MULTIPLE BIFURCATIONS OF A CYLINDRICAL DYNAMICAL SYSTEM

Transcription

1 Journal of Theoretical and Applied Mechanics, Sofia, 2016, vol 46, No 1, pp MULTIPLE BIFURCATIONS OF A CYLINDRICAL DYNAMICAL SYSTEM Ning Han, Qingjie Cao Centre for Nonlinear Dnamics Research, Harbin Institute of Technolog, School of Astronautics, Harbin China, s: han_ning@outlookcom, qjcao@hiteducn [Received 05 June 2015, Accepted 29 Februar 2016] Abstract This paper focuses on multiple bifurcations of a clindrical dnamical sstem, which is evolved from a rotating pendulum with SD oscillator The rotating pendulum sstem exhibits the coupling dnamics propert of the bistable state and conventional pendulum with the homoclinic orbits of the first and second tpe A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected b using analtical analsis and numerical calculation It is found that the homoclinic orbits of the second tpe can bifurcate into a pair of rotational limit ccles, coexisting with the oscillating limit ccle Additionall, the results obtained herein, are helpful to explore different tpes of limit ccles and the complex dnamic bifurcation of clindrical dnamical sstem Ke words: Clindrical dnamical sstem, rotating pendulum with SD oscillator, homoclinic bifurcation, saddle-node bifurcations of periodic orbits, limit ccle 1 Introduction The clindrical dnamical sstem [1, 2, 3] has received a resurgence of attention in recent ears, owing to the wide range of application in civil, militar and industrial applications, such as horologium, robot [4], pendulum absorber [5], gas turbine rotors [6], as well It is well known, that the conventional pendulum is a tpicall clindrical dnamical sstem, which is among * Corresponding author qjcao@hiteducn The authors would like to acknowledge the financial support from the Natural Science Foundation of China (Grant Nos and ) and the National Basic Research Program of China (Grant No 2015CB057405)

2 34 Ning Han, Qingjie Cao the most widel investigated motions in phsics and man nonlinear phenomena in the real world Galileo found the law of isochronism through classical experiments in the 16th centur, which marked the beginning of the scientific research of clindrical pendulum In the 17th centur, Hugens discovered the periodicit of large swings, deviating from the law isochronism, which is one of the earliest records of clindrical pendulum In modern time, man kinds of pendulum-like sstem or sstems containing the pendulum have been proposed and investigated, which greatl enrich the research content of clindrical dnamical sstem For example, the chaotic dnamics and subharmonic bifurcations in a pendulum-like sstem were studied in [7] The approximate analtical solutions for oscillator and rotational motion of a parametric pendulum were discussed in [8] The bifurcation in an inverted pendulum with the high frequenc excitation was proved b using analtical and experimental investigations in [9] Subharmonic and homoclinic bifurcations in a parametricall forced pendulum was studied in [10] Recentl, a rotating pendulum linked b an oblique spring with fixed end has been proposed and investigated in [11] This sstem is a clindrical dnamical sstem with irrational characters, whose free motion is similar to the conventional pendulum coupled with SD oscillator [12, 13, 14] of the homoclinic orbits of the first and second tpe The authors introduce a clindrical approximate sstem, for which the analtical solutions can be obtained successfull to reflect the nature of the rotating pendulum sstem, without barrier of the irrationalities [15] Then, the chaotic boundar of discontinuous case with a pair of heteroclinic-like orbits [16] are obtained b using semi-analtical method Throughout the hot research topics of clindrical dnamical sstem, ver few attempts have been made in reporting the oscillating and rotational limit ccles of the autonomous sstem, which is of fundamental importance to reveal the essential phenomenolog of clindrical dnamical sstem from the view of nonlinear science and ma be helpful to explore directl the bifurcation of the clindrical dnamical sstem avoiding Talor s expansion The aim of this paper is to start exploring multiple bifurcations of a clindrical dnamical sstem, which is evolved from a rotating pendulum linked b a vertical spring with sliding end under a nonlinear perturbation The other motivation of this work is to provide a tpical example to reveal the clindrical phenomena b means of the results obtained herein Particularl, it is found that the homoclinic orbits of the second tpe can bifurcate into a pair of rotational limit ccles coexisting with the oscillating limit ccle This paper is organized as follows: In Section 2, the mathematical model for the clindrical pendulum sstem with a nonlinear perturbation is developed

