BIFURCATIONS OF PERIODIC ORBITS IN THREE-WELL DUFFING SYSTEM WITH A PHASE SHIFT

Transcription

1 J Syst Sci Complex (11 4: BIFURCATIONS OF PERIODIC ORBITS IN THREE-WELL DUFFING SYSTEM WITH A PHASE SHIFT Jicai HUANG Han ZHANG DOI: 1.17/s Received: 9 May 8 / Revised: 5 December 9 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 11 Abstract Bifurcations of periodic orbits of three-well Duffing system with a phase shift are investigated in detail. The conditions of the existence bifurcations for harmonics, subharmonics (-order, 3- order m-order superharmonics under small perturbations are given by using second-order averaging method Melnikov s method. The influence of the phase shift on the dynamics is also obtained. Key words Bifurcations, Melnikov s method, periodic orbits, second-order averaging, three-well Duffing system. 1 Introduction In this paper, we consider a three-well Duffing system with a phase shift ẋ = y, ẏ = x(x a (x 1 δ 1 y + γ 1 cos(ωt + φ, (1 <a<1, δ 1,γ 1,ω are real parameters. Physically, δ 1 can be regarded as dissipation or damping, γ 1 as the amplitudes of the forcing, ω as the frequencies, for this reason, δ 1,γ 1,ω, a measures the ratio between the stable unstable equilibrium positions of the beam, while the φ sts for a phase shift. Equation (1 is a ubiquitous nonlinear differential equation, many nonlinear oscillators in physical, engineering even biological problems can really be described by the model or analogous ones. The Duffing equations with single-well or two-well potentials have been extensively investigated by using analytic methods numerical simulations. For examples, Moon [1], Holmes [ 3], Wiggins [4], Lakshmanan Murali [5], Yagasaki [6 7], Bunz, et al. [8], Parlitz Lauterborn [9], Rio, et al. [1] described the rich variety of bifurcation chaos phenomena Jicai HUANG School of Mathematics Statistics, Central China Normal University, Wuhan 4379, China. huangjicai@yahoo.com.cn. Han ZHANG College of Information Technical Science, Nankai University, Tianjin 371, China. This research was supported by the National Natural Science Foundation of China under Grant No. 176, CCNU Project under Grant No. CCNU9A13, Tianjin Fund for Natural Sciences 7JCYBJC147. This paper was recommended for publication by Editor Jinhu LÜ.

2 5 JICAI HUANG HAN ZHANG exhibited by them. Besides, Yagasaki [11 1] succeeded in analyzing the ultra-subharmonic resonance of order as well as the 4-order subharmonic superharmonic resonance by using higher-order averaging method up to 4-order, gave an extended version of the subharmonic Melnikov method applied it to study degenerate resonance at cusp bifurcations. Li Moon [13],Chacón Bejarano [14] studied the three-well Duffing system with one external forcing, they provided the necessary conditions for chaos based on both homoclinic heteroclinic bifurcation, plotted homoclinic heteroclinic bifurcation curves chaotic attractors by numerical simulations. Nayfeh Balachran [15] considered the three-well nonlinear system introduced the concept of the Melnikov equivalent damping as a global measure of the system. Cao Jing [16 17] discussed the chaotic dynamics bifurcations of periodic orbits in a Josephson equation with a phase shift. Huang Jing [18], Jing, et al. [19], studied the three-well Duffing system with one external forcing two external forcing, they provided the threshold values of chaotic motion under periodic quasiperiodic perturbations a control strategy of chaos by adjusting some parameters. This paper is organized as follows. In Section, we give the conditions of the existence bifurcations of harmonic resonance. The conditions of the existence bifurcations of -order subharmonics 3-order subharmonics are provided in Section 3. By using Melnikov method, the m-order subharmonics are discussed in Section 4. The -order 3-order superharmonics are discussed in Section 5. Finally, the influence of shifted phase on the dynamics of system (1 is concluded in Section 6. Primary Resonance Bifurcation In the section we study the existence of the primary resonance ω ω using the -order averaging method. Introduce a small parameter ε such that <ε 1 replace δ 1 γ 1 by εδ ε 3 γ, respectively, then Equation (1 can be rewritten as ẋ = y, ( ẏ = f(x εδy + ε 3 γ cos(ωt + φ, f(x =x(x a (x 1. Assume that ω ω = O(ε, εω = ω ω. Letting x = x + εz, (x, is a center of Equation (1 at ε =,ω = f (x, it is easy to see that: ω = a if x =; ω = (1 a ifx = ±1. Then Equation ( can be rewritten as z + ωz = εa z + ε[γ cos(ωt + φ a 3 z 3 δż]+o(ε 3, (3 a i = 1 d i f i! dx i (x, i =1,, 3 a 1 = a, a =, a 3 = 6 6a if x =; a 1 = a, a =14 6a, a 3 =54 6a if x =1; a 1 = a, a = a, a 3 =54 6a if x = 1.

