects and chaos... 873 very small, between ϑ = 0.95 and ϑ = 0.96 (dashed line in Fig.2). Next, the solution transits to a stable focus. The synchronisation region starts in the point where the steady state is stable and the systemvibrates periodically with the single ϑ frequency. Above the frequency ϑ = 1.0486 (the second bifurcation point) an unstable saddle point appears in the synchronisation region. Next, after passing ϑ = 1.1 an additional quasi-periodic solution represented by a stable quasi-periodic limit cycle occurs. In the region ϑ ∈ (1.1, 1.3)we obtain periodic and quasi-periodic solutions, dependingon initial conditions. The saddle points divide the phase space into two basins of an attractor. Above ϑ = 1.3 the system leaves the synchronisation region and vibrates quasi-periodically. From practical point of view, it is important to determine the parameters influencing the synchronisation width and vibration amplitudes. Analytical dependencies enable us to find the critical value of µ and α parameters where the bifurcation points disappear. The grey area in Fig.3 corresponds to condition (3.24) and for such parameters that the resonance curve has a closed form and no points belonging to the ϑ axis. For small values of the parameters α, µ the limit region in Fig.3 is determined by the slope of a straight line µ/α=2. Fig. 4. Influence of α and µ parameters on the resonance curve around the main parametric resonance The influence of the α parameter on the resonance curve is presented in Fig.4a. For α=0.01, µ=0.2 the resonance curve has two bifurcation points, 874 J.Warmiński for α= 0.1 the bifurcation points are very close one to another (α/µ = 2), for α = 0.2, α = 0.3 the resonance curve has a closed form and lies above the ϑ axis (grey region in Fig.3). An increase in the parameter α, which is connected with self-excitation, causes an increase in the vibration amplitude and reduction in thewidth of the synchronisation region. For greater values of α the synchronisation area moves along a skeleton line towards higher values of the excitation frequency ϑ. The influence of the parametric excitation is different. In Fig.4b we can observe resonance curves for the constant value α = 0.1 and variable µ parameter. Generally, decreasing of the parametric excitation causes decreasing of the vibration amplitude and decreasing of the width of the synchronisation region. Nevertheless, the centre of the resonance remains permanently in the same place. Fig. 5. Bifurcation diagram versus ϑ parameter The bifurcation diagram versus the ϑ parameter (Fig.5) obtained from a numerical simulation confirms the analytical results. The black regions cor- respond to quasi-periodic motion, while the single lines correspond to syn- chronisation regions. We can notice three resonance areas in the considered ϑ interval, i.e. the main parametric resonance around ϑ ≈ 1.0, and the two second order resonances around ϑ ≈ 1/2 and ϑ ≈ 2. The most domina- ting synchronisation region is near the main parametric resonance. The two additional regions are almost invisible. The synchronisation width and ampli- tude are much smaller than for the main parametric resonance. Therefore, in Synchronisation effects and chaos... 875 Section 6, we decided to examine the influence of the external force on the vibrating system exactly in the main parametric resonance region. 5. Chaotic vibrations Analysis of chaotic vibrationswas based on numerical simulations byusing the Dynamics package (Nusse and York, 1994). Numerical calculations of the parametric and self-excited system were made for the parameters α = 0.01, β =0.05, γ = 0.1, ϑ= 1.0, but for larger values of the parametric excitation amplitude µ. First, a bifurcation diagram andmaximal Lyapunov’s exponent in the region µ∈< 0,2> for the systemwithout an external excitation q=0, was calculated (Fig.6). Fig. 6. Bifurcation diagram (a) andmaximal Lyapunov’s exponent (b) versus µ parameter For small values of the parametric excitation the oscillator vibrates quasi- periodically (black regions in Fig.6a). Maximal Lyapunov’s exponent for this type of motion is equal to zero (Fig.6b). For µ ≈ 0.08 