3 Multiple Bifurcations of a Clindrical Dnamical Sstem 35 In the following section, Section 3, the unperturbed dnamics of the pendulum sstem is directl analzed b using nonlinear dnamical technique While in Section 4, a double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are investigated b using analtical analsis and numerical calculation Finall this paper is ended with conclusions 2 Clindrical dnamical sstem Consider the simple model which comprises a rotating pendulum linked b a vertical spring with sliding end, as shown in Fig 1 It is assumed to oscillate or rotate in the vertical plane The equation of motion is given b: (1) ml x mgsinx ksinx(h l+lcosx) = 0, where the dot denotes derivative with respect to t, x is the angular displacement, m andlare the lump mass and pendulum length, k andlare the stiffness and relax length of the spring, h is the height between A and B Again, sstem (1) can be written as: (2) x sinx(α+βcosx) = 0, α = 1 L (g + km (h l) ), β = k m From the mathematical perspective, parameters α and β can be regarded as mutuall independent From a phsical standpoint, parameter α defines the geometr of the phsical model and β mainl reflects the sstem stiffness Fig 1 The model of a rotating pendulum linked b a vertical spring with sliding end

4 36 Ning Han, Qingjie Cao Assuming that sstem (1) is perturbed b a nonlinear perturbation ( c + 2 dcos x)ẋ, the forced dissipative sstem can be derived as a two dimensional one: (3) x =, = sinx (α+βcosx) ( ξ +δcos 2 x), ξ = c ml, δ = d ml, where ξ and δ are positive small parameters Letting X = cosx,y = sinx,z =, sstem (3) can be re-written in the following clindrical form: (4) X = YZ, Y = XZ, Z = Y(α+βX) ( ξ +δx 2 )Z, X 2 +Y 2 = 1 It is noticing that the perturbation term ( ξ +δx 2 )Z in sstem (4) is similar to the Vanderpol oscillator [17, 18] 3 Unperturbed dnamics In this section, the unperturbed dnamics of the rotating pendulum sstem including the equilibria, Hamiltonian function, homoclinic orbits and phase portraits are discussed b using nonlinear dnamical technique, respectivel The unperturbed sstem (2) can be written as a two dimensional one: (5) x =, = sinx (α+βcosx), α < β, of which the equilibria can be easil obtained b letting sinx (α+βcosx) = 0: (6) (x 1, 1 ) = (0,0), (x 2,3, 2,3 ) = (±π,0), (x 4,5, 4,5 ) = (±arccos( α/β),0), where (x 4,5, 4,5 ) exist onl for α/β < 1 It is worth reiterating here that sstem (2) bears significant similarities to conventional pendulum sstem for α β The Hamiltonian function of sstem (5) can be obtained in the following form: (7) H(x,) = 2 2 +αcosx β 2 sin2 x α and the three-dimensional surface of Hamiltonian function in {x,,h(x,)} space is plotted for α = 1,β = 2, as shown in Fig 2(a) With the help of Hamiltonian function (7), the trajectories of sstem (5) can be classified b means of H(x,) = h In order to understand the trajectories of phase portraits as the parameter h var, we define the trajectories as Γ h = {(x,) H(x,) = h} Γ h represents a pair of closed orbits for h ( (α + β) 2 /2β,0); When h =

5 Multiple Bifurcations of a Clindrical Dnamical Sstem 37 Fig 2 When α = 1 and β = 2: (a) the surface diagram of Hamiltonian function for the unperturbed sstem (5), (b) phase portrait for sstem (5): Γ h = {(x,) H(x,) = h}, (c) the analtical solution x(t) of the homoclinic orbits of the first and second tpe and (d) corresponding the solution of velocit (t) (α +β) 2 /2β, Γ h shrinks into the pair of equilibria (x 4,5, 4,5 ) On the other side, Γ h extends into a double homoclinic orbits connecting the equilibrium (x 1, 1 ) for h = 0 While h (0, 2α), Γ h gives the large closed orbit encircling the equilibria (x 1, 1 ), (x 4,5, 4,5 ) and Γ h extends into a double connected homoclinic orbits of second tpe connecting the equilibria (x 2, 3 ) and (x 3, 3 ), when h = 2α It is worth noticing, that this large closed orbit for Γ h (0, 2α) does shrink itself into the double homoclinic orbits of first tpe as h 0 +, on the contrar, it does extend itself into the double homoclinic orbits of second tpe as h ( 2α) While h > 2α, the double homoclinic orbits of second tpe Γ h= 2α divide into a pair of wavelike trajectories near the homoclinic orbits of second tpe Take α = 1 and β = 2 for example, the detailed classification for the trajectories of phase portraits are shown in Fig 2(b)