3 BIFURCATIONS OF PERIODIC ORBITS IN A DUFFING SYSTEM 51 When ω kx, we use the van der Pol transformation ( ( ωt u cos k ( ωt = v k ω sin ( ( k ωt sin k ż ( ωt k ω cos z k upon using transformation (4 with k = 1, Equation (3 becomes ( = u v 1 ω [εωz + ( sinωt ε( a z +ε(γcos(ωt + φ a 3 z 3 δż] cosωt, (4 + O(ε 3, (5 in which z,ż can be written as functions of u, v t via Equation (4, Equation (5 is in the correct form for averaging. Carrying out averaging up to -order for Equation (5, we give the averaged equation: u = ε [ δω u + Ωv b (u + v v + γsinφ], ω v = ε (6 [ δω v Ωu + b (u + v u γcosφ], ω by simple calculations we can see b = 9a 1a 3 1a 1a 1, b = 9 (1 + a < ifx = b = 47 15a +63a 4 6( 1+a < ifx = ±1. In polar coordinates r = u + v θ=arctan(v/u, Equation (6 becomes ṙ = ε ( δω r + γsin(φ θ, r θ = ε ( Ωr + b r 3 γcos(φ θ. (7 ω ω Fixed points of Equation (7 satisfy b x3 b Ωx +(Ω + δ ω x γ =, (8 x = r.letg(x =b x 3 b Ωx +(Ω + δ ω x γ. The stability of fixed points of Equation (7 is determined by the sign of roots for λ +δω λ + h(x =, h(x =g (x =3b x 4Ωb x + Ω + δ ω. Let x = y +(Ω/3b, then Equation (8 becomes y 3 + py + q =,

4 5 JICAI HUANG HAN ZHANG p = 3δ ω Ω 3b, q = 7b γ +Ω 3 +18δ Ωω 7b 3. The discriminate of Equation (8 is given by ( p 3 ( q = + =7b 3 γ 4 4b Ω(Ω +9δ ω γ +4δ ω (Ω + δ ω. (9 Let denote the discriminate of =,then =16b (Ω 3δ ω 3. We have the following conclusion: i If Ω < 3δω, then there exist two positive real roots of =as γ 1 = Ω(Ω +9δ ω (Ω 3δ ω Ω 3δ ω 7b ; (1 ii If Ω > 3δω, then there exist not any positive real root of =; iii If Ω = 3δω, then there exists one positive real root of =: γ 1 = 8 3δ 3 ω 3 9b, Ω 1 = 3δω. By the above analysis we have the following conclusion: A supercritical saddle-node bifurcation of fixed points occurs at γ = γ+1 the subcritical saddle-node bifurcation at γ = γ 1. The two curves meet at (Ω 1,γ1inthe(Ω,γ -space. Moreover, by the Dulac s criterion, it is known that the averaged Equation (6 has no closed orbit. By the analysis of stability for the fixed points of Equation (6, Dulac s criterion the averaging theorem in [], we can give the following Theorem. Theorem 1 For Equation (, we have: i For Ω < 3δω <ε 1, there are two stable resonant harmonic solutions one unstable resonant harmonic solution when γ+1 <γ 1 <γ 1, there is a stable resonant harmonic solution when γ1 >γ 1 or γ1 <γ+1. A stable harmonic appears near the supercritical bifurcation curve γ+1 a stable harmonic disappears near the subcritical bifurcation curve γ 1. ii For Ω > 3δω <ε 1, there exists one stable harmonic; iii For Ω = 3δω there exists one stable harmonic if (Ω,γ Ω,γ (Ω 1,γ1, one unstable harmonic if (Ω,γ =(Ω 1,γ1. iv The harmonic solutions of Equation ( is approximately given by x(t =x + εr s cos(ωt + θ s +O(ε, (11 (r s,θ s is given by the equilibrium solutions of Equation (7.