the system transits from quasi-periodic to periodic motion represented by two lines on the bi- furcation diagram. There, the Hopf bifurcation takes place. In a wide region µ ∈ (0.08, 1.3) the system vibrates purely periodically. There are no quali- tative changes in the bifurcation diagram in this region. Near µ = 1.3 we observe the first ”pitchfork” bifurcation. A further increase in the µ parame- 876 J.Warmiński ter, causes next pitchfork bifurcations, and around µ= 1.40 after a cascade of bifurcations, the system passes from regular to chaotic motion. In the re- gion µ ∈ (1.41, 1.57) maximal Lyapunov’s exponent is positive (Fig.6b). It means that the system vibrates chaotically. Near µ= 1.57 after the ”crisis” bifurcation the system directly jumps from chaotic motion to periodic one. Fig. 7. Basins of attraction evolution near chaotic region; (a) µ=1.30, (b) µ=1.43, (c) µ=1.45, (d) µ=1.49, (e) µ=1.55, (f) µ=1.60 Topological changes of the attractor during the transition through chaotic motion are presented in Fig.7. We can observe the attractors as well as evo- lution of their basins. In the region of regular motion for µ = 1.3 (Fig.7a) we obtain regular attractors as two points with two different basins.Whenwe come into the chaotic area those regular points change into two small separate strange chaotic attractors and their basins become strongly mixed (Fig.7b). Next, the two attractors join themselves into one large strange chaotic attrac- Synchronisation effects and chaos... 877 tor. There is only one basin in this area (Fig.7c). After exceeding µ=1.486, apart from the chaotic attractors there appear two regular ones. The pha- se space is divided into three areas, two for regular motion and the third in the centre for chaotic one (Fig.7d). Regular and chaotic motion coexists with each other. Two ”tongues” in the central area appear. We observe a similar situation in Fig.7e, however the influence of regular motion increases. After exceeding the parameter value µ=1.58 the chaotic region is completely de- stroyed by the two ”tongues” andwe observe only the regularmotion (Fig.7f). The central region is mixed by two regular basins of the attractors. Fig. 8. Regions of appearance of regular and chaotic motion The limit value of the amplitude and frequency of the parametric excita- tion where chaotic motion does not take place is presented in Fig.8. Below the limit line the chaotic motion does not appear. The plane above that li- ne is divided into smaller chaotic and regular regions. It was difficult to find precisely all chaotic subregions and, therefore, a criterion for regular motion was applied. Basing on the numerical investigations we can conclude that the chaotic motion can appear only if the value of the parametric excitation is greater than 1, µ> 1. 6. Influence of external force on regular and chaotic motion In Section 4 we showed that the synchronisation area near the main pa- 878 J.Warmiński rametric region has themost importantmeaning. In this section we will inve- stigate behaviour of the system when it is additionally forced by an external harmonic force. Let us assume that the amplitude of the external force q is not large and that the frequency ω is equal to ϑ, i.e. the external frequency is half of the parametric frequency value (2.4). An approximate solution of the system with external excitation has form (3.1), however the amplitude and phase are descried by (3.17). In the first order perturbation analysis the modulation equations take the form ȧ= 1 2 εα̃a− 1 8 εβ̃a3− 1 4ϑ εµ̃asin2Φ− 1 2ϑ εq̃ sinΦ (6.1) Φ̇=− 1 2ϑ (−1+ϑ2)+ 3 8ϑ εγ̃a2− 1 4ϑ εµ̃cos2Φ− 1 2ϑa εq̃cosΦ Wewill present the influence of the external harmonic force on the system for the following numerical data α=0.1 β=0.05 γ=0.1 µ=0.2 q=0.2 The resonance curves obtained from (3.19) near the main parametric re- sonance are presented in Fig.9. Fig. 9. Synchronisation area around the main parametric resonance; systemwith external excitation q=0.2 Whena small external force additionally excites the system,wecanobserve significant differences in the synchronisation region. The amplitude-frequency Synchronisation effects and chaos... 