6 38 Ning Han, Qingjie Cao The double connected homoclinic orbits of first tpe denoted b hom1, for h = 0, can be written in the parametric form as: (8) ( ) x hom1 ± (t),± hom1 (t) = ( α ( )) ±2cot 1 α+β cosh α+β t, 2 αsinh ( α+β t ) 1+ α α+β cosh2( ), α+β t and is plotted for α = 1 and β = 2 in Fig 2(c) and (d) with dashed line Similarl, the double connected homoclinic orbits of second tpe marked b hom2, for h = 2α, can be written in the parametric form as: (9) ( ) x hom2 ± (t),± hom2 (t) = ( α ( )) ±2tan 1 β α sinh β α t, 2 αcosh ( β α t ) 1+ α β α sinh2( ), β α t and is displaed for α = 1 and β = 2 in Fig 2(c) and (d) with solid line For ξ = δ = 0, the clindrical form (4) is given b (10) X = YZ, Y = XZ, Z = Y(α+βX), X 2 +Y 2 = 1 The equilibria of sstem (10) can be easil derived b letting Ẋ = 0, Y = 0,Ż = 0, and written as: ( (11) (X 1,2,Y 1,2,Z 1,2 ) = (±1,0,0), (X 3,4,Y 3,4,Z 3,4 ) = α ) β β, ± 2 α 2,0 β Multipling both sides of the second and third equation b the first equation in sstem (10) leads to: (12) XẊ = Y Y, ZŻ = (α+βx)ẋ Integrating both sides of sstem (12), the Hamiltonian function of sstem (10) can be derived: (13) H(X,Y,Z) = Z2 2 +αx + β 2 X2, X 2 +Y 2 = 1

7 Multiple Bifurcations of a Clindrical Dnamical Sstem 39 The phase portraits plotted via Hamiltonian function (13) H(X,Y,Z) = h is shown for α = 1 and β = 2 in Fig 3(a) It is more interesting that a pair of homoclinic orbits of the first tpe connecting the saddle point (X 1,Y 1,Z 1 ) and the second tpe connecting the saddle point (X 2,Y 2,Z 2 ), (hom1 and hom2), are co-existed in Fig 3(a) The homoclinic orbits of the first tpe connected Fig 3 When α = 1 and β = 2: (a) clindrical phase portraits: a pair of homoclinic orbits of the first (hom1) and the second tpe (hom2) with saddles at (±1,0,0) and centers at (1/2,± ( ) 3/2,0) (b)-(d) the solutions X hom1,2 ± (t),y hom1,2 ± (t),z hom1,2 ± (t) for a pair of homoclinic orbits of the first (dashed line) and the second tpe (solid line) the equilibrium (X 1,Y 1,Z 1 ), for h = 0, can be written in the parametric form as: (14) ( X hom1 ± (t),y hom1 ± (t),z hom1 ) ± (t) ( ( = cos x hom1 ± (t) ) ( ) ),±sin x hom1 ± (t),± hom1 (t)