5 BIFURCATIONS OF PERIODIC ORBITS IN A DUFFING SYSTEM Order 3-Order Subharmonic Resonance Bifurcations In this section we consider the -order subharmonic resonance ω ω set εω = (ω 4ω/4. Replace δ 1 γ 1 by εδ εγ ( <ε 1 in Equation (1, respectively. Using regular perturbation methods, one obtains harmonics of the adjusted Equation (1 as [ γ ] x(t =x ε ω ω cos(ωt + φ+o(ε. To investigate stability of the harmonic x(t, one can set Γ = γ 3a 1 = x = x(t+ εz = x + εz εγ cos(ωt + φ+o(ε, (1 γ (ω ω + O(ε, so that the adjusted Equation (1 becomes z + ω z = εa z + ε[a Γ zcos(ωt + φ a 3 z 3 δż]+o(ε 3, (13 ω =ω + O(ε. Using Van der Pol transformation (4 with k = in Equation (13, we get Equation (13 as ( = u v ω [εωz + ε( a z +ε(a Γ zcos(ωt + φ a 3 z 3 δż] sinωt cos ωt + O(ε 3, (14 in which z,ż can be written as functions of u, v t via Equation (4 with k =, Equation (14 is in the correct form for averaging. Carrying out averaging up to -order for Equation (14, one has u = ε [ (δω a Γ sinφu +(Ω a Γ cosφv b (u + v v], ω v = ε (15 [ (δω + a Γ sinφv (Ω + a Γ cosφu + b (u + v u]. ω By calculation, it is easy to see that: a =ifx =; a =14 6a > ifx =+1; a = a < ifx = 1. Furthermore, b is always negative. In polar coordinates r = u + v θ=arctan(v/u, Equation (15 becomes ṙ = ε ω ( δω + a Γ sin(φ θr, θ = ε ω ( Ω + b r a Γ cos(φ θ. (16 Fixed points of Equation (16 satisfy the following equation: b r4 b Ωr + Ω +(δω (a Γ =. (17 By the analyses of the roots of Equation (17 stability of the fixed points of Equation (16, using the averaging theorem, we have the following conclusions.

6 54 JICAI HUANG HAN ZHANG Theorem The saddle-node bifurcations of subharmonics occur near the curve SN1 : γ = 9a 1 a (δω, Ω < ; supercritical period doubling bifurcations of harmonics occur near the curve PD1 : γ = 9a 1 a [Ω +(δω ], Ω > ; subcritical period doubling bifurcations of harmonics occur near the curve PD : γ = 9a 1 a [Ω +(δω ], Ω <. Each pair of fixed points (r ±,θ ± (r ±,θ ± + π of Equation (16 correspond to a single -order subharmonic of Equation (1, which is approximately given by x(t =x + [( ω ] εr ± cos t + θ ± + O(ε, 1 r ± = (Ω ± (δω b +(a Γ, θ + = 1 ( ( δω φ arcsin, θ = π a Γ θ +, for x =1, θ = 1 ( φ arcsin ( δω a Γ, θ + = π θ, for x = 1. In the following part, we consider the 3-order subharmonic resonance ω 3ω set ε Ω = (ω 9ω /9. Replacing δ 1 γ 1 by ε δ εγ( < ε 1 in Equation (1, respectively. Using regular perturbation methods, one obtains a periodic solution of period π/ω of the adjusted Equation (1 as Let Γ = x(t =x + εx 1 (t+ε x (t+o(ε 3, x 1 (t = γ ω ω cos(ωt + φ, a γ x (t = a 1 (ω ω + a γ (ω ω (4ω ω + φ. cos(ωt x = x(t+εz = x + ε(z Γ cos(ωt + φ ε a Γ γ 8a 1 = 7a 1 (35 cos(ωt + φ + O(ε 3, γ (ω ω + O(ε. The adjusted Equation (1 becomes z + ω z = εa (z Γ cos(ωt + φz [ a +ε Γ (35 cos(ωt + φ(z Γ cos(ωt + φ 35a 1 ] a 3 (z Γ cos(ωt + φ 3 δ(ż + ωγ sin(ωt + φ + O(ε 3. (18