879 characteristic has not a typical shape. There is an additional internal loop in the considered region. Stability analysis shows that only the upper part of the internal loop is stable. There are five possible steady solutions inside the synchronisation regions. Nevertheless, only the two upper solutions are stable. This effect was also observed by Szabelski and Warmiński (1995a,b), in Rayleigh’s model. Outside the synchronisation regions the system vibrates quasi-periodically. The two points, according to the extreme defection of the system (Fig.9)mark this type ofmotion. These stable additional solutions are visible also in the bifurcation diagrams versus the frequency ϑ (Fig.10). Fig. 10. Bifurcation diagram versus frequency ϑ, systemwith external excitation q=0.2 The single line in Fig.10 corresponds to synchronised vibrations and the black regions to quasi-periodic motion. The additional stable solution is loca- ted in the lower part of the plot. It means, that this solution has a different phase with respect to the upper solution. We can also notice that the exter- nal force causes an increase in the vibration amplitude in the synchronisation area and an increase in the vibration modulation outside this region. Fig.11 presents the resonance curves versus the amplitude of the external force. One can notice that the internal loop in the synchronisation region near the main parametric resonance appears only as a result of action of a small external force. The limit value for the van der Pol model is q = 0.7. For larger values 880 J.Warmiński Fig. 11. Influence of the external force on the synchronisation area near the main parametric resonance Fig. 12. Systemwith external excitation; (a) bifurcation diagram, (b) Lyapunov’s exponent versus µ parameter Synchronisation effects and chaos... 881 of the external force the internal loop completely disappears. Increasing of the amplitude of the external force makes the vibrations stable in a wider interval of ϑ (synchronisation region iswider)andvibrationamplitude slightly increasing. The transition of the system to chaos with an additional external force is visible in the bifurcation diagrams in Fig.12. In Fig.12a the generalised coor- dinate X versus the µ parameter was plotted for q =0.2, ϑ=1.0. Fig.12b presents the diagram of the correspondingLyapunov exponent. In comparison with Fig.6 we can observe a symmetry breaking in the system. Bifurcations appear in two different regions of the µ parameter, shifted relatively one to another. The system can vibrate along the upper or lower curve, depending on initial conditions. The region of quasi-periodic motion, which occurred for a small value of the µ parameter for the system without an external force (Fig.6), disappeared in Fig.12. Lyapunov’s exponent is negative in that re- gion. Positive values of the Lyapunov exponent emerge around µ = 1.45, similar to the system without an external force. Nevertheless, an additional external force decreases the chaotic area and divides it into two parts. This is an effect of asymmetry of the system as a result of the external force action. The influenceof the external force on the chaoticmotion canbe investigated by observing the behaviour of one chosen strange chaotic attractor (for µ=1.45, Fig.13) when the system is additionally excited by an external harmonic for- ce. The amplitude of that force was changed in the interval q ∈< 0,1 >. In Fig.13 the coordinate X and Lyapunov exponent versus q is presented. Fig. 13. Influence of the external force on chaotic vibration; (a) bifurcation diagram, (b) Lyapunov exponent versus q parameter 882 J.Warmiński We can see that the external force removed chaos from the system. The strange attractor transits from chaos to regularmotion via quasi-periodicmo- tion.Maximal Lyapunov’s exponent gradually tends from positive to negative values (Fig.13b). 