8 40 Ning Han, Qingjie Cao The homoclinic orbits of the second tpe connected the equilibrium (X 2,Y 2, Z 2 ), for h = 2α, can be written in the parametric form as (15) ( X hom2 ± (t),y hom2 ± (t),z hom2 ) ± (t) ( ( = cos x hom2 ± (t) ) ( ) ),±sin x hom2 ± (t),± hom2 (t) Based upon Eq (14) and Eq (15), a pair of homoclinic orbits of the first (dashed line) and the second tpe (solid line) are plotted in Fig 3(b), (c) and (d) 4 Perturbed dnamics In this section, a double Andronov-Hopf bifurcation [19], two saddlenode bifurcations of periodic orbits [20] and a pair of homoclinic bifurcations [21] are investigated for a given set of parameters b using analtical analsis and numerical calculation 41 Double Andronov-Hopf Bifurcation Andronov-Hopf bifurcation of nonlinear dnamical sstem has been paid close attention widel over the last decade It is a phenomenon, which consists of a famil of limit ccles bifurcating from an initiall stable (or nonhperbolic) equilibrium Then, we need to analze the equilibrium points and their stabilit for the perturbed sstem (3) For α < β, the Jacobian matrix of the unperturbed sstem (3) at equilibria (x i, i ) i=1,2 5 can be derived as [ J (xi, i ) 1,2 5 = 0 1 αcosx i +βcos2x i δ i sin2x i ξ δcos 2 x i The characteristic equation of Jacobian matrix ields: λ 2 (ξ δcos 2 x i )λ (αcosx i +βcos2x i δ i sin2x i ) = 0 It is clear, [ that equilibrium (x 1, 1 ) is a saddle point with the eigenvalues λ 1,2 = (ξ δ)± ] (ξ δ) 2 +4(α+β) /2, and (x 2,3, 2,3 ) are saddle points [ with the eigenvalues λ 1,2 = (ξ δ)± ] (ξ δ) 2 +4(β α) /2, respectivel The Jacobian matrix for sstem (3) at (x 4,5, 4,5 ) is given b: 0 ) 1 J (x4,5, 4,5 ) = αβ (2 4α2 β 2 ξ α2 β 2δ ]

9 Multiple Bifurcations of a Clindrical Dnamical Sstem 41 The characteristic equation ields: λ 2 (ξ α2 αβ )λ+2 β 2δ 4α2 β 2 = 0 Letting p = (ξ δα 2 /β 2 ), q = 2 α/β 4α 2 /β 2, ξ 1 and δ 1, the eigenvalues of the characteristic equation can be obtained as: λ 1,2 = p±i 4q p 2 2 It is worth pointing out, that the equilibria (x 4,5, 4,5 ) are stable focus forp 0 On the other hand, the equilibria (x 4,5, 4,5 ) are unstable focus when p > 0 It is found that sstem (3) has two semi-simple eigenvalues λ 1,2 = ± q i for p = 0, the double Andronov-Hopf bifurcation occurs on ξ/δ = α 2 /β 2 It is a necessar condition for Andronov-Hopf bifurcation 42 Homoclinic Bifurcation Here, we provide the details of Melnikov analsis for the sstem (3), expositor discussions of theor can be found in [22, 23, 24] Then, we introduce following notation in sstem (3): [ F(X) = and sinx(α+βcosx) [ Tr(DF) = Tr ] [, G(X) = 0 (ξ δcos 2 x) 0 1 cosx(α+βcosx) βsin 2 x 0 ] [ x, X = ] = 0, F(X) G(X) = (ξ δcos 2 (x hom ± (t)))( hom ± (t)) 2, where u v = u 1 v 2 u 2 v 1 for an u = (u 1,u 2 ) T and v = (v 1,v 2 ) T, the corresponding Melnikov function for the homoclinic orbits of the first and second tpe is given b: (16) M hom (ξ,δ) = Eq (16) can be rewritten as + (ξ δcos 2 (x hom ± (t)))( hom ± (t)) 2 dt (17) M hom1,2 (ξ,δ) = ξi hom1,2 1 δi hom1,2 2, ],

10 42 Ning Han, Qingjie Cao where I hom1 1 = = + + ( hom1 ± (t)) 2 dt 2 αsinh ( α+β t ) 1+ α α+β cosh2( ) α+β t 2 dt = 4 α+β +4 α ln ( βα ) + 1 β, β α (18) I hom1 2 = = + + cos 2 (x hom1 ± (t))( hom1 ± (t)) 2 dt cos 2 ( ±2cot 1 ( α α+β cosh ( α+β t ))) 2 αsinh ( α+β t ) 1+ α α+β cosh2( ) α+β t 2 dt = A ( 3(1+B) 25 2(2+8B +3B 2 ) 1+B ( )) 2+B 2 1+B +3B(2+2B +B 2 )ln B A = α+β, B = α α+β