7 BIFURCATIONS OF PERIODIC ORBITS IN A DUFFING SYSTEM 55 Using the Van der Pol transformation (4 with k = 3 in Equation (18 carrying out averaging up to -order, so that the averaged equation corresponding to Equation (18 becomes u = ε ω [ δω u +(Ω b 1 Γ v b (u + v v + b Γ ((u v sinφ uv cos φ], v = ε ω [ δω v (Ω b 1 Γ u + b (u + v u b Γ ((u v cosφ +uv sin φ], b 1 = 15a 1a 3 6a 1a 1, b = 3a 1a 3 +a 4a 1, (19 b 1 = 9 9a, b = 9 9a if x =; b 1 = 3( a +3a 4 5( 1+a, b = a +7a 4 ( 1+a if x = ±1. In polar coordinates r = u + v θ=arctan(v/u, Equation (19 become ṙ = ε ω ( δω b Γ r sin(3θ φr, θ = ε ω ( Ω + b r + b 1 Γ b Γ rcos(3θ φ. Equation (19 always has the trivial fixed point (, which is always stable, the nontrivial fixed points satisfy the following equation: ( b r 4 +(b b 1 Γ b Ω b Γ r +(δω +(b 1 Γ Ω =. (1 If Ωb +(b b b 1 Γ > b (δω +(Ω b 1 Γ, ( then there are nontrivial fixed points (r, θ =(r ±,θ ± +(iπ/3(i =, 1, for Equation (, r ± = b Ω +(b b b b 1 Γ ± (b 4b b 1 b Γ 4 +4b b ΩΓ 4b δ ω, θ ± = 1 3 ( φ arc sin δω b Γ r ± for Ω + b 1Γ + b r ± b Γ r ± >, θ ± = π 3 1 ( φ arc sin δω for Ω + b 1Γ + b r± <. 3 b Γ r ± b Γ r ± A simple calculation shows that condition ( is equivalent to (b 4b b 1 b Γ 4 +4b b ΩΓ 4b δ ω > Ωb +(b b b 1 Γ >.

8 56 JICAI HUANG HAN ZHANG And it is easy to see that b 4b b 1 = 567/4 567a / 567a 4 /4 < for x =, b 4b b 1 = forx = ±1 a.8693, b 4b b 1 > b 4b b 1 < for x = ±1 <a<.8693, for x = ± <a<1. The stability of fixed points of Equation (3 is determined by the sign of roots for the following equation λ traλ +deta =, tra = δω, deta =3b r 4 + δ ω + b 1Γ 4 + Ω 4b Γ r +4b 1 Γ b r Ωb 1 Γ 4Ωb r +4b r 3 b Γ cos(3θ φ. By the above analysis for Equation (, we can know that (, is always stable, (r +,θ + +iπ/3,i =, 1,, arestable(r,θ +iπ/3,i =, 1,, are unstable, the following conclusion. Lemma 1 i If x = ±1, <a<.8693, Ω < 9 51 ( a +8189a 4 64a a 8 γ (47 15a +63a 4 ( 1+a γ >γ 3 1, then there exist six nontrivial fixed points one trivial fixed point (,, S 1 : γ1 = 18a 1 [ b b Ω b b Ω +(b 4b b 1 δ ω ] b (b 4b ; b 1 ii If x = ±1,a.8693, Ω <.3456γ,γ >γ, then there exist six nontrivial fixed points one trivial fixed point (,, S : γ.34 δ Ω, Ω < ; iii If x = ±1,.8693 <a<1, Ω < 9 51 ( a a 4 64a 6 + ( 963a 8 γ (47 15a +63a 4 ( 1+a or 3 x =, <a<1, Ω< 7(1+a γ 56a,γ 4 1 <γ <γ3, then there exist six nontrivial fixed points one trivial fixed point (,, S 3 : γ3 = 18a 1 [ b b Ω + b b Ω +(b 4b b 1 δ ω ] b (b 4b, Ω <. b 1 Theorem 3 i If the condition i in Lemma 1 is satisfied, then there exist two resonant 3-order subharmonics a stable nonresonant harmonic, in which one subharmonic is stable, while the other unstable, the supercritical saddle-node bifurcations of subharmonic occur near the curve S 1. ii If the condition ii in Lemma 1 is satisfied, then there exist two resonant 3-order subharmonics a stable nonresonant harmonic, in which one subharmonic is stable, while the other unstable, the supercritical saddle-node bifurcations of subharmonic occur near the curve S.