7. Summary and conclusion The interaction between parametric and self-excited vibrations leads to synchronisation phenomena around the parametric resonances. In this paper vibrations near the main parametric resonance around ϑ≈ 1 are considered. Then, self-excited vibrations are pulled in by the parametric ones and the re- sponseof the systembecomespurelyharmonic.Outside that region the system vibrates quasi-periodically with a modulated amplitude. The synchronisation effect is most visible around the main parametric resonance. That region is the widest and the vibration amplitudes are the largest. It is also presented that the second order resonances are less important from practical point of view. Increasing of the parametric excitation µ over 1 (µ> 1) can introduce van der Pol-Mathieu’s oscillator to chaoticmotion. Coexistence of chaotic and regularmotions for the same parameter, depending on initial conditions, is al- so possible. The additional external force causes qualitative and quantitative changes in the synchronisation area around the main parametric resonance. For a small amplitude of the external force, there appears an additional inter- nal loop in that region.Only the upperpart of the loop is stable, the lower one is unstable. This result is similar to Rayleigh-Mathieu’s oscillator obtained by Szabelski andWarmiński (1995a). The external force removes chaos from the considered system. Then we observe soft transition from chaotic to periodic motion through quasi-periodic one. References 1. Awrejcewicz J., 1990, Bifurkacje i chaos w układach dynamicznych, Zeszyty Naukowe 383, Rozprawy Naukowe, 127, Politechnika Łódzka 2. Awrejcewicz J., 1996, Drgania deterministyczne układów dyskretnych, Wy- dawnictwaNaukowo-Techniczne,Warszawa Synchronisation effects and chaos... 883 3. Guckenheimer J., HolmesP., 1997,NonlinearOscillations,Dynamical Sys- tems, and Bifurcatins of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag 4. Hayashi Ch., 1964,Nonlinear Oscillations in Physical Systems, McGraw-Hill, Inc 5. Kapitaniak T., Steeb W.H., 1991, Transition to Hyperchaos in Coupled Generalized Van der Pol’s Equations,Physics Letters A, 152, 33-37 6. LitakG., Spuz-SzposG., SzabelskiK.,Warmiński J., 1999,Vibration of Externally Forced Froude Pendulum, International Journal of Bifurcation and Chaos, 9, 3, 561-570 7. Nayfeh, A.,H., 1981, Introduction to Perturbation Techniques, John Wiley and Sons, NewYork 8. Nusse H., York A., 1994,Dynamics, Numerical Explorations, Springer Ver- lag, NewYork 9. Sanchez N.E., Nayfeh A.H., 1990, Prediction of Bifurcations in a Parame- trically Excited Duffing Oscillators, Int. Journal Non-Linear Mechanics, 25, 2/3, 163-176 10. Steeb W.H., Kunick A., 1987, Chaos in Limit Cycle Systems with External Periodic Excitations, Int. J. Non-Linear Mechanics, 349-361 11. SzabelskiK.,Warmiński J., 1995a,TheParametricSelf ExcitedNon-Linear System Vibrations Analysis with the Inertial Excitation, Int. Journal of Non- Linear Mechanics, 30, 2, 179-189 12. Szabelski K., Warmiński J., 1995b, The Self-Excited System Vibrations with theParametric andExternalExcitations,Journal of Sound andVibration, 187, 4, 595-607 13. Tondl A., 1978,On the Interaction Between Self-Excited and Parametric Vi- brations, NationalResearch Institute forMachineDesign,Monographs andMe- moranda, No. 25, Prague 14. YanoS., 1989,Considerations onSelf-AndParametricallyExcitedVibrational Systems, Ingenieur-Archiv, 59, 285-295 884 J.Warmiński Zjawiska synchronizacji i chaos w oscylatorze van der Pola-Mathieu Streszczenie W pracy przedstawiono efekty oddziaływania drgań parametrycznych i samo- wzbudnych w układzie o jednym stopniu swobody. Analizę przeprowadzono dla nie- liniowego oscylatora z samowzbudzeniem typu van der Pola oraz wzbudzeniem para- metrycznym typu Mathieu, biorąc pod uwagę zarówno drgania regularne, jak i cha- otyczne. Określono obszary synchronizacji drgań w otoczeniu głównego rezonansu parametrycznego oraz wyznaczono warunki, w których następuje przejście do ruchu chaotycznego. Przedstawiono również wpływ siły zewnętrznej na drgania regularne i chaotyczne układu. Wykazano, że małe wymuszenie zewnętrzne powoduje istotne zmiany jakościowe i ilościowe w obszarze głównego rezonansu parametrycznego oraz żewymuszenie zewnętrznepowoduje eliminację drgańchaotycznych i przejście układu do ruchu regularnego. Manuscript received November 26, 2000; accepted for print December 27, 2000