11 Multiple Bifurcations of a Clindrical Dnamical Sstem 43 and I hom2 1 = = + ( hom2 + ± (t)) 2 dt 2 αcosh ( β α t ) 1+ α β α sinh2( ) β α t = 4 β α 4 α β ln ( βα + 1 β α ) 2, dt (19) I hom2 2 = = = + + cos 2 (x hom2 ± (t))( hom2 ± (t)) 2 dt cos 2 ( ±2tan 1 ( α ( ))) β α sinh β α t 2 αcosh ( β α t ) 2 1+ α β α sinh2( ) dt β α t C ( 3(1 D) 25 2(2 8D +3D 2 ) 1 D ( )) 2 D 2 1 D 3D(2 2D +D 2 )ln D C = α β α, D = β α It is worth pointing out, that the homoclinic bifurcations of the first and second tpe occur on M hom1 (ξ,δ) = 0 and M hom2 (ξ,δ) = 0, respectivel 43 Closed Orbits Bifurcation In order to analze multiple bifurcations and different tpes of limit ccles of sstem (3), we need to discuss the number of zero solutions of subharmonic Melnikov function Let (x(t), (t)) represent the closed periodic orbits inside or outside a pair of homoclinic loops Γ h=0 and Γ h= 2α The Hamiltonian function is defined b Eq (7), now the closed orbit H(x(t),(t)) = h is satisfied with period T Thus, the subharmonic Melnikov function can be obtained as: T (20) M(h) = (ξ δcos 2 (x(t)))((t)) 2 dt = (ξ δcos 2 x)dx Let M(h) = 0, we have: 0 (21) P(h) = ξ δ = I 2(h) I 1 (h), H=h

12 44 Ning Han, Qingjie Cao where: (22) I 1 (h) = I 2 (h) = H=h H=h b dx = 2 a cos 2 xdx = 2 The function P(h) can be rewritten as: (x)dx, b (23) P(h) = ξ δ = b a (x)cos2 xdx b a (x)dx a (x)cos 2 xdx For a given closed orbit to remain after perturbation, we onl require M(h) = 0 or P(h) = 0 When H(x(t), (t)) = h, the Hamiltonian function (7) can be expressed as a function of x: (24) (x) = ± 2h 2αcosx+βsin 2 x+2α For (x) = 0, the trajector of sstem (5) intersects the x-axis at (a i,b i ) i=1,2,3 Furthermore, the turning points (a i,b i ) i=1,2,3 can be obtained and classified in the following form: ) (i) h [ (α+β)2,0, 2β ( α ) (α+β) a 1 = arccos 2 +2βh, β (ii) h [0, 2α), (iii) h [ 2α,+ ), b 1 = arccos ( α+ (α+β) 2 +2βh β ( α ) (α+β) a 2 = arccos 2 +2βh, β ( α ) (α+β) b 2 = arccos 2 +2βh ; β a 3 = π, b 3 = π Take α = 1 and β = 2 for example, the turning points (a i,b i ) i=1,2,3 are marked with solid circles in the Fig 2(b) 44 Multiple Bifurcations and Limit ccles Then, we can analze the zeros of P(h) and the associated level curves P(h) = ζ to investigate the number of limit ccles b using numerical integral ) ;

13 Multiple Bifurcations of a Clindrical Dnamical Sstem 45 method for α = 1 and β = 2 Therefore, the zeros of M(h) can be determined b the intersection points of P(h) and the level line P(h) = ζ, as shown in Fig 4 {h,p(h)} Take α = 1 and β = 2 for example, more detailed classification is as follows: (1) When ζ [0,025), there is no point of intersection between the curve P(h) and the level curve P(h) = ζ, which indicates that none closed orbit exists for ζ [0,025); (2) For ζ [025,b 1 ), the curve P(h) intersects the level curve P(h) = ζ at unique point for h ( 025,a 1 ) This implies, for ζ (025,b 1 ), there exist two closed orbits for h ( 025,a 1 ), encircling the singular points (x 4, 4 ) and (x 5, 5 ), respectivel; (3) Enlarging ζ to b 1 with P (a 6 ) = 0 (P (h) = P(h)/ h), the curve P(h) and the level curve P(h) = ζ intersect at two points for h = a 1,h = a 6 This suggests, for ζ = b 1, there exist three closed orbits, two for h = a 1, encircling the singular points (x 4, 4 ) and (x 5, 5 ), respectivel and another one for h = a 6 enclosing the singular points (x 1, 1 ) and (x 4,5, 4,5 ); (4) When ζ (b 1,P(0)), the curve P(h) meets the level curve P(h) = ζ at three points for h (a 1,0), h (a 5,a 6 ) and h (a 6,a 7 ) There exist four closed orbits, two large closed orbits, encircling the singular points (x 1, 1 ) and (x 4,5, 4,5 ) for h (a 5,a 6 ) and h (a 6,a 7 ) and two small 045 b2 ( a3, b2) ( a2, P(2)) ( a4, P(2)) ( a8, b2) (2, p(2)) 040 b1 (0, P(0)) a b (, ) 1 1 ( a5, P(0)) ( a6, b1) ( a7, P(0)) P( h) 035 lhopf 030 lc1 lhom1 lhom2 lc Fig 4 The graph of curve P(h) with associated preserved level curves P(h) = ζ h

