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bifurcation!0)67434#i0)06945;!0)67434!i0)06945!0)14346#i0)19808;!0)14346!i0)19808!0)41499#i0)21141;!0)41499!i0)21141!0)21215#i0)06682;!0)21215!i0)066822)83 !1)02718#i0)60448;!1)02718!i0)60448 Naimark}Sacker bifurcation!0)72867#i0)19399;!0)72867!i0)19399!0)23703#i0)49386;!0)23703!i0)493860)01937#i0)37864; 0)01937!i0)37864!0)16616#i0)30358;!0)16616!i0)303586)22 !0)77510#i0)54037;!0)77510!i0)54037 Periodic solution is stable!0)71783#i0)39661;!0)71783!i0)396610)47102#i0)65186; 0)47102!i0)651860)49991#i0)40466; 0)49991!i0)404660)38520#i0)51588; 0)38520!i0)51588s is stable when any disturbance eu yields a / such that E/E(DeD. This is ful"lled if alleigenvalues of DP(s) are inside the unit circle. Since Q"P!I, DP can be obtained byDP(s)"DQ(s)#I. (12)The eigenvalues of DP(s) are the Floquet multipliers [8], or the characteristic multipliers.Therefore, it is possible to use Floquet theory to discuss the stability of the periodicsolution. If the Floquet multipliers of the system are j1, j2,2, j10, concerning bifurcationand stability of equation (6), there are the following conclusions:(1) When DjiD(1(i"1, 2,2, n and ns"R), the stable periodic solution of equation (6)is asymptotically stable.(2) If there is one jjwhich passes the unit circle outwards through the point of!1 andother DjiDiOj(1(i"1, 2,2, n), the stable periodic solution will have the period-doublingbifurcation.(3) If there is one jjwhich passes the unit circle outwards through the point of#1 andother DjiDiOj(1(i"1, 2,2, n), the stable periodic solution will have the saddle-nodebifurcation.(4) If there is a pair of conjugate complex characteristic multipliers jj"a$ib whichpass the unit circle outwards and other DjiDiOj(1(i"1, 2,2, n), the stable periodicsolution will have the Naimark}Sacker bifurcation [9] and the bifurcation will lead to aninvariant torus.The above-discussed shootingmethod is used to obtain the periodic solutions of equation(6) for some rotating speeds. The Floquet multipliers at di!erent rotating speeds areobtained as shown in Table 1. It can be seen that at some rotating speeds, the system will 4. NUMERICAL SIMULATIONThe shooting method is to some extent still an approximate method for analyzinga non-linear system. In order to further observe the dynamic behavior of the system, thefourth order Runge}Kutta method was then used to integrate equation (6). In this section,the rotating speed and imbalance were used as the control parameters to perform a detailedinvestigation on bifurcation, chaos, and the routes to or out of chaos.During integration, a smaller marching step is chosen to ensure a stable solution and toavoid the numeric pergence at the point where damping and sti!ness coe$cients arediscontinuous. Generally, long time-marching computation is needed to obtaina convergent orbit. In the case of a strongly stable motion with heavy damping, severalhundred periods of integration may be enough while for some other cases several thousandperiods are necessary.To illustrate the motion behavior of the system, the orbit, the Poincare's map and thebifurcation diagram are used. A Poincare's section is a stroboscopic picture of a motion andconsists of the time series at a constant interval of ¹ with ¹ being the period of excitation.The corresponding Poincare's map is a combination of those return points and afteriterating enough times these points may converge at a subset which is often called anattractor. Examination of the distribution of return points on the Poincare'smap can revealthe nature of motion. In the case of a periodic motion, the N discrete points on thePoincare's map indicate that the period of motion is N¹. In the case of a quasi-periodicmotion, return points appear to "ll up a closed curve in the Poincare's map. For a chaoticmotion, return points form a geometrically fractal structure. The bifurcation diagram isanother type of plot to re#ect the motion change. To compute a bifurcation diagram,a control parameter was varied at a constant step. The variation of the > (y2/cl) co-ordinateof the return point in the Poincare's map versus the control parameter to form a bifurcationdiagram was then plotted. In this paper, the motion of the disk position in the form of x2/cland y2/clwas recorded to form orbits and the corresponding Poincare's maps andbifurcation diagrams. In some cases, the orbits at the left-bearing position in the form ofx3/cland y3/clwere also presented.In order to judge whether a motion is chaotic or not, a method discussed by Wolf et al.[12] was used to calculate the maximum Lyapunov exponent. The program contained inreference [12] for ordinary di!erential equations plus the IMSL-routine DVERK was usedto perform this calculation. Also to describe the fractal behavior of the attractorquantitatively the information dimension was calculated for some cases. The informationdimension is one of themany de"nitions of the fractal dimension thatmeasures the extent towhich the points "ll a subspace as the number of points becomes large [13].It has to be pointed out that in "gures presented in this paper the same amount of data of80 periods are used in all orbit plots but with di!erent scales and also the di!erent scales areused in the Poincare's maps in order to amplify the attractors and to re#ect the shape of anattractor adequately.Figure 3 is the bifurcation diagram of the system by using the rotating speed as thecontrol parameter where for every rotating speed 100 points are included. It can be seen thatat very low rotating speeds the motion is synchronous with period-2. The bifurcation mapshows two points for every rotating speed. But by examining the Poincare's map carefully itis found that the motion is actually quasi-periodic and the Poincare's return points move very slowly. It is di$cult to see the return points to form a closed curve because the dataamount is too large. As the rotating speed increases further the attractor perges and atabout u/u1"1)70 the motion becomes chaotic.