9 BIFURCATIONS OF PERIODIC ORBITS IN A DUFFING SYSTEM 57 iii If the condition iii in Lemma 1 is satisfied, then there exist two resonant 3-order subharmonics a stable nonresonant harmonic, in which one subharmonic is stable, while the other unstable, the supercritical saddle-node bifurcations of subharmonic occur near the curve S 1 for Ω <, the subcritical saddle-node bifurcations of subharmonic occur near the curve S 3. Each triple of fixed points (r ±,θ ± + iπ 3,i =, 1,, 4, corresponds to a single 3-order subharmonic of Equation (1, which is approximately given by x = x + [ ωt ] εr ± cos 3 + θ ± + O(ε. 4 m-order Subharmonics Bifurcation In this section we investigate the existence of m-order subharmonics of Equation (1 by using Melnikov s method for subharmonic which is defined in [4, 6]. Consider the perturbation system: ẋ = y, ẏ = x(x a (x 1 ε[δ 1 y γ 1 cos(ωt + φ]. (3 Let q α (t =(x α (t,y α (t(α (α 1,α denote a one-parameter family of periodic orbits with period πm nω of Equation (3 for ε =,α 1 α are constants, m n are relatively prime. In [6], it has been proved that M m/n (t can have simple zero only if n =1, so the Melnikov function for q α (t of Equation (3 is given by B m (ω = M m (t = πm ω πm ω y α(t [ δ 1 y α(t + γ 1 cos(ω(t + t +φ]dt = γ 1 A m (ωcos[ωt + Θ m (ω] δ 1 B m (ω, (4 [y α (t] dt >, A m (ω, φ = [C m (ω, φ] +[S m (ω, φ], Θ m (ω, φ = arctan Sm (ω, φ C m (ω, φ, C m (ω, φ = πm ω y α (tcos(ωt + φdt, S m (ω, φ = πm ω y α (tsin(ωt + φdt. By Melnikov Theorem, we can give the conclusion. Theorem 4 If γ 1 > Bm (ω δ 1 A m (ω, φ, (5 then M m (t has simple zeros there exists subharmonics of πm ω of Equation (3. Moreover, m-order subharmonics are created occurs at γ 1 = Bm (ω δ 1 A m (ω, φ + O(ε Rm (ω, φ+o(ε. (6

10 58 JICAI HUANG HAN ZHANG 5 Superharmonics Resonance Bifurcation In this section, we consider superharmonic resonance using the -order averaging method. For the case of -order superharmonic resonance ω ω,onesetsε Ω =4ω ω. Replace δ 1 γ 1 by ε δ ε 3/ γ ( <ε<1 in Equation (1, respectively. Let Γ = 4γ 3a 1 (= x = x + ε 3 Γ cos(ωt + φ+εz, γ + O(ε. The adjusted Equation (1 becomes ω ω z + ω z = εa (z + εγ cos(ωt + φ ε (a 3 z 3 + δż+o(ε 5. (7 Using the Van der Pol transformation (4 with k = 1 in Equation (7 carrying out averaging up to -order, the averaged equation corresponding to Equation (7 becomes [ u = ε δω u + Ωv b (u + v v a ] ω Γ sinφ, [ v = ε δω v Ωu + b (u + v u + a ] (8 ω Γ cosφ. In polar coordinates r = u + v θ=arctan(v/u, Equation (8 becomes [ ṙ = ε δω r + a [ ω Γ sin(θ φ ], r θ = ε Ωr + b r 3 + a ] ω Γ cos(θ φ. (9 The fixed points of Equation (9 satisfy the following equation: b x 3 b Ωx +(Ω + δ ωx a 4 Γ 4 =, (3 x = r.letg(x =b x 3 b Ωx +(Ω + δ ωx γ. The stability of fixed points of Equation (9 is determined by the sign of roots for the following equation: λ +δω λ + h(x =, h(x =g (x =δ ω + Ω 4Ωb x +3b x. The discriminate of Equation (33 is given by = 14b a4 γ8 96a 4 1 b a (Ω 3 +18Ωδ ω γ4 + 43a 8 1 δ ω (Ω + δ ω. (31 Let denote the discriminate of =,then = (19b a a 4 1 (Ω 3δ ω 3. Combining the above analyses we have the following Theorem. Theorem 5 For Ω < 3δω, a supercritical saddle-node bifurcations of -order superharmonics occur near the curve γ1 4 = 3a4 1 3b a [Ω(Ω +9δ ω+(ω 3δ ω Ω 3δ ω ], (3