14 46 Ning Han, Qingjie Cao closed orbits, encircling the singular points (x 4,5, 4,5 ) for h (a 1,0), respectivel; (5) For ζ = P(0), P(h) and P(h) = ζ intersect at three points for h = 0, h = a 5 and h = a 7 This indicates, for ζ = P(0), there exist two large orbits, encircling the singular points (x 1, 1 ) and (x 4,5, 4,5 ) for h = a 5, h = a 7 and the homoclinic orbits of first tpe for h = 0; (6) Then ζ (P(0),P(2)), P(h) and P(h) = ζ meet at three points for h (0,a 2 ), h (a 4,a 5 ) and h (a 7,2) This means, that there exist three large orbits, encircling the singular points (x 1, 1 ) and (x 4,5, 4,5 ); (7) When ζ = P(2), P(h) intersects P(h) = ζ at three points for h = a 2, h = a 4 and h = 2 This implies, for ζ = P(2), there exist two large closed orbits, encircling the singular points (x 1, 1 ) and (x 4,5, 4,5 ) and the homoclinic orbits of the second-tpe, connecting the singular points (x 2, 2 ) and (x 3, 3 ); (8) While ζ (P(2),b 2 ), P(h) and P(h) = ζ intersect at three points for h (a 2,a 3 ), h (a 3,a 4 ) and h (2,a 8 ) This means, for ζ (P(2),b 2 ), there exist two large rotational closed orbits and two large closed orbits, encircling the singular points (x 1, 1 ) and (x 4,5, 4,5 ); (9) When ζ = b 2 with P (a 3 ) = 0, P(h) intersects P(h) = ζ at two points for h = a 3 and h = a 8 It is found, that there exist two large rotational closed orbits and one large closed orbit, encircling the singular points (x 1, 1 ) and (x 4,5, 4,5 ); (10) For ζ (b 2,+ ), P(h) and P(h) = ζ intersect at one point for h (a 8,+ ) This implies, for ζ (b 2,+ ), there exist two large rotational closed orbits for h (a 8,+ ) Then, we define: (25) Σ = l hopf l c1 l hom1 l hom2 l c2, as the bifurcation diagrams for sstem (3) in parameter space {ξ,δ} plane To illustrate the above theoretical analsis, the bifurcation diagram Σ is plotted in Fig 5 (Σ): (1) The double Andronov-Hopf bifurcation l hopf with solid line: δ 4ξ = 0; (2) The first saddle-node bifurcation of periodic orbits bifurcation l c1 marked dashed line: δ c 1 ξ = 0,c 1 = 1/b ;

15 Multiple Bifurcations of a Clindrical Dnamical Sstem 47 (Σ) δ lhopf (a) I (b) II lc1 I II III lhom1 l hom2 IV V lc2 VI (c) lc1 -π ξ (d) III x π (e) -π lhom1 x π hom1 hom1 (f) -π IV x π (g) -π l hom2 x π -π (h) V x Rotational Limit Ccle π hom2 hom2 Rotational Limit Ccle (i) -π lc2 x Rotational Limit Ccle π (j) -π VI x Rotational Limit Ccle π -π x π Rotational Limit Ccle Rotational Limit Ccle -π π -π π x x Fig 5 Bifurcation diagrams (Σ) and the corresponding perturbed dnamics (a) (j) (3) The homoclinic bifurcation of the first tpe l hom1, marked dotted line: ( ( 2ln 3 2 ( ) ) 5 2 ( 2 +4 ξ 8 ln 3 2 ) ) δ = 0; 6 (4) The homoclinic bifurcation of the second tpe l hom2 with thick dotted

16 48 Ning Han, Qingjie Cao Fig 6 (a) A pair of small oscillating limit ccles for double Andronov-Hopf bifurcation; (b) the homoclinic orbits of first tpe and two large oscillating limit ccles; (c) the homoclinic orbits of second tpe and two large oscillating limit ccles; (d) the coexistence of large oscillating limit ccles and rotational limit ccles line, 2 ln ξ! δ = 0; ln (5) The second saddle-node bifurcation of periodic orbits bifurcation lc2 denoted thick dashed line, δ c2 ξ = 0, c2 = 1/b2 228 It is worth pointing out, that the bifurcation curves divide the two dimensional parameter space {ξ, δ} into six persistence regions, denoted as I, II, III, IV, V and VI, respectivel Meanwhile, the stabilit, the number and position of the limit ccles in ever region and bifurcation curves are displaed in Fig 5(a) (j) When (ξ, δ) locates into region I, three saddles (x1,2,3, 1,2,3 ) and two

17 Multiple Bifurcations of a Clindrical Dnamical Sstem 49 stable focusses(x 4,5, 4,5 ) are displaed without limit ccle, as shown in Fig 5(a) When (ξ,δ) touches the branch l hopf, there appears a double Andronov-Hopf bifurcation at the equilibria (x 4,5, 4,5 ) in Fig 5(b), which bifurcates into a pair of small oscillating limit ccles (stable), encircling the equilibria (x 4, 4 ) and (x 5, 5 ), respectivel, when (ξ,δ) enters into region II While (ξ,δ) touches the branch l c1, there appears one saddle-node bifurcation of periodic solution in Fig 5(c) It is found, that there exist semi-stable large oscillating limit ccles, encircling the equilibria (x 1, 1 ) and (x 4,5, 4,5 ), marked dotted line, coexisted with the pair of small oscillating limit ccles, encircling the equilibria (x 4, 4 ) and (x 5, 5 ), respectivel As (ξ,δ) crosses the branch l c1 and locates into region III, the semistable limit ccles bifurcate into two large oscillating limit ccles, encircling the equilibria (x 1, 1 ) and (x 4,5, 4,5 ) One of them is stable denoted b solid line, the other is unstable marked b dashed line Meanwhile there are a pair of stable small oscillating limit ccles, encircling the equilibria (x 4, 4 ) and (x 5, 5 ), respectivel, as shown in Fig 5(d) A pair of stable small oscillating limit ccles encircling the equilibria (x 4, 4 ) and (x 5, 5 ) extend into a double-connected homoclinic orbits of the first tpe (hom1), encircling the pair of unstable focusses in Fig 5(e) A pair of large oscillating limit ccles encircling the equilibria (x 1, 1 ) and (x 4,5, 4,5 ) can be plotted outside of the homoclinic orbits of the first-tpe with solid line (stable) and dashed line (unstable) When (ξ,δ) across l hom1 into region IV, this double-connected homoclinic orbits of the first tpe bifurcates into a stable large oscillating limit ccle, encircling the pair of the unstable focuses and the saddle, embraced b a pair of limit ccles with solid line (stable) and dashed line (unstable), as shown in Fig 5(f) When (ξ,δ) crosses the branch l hom2, the stable large limit ccle on the furthest exterior in Fig 5(f) extend into a double-connected homoclinic orbits of the second tpe (hom2) connecting the saddle (x 2, 2 ) and (x 3, 3 ) in Fig 5(g) It is noting, that there are a pair of oscillating limit ccles with solid line (stable) and dashed line (unstable) encircling the equilibria (x 1, 1 ) and (x 4,5, 4,5 ) inside the double-connected homoclinic orbits of the second tpe When (ξ,δ) crosses the branch l hom2 and locates into the region V, branch l c2 and VI, the double-connected homoclinic orbits of the second tpe bifurcate into a pair of rotational limit ccles above and below the homoclinic orbits, as shown in Fig 5(h) (j) It is clear, that a pair of oscillating limit ccles with solid line (stable) and dashed line (unstable) encircling the equilibria (x 1, 1 ) and (x 4,5, 4,5 ) in Fig 5(h) concentrate in a semi-stable oscillating limit

18 50 Ning Han, Qingjie Cao ccles with dotted line in Fig 5(i) and then disappear in Fig 5(j) In order to more deepl understand multiple bifurcations and different tpes of limit ccles, we introduce clindrical phase portraits to clearl describe the double Andronov-Hopf bifurcation, the homoclinic bifurcations and the coexistence of oscillating and rotational limit ccles, b means of the clindrical sstem (4), as shown in Fig 6 Due to the double Andronov-Hopf bifurcation, a pair of stable limit ccles can be found near equilibria (1/2, 3/2,0) and (1/2, 3/2,0) in Fig 6(a) Figure 6(b) shows the homoclinic bifurcation of first tpe, including a homoclinic orbits of first tpe connecting the equilibria (1,0,0) and two limit ccles, encircling the equilibria (1,0,0), (1/2,± 3/2,0) Figure 6(c) displas the homoclinic bifurcation of second tpe, including a homoclinic orbits of second tpe connecting the equilibria ( 1, 0, 0) and two limit ccles encircling the equilibria (1,0,0), (1/2,± 3/2,0) It is most interesting that the homoclinic orbits of the second tpe can bifurcate into a pair of rotational limit ccles around the unit clinder, coexisting with a stable limit ccles with thick solid line and a unstable limit ccles with thick dash line, as shown in Fig 6(d) Table 1 The number of limit ccle and their stabilit in ever region (ξ, δ) I II III IV V VI Limit Ccle S 2 S Limit Ccle SU 3 SUS 2 US 0 Limit Ccle SS 2 SS Table 2 The number of limit ccle and their stabilit in bifurcation curves (ξ, δ) l hopf l c1 l hom1 l hom2 l c2 Limit Ccle S Limit Ccle S 2 SU 2 SU 1 S Limit Ccle SS Double Hom S 0 0 Double Hom S 0 The number and the stabilit of limit ccles in ever region and bifurcation curve are listed in Tables 1 and 2, respectivel The limit ccle 1, limit ccle 2 and limit ccle 3 represent three kinds of limit ccles, small oscillating limit ccle, encircling one singular point, large oscillating limit ccle, encircling three singular points and rotational limit ccle, surrounding unit clinder, respectivel The double hom1 and double hom2 are on behalf of the double

19 Multiple Bifurcations of a Clindrical Dnamical Sstem 51 homoclinic orbits of first and second tpe Meanwhile, the stabilit of limit ccles could be divided into stable, unstable and semi-stable, denoted b S, U and S in Tables 1 and 2, respectivel 5 Conclusions In this paper, we have studied multiple bifurcations of a tipicall clindrical dnamical sstem, whose unperturbed sstem bears the coupling dnamics propert of the bistable state and the conventional pendulum with the homoclinic orbits of the first and second tpe In the perturbed sstem, a double Andronov-Hopf bifurcation, a pair of homoclinic bifurcations and two saddle-node bifurcations of periodic orbits have been investigated b using the analtical analsis and numerical calculation The number, position and stabilit of all the oscillating and rotational limit ccles are given as the perturbed parameters var It has been shown, that the homoclinic orbits of the second tpe can bifurcate into a pair of rotational limit ccles, coexisting with the oscillating limit ccle for a given set of parameters Meanwhile, we have discovered that there are four limit ccles in this clindrical dnamical sstem The contribution of this work is to provide a tpical example to reveal the complex dnamic bifurcation and different tpes of limit ccles of clindrical dnamical sstem All the results related to the perturbed sstem are obtained b the fourth order Runge-Kutta method, which ensures the accurac of the computation R EFERENCES [1] Guckenheimer, J, P Holmes Nonlinear Oscillations, Dnamical Sstems, and Bifurcations of Vector Fields, New York, Springer, 1983 [2] Yagasaki, K A Simple Feedback Control Sstem: Bifurcations of Periodic Orbits and Chaos Nonlinear Dn, 9 (1996), [3] Colonius, F, L Grüne Dnamics, Bifurcations and Control, Berlin, Springer- Verlag, 2002 [4] Kim, Y H, S H Kim, Y K Kwak Dnamic Analsis of a Nonholonomic Two- Wheeled Inverted Pendulum Robot J Intell Robot Sst, 44 (2005), [5] Anh, N D, H Matsuhisa, L D Viet, M Yasuda Vibration Control of an Inverted Pendulum Tpe Strcture b Passive Mass-Spring-Pendulum Dnamics Vibration Absorber J Sound Vib, 307 (2007), [6] Sorokin, A, F Arnold Electricall Charged Small Soot Particles in the Exhaust of an Aircraft Gas-Turbine Engine Combustor: Comparison of Model and Experiment Atmos Environ, 38 (2004), No 17,

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