11 BIFURCATIONS OF PERIODIC ORBITS IN A DUFFING SYSTEM 59 subcritical saddle-node bifurcations near the curve γ 4 = 3a4 1 3b a [Ω(Ω +9δ ω (Ω 3δ ω Ω 3δ ω ]. (33 A stable resonant superharmonic appears near the curve (3 a stable nonresonant superharmonic disappears near the curve (33. Now, we discuss the 3-order superharmonic resonance bifurcation. For the 3-order superharmonic resonance 3ω ω,onesetsε Ω =9ω ω. Replace δ 1 γ 1 by ε δ εγ( <ε 1, respectively. Let Γ = 9γ 8a 1 (= x = x + εγ cos(ωt + φ+εz, γ + O(ε. Equation ( becomes ω ω z + ω z = εa (z + Γ cos(ωt + φ ε [a 3 (z + Γ cos(ωt + φ 3 +δ(ż ωγ sin(ωt + φ] + O(ε 3. (34 Use the Van der Pol transformation (4 with k = 1 3 in Equation (34 carrying out averaging up to -order, the averaged equation corresponding to Equation (34 becomes u = ε ω [ δω u +(Ω b 3 Γ v b (u + v v b 4 Γ 3 sin3φ], v = ε ω [ δω v (Ω b 3 Γ u + b (u + v u + b 4 Γ 3 cos3φ], b 3 = 15a 1a 3 16a 7a 1, b 4 = 5a 1a 3 18a a 1, b 3 = 9 9a, b 4 = 3(1 + a if x =; (35 b 3 = a + 639a 4 35( 1+a, b 4 = 3(49 a +49a 4 1( 1+a if x = ±1. In polar coordinates r = u + v θ=arctan(v/u, Equation (35 becomes ṙ = ε ω [ δω r + b 4 Γ 3 sin(θ 3φ], r θ = ε ω [ (Ω b 3 Γ r + b r 3 + b 4 Γ 3 cos(θ 3φ]. The fixed points of Equation (36 satisfy (36 b x3 b (Ω b 3 Γ x +((Ω b 3 Γ + δ ω x b 4 Γ 6 =, (37 x = r.letg(x =b x3 b (Ω b 3 Γ x +((Ω b 3 Γ + δ ω x b 4 Γ 6. The stability of fixed points of Equation (36 is determined by the sign of roots for the following equation λ +δω λ + h(x =,

12 53 JICAI HUANG HAN ZHANG Note that if h(x =g (x =δ ω + Ω Ωb 3 Γ 4Ωb x + b 3Γ 4 +4b 3 Γ b x +3b x. ( b 3 3b b b b 4 4γ b a 1Ωb 3b 4γ 1 +( b 3 δ ω a 4 1b b b 4 3δ ω a Ω a 4 1b 3 b b 4γ 8 +( Ωa 6 1 b3 3 δ ω Ω 3 a 6 1 b b Ωa6 1 δ ω b b 4 γ6 +( Ω a 8 1 b 3 δ ω b 3 δ4 ω 4 a8 1 γ4 +( Ω 3 a 1 1 b 3δ ω Ωa1 1 δ4 ω 4 b 3γ +a 1 1 ( δ6 ω Ω 4 δ ω Ω δ 4 ω 4 =, then Equation (37 has a multiple root. By numerical simulation, For parameter values (38 there are two positive real roots of Equation (38: a =.5, δ =.1, Ω = 5, (39 γ 1.186, γ.697 if x =; γ , γ if x = ±1. Thus, we obtain the following theorem. Theorem 6 There are two saddle-node bifurcations of 3-order superharmonics occurring near γ 1 γ for the fixed parameter value (39, one is subcritical the other is supercritical. A stable resonant superharmonic appears near the supercritical saddle-node bifurcation a stable non-resonant superharmonic appears near the subcritical saddle-node bifurcation. 6 Conclusions The conditions of the existence bifurcations of harmonics, subharmonics superharmonics for a three-well Duffing system with a phase shift are investigated by using the -order averaging method Melnikov method. Comparing the above analyses with the first author s another paper (see [18], we can get the following conclusions about the influence of phase shift on the system dynamics: The variety of the phase shift doesn t effect the number of periodic orbits by the -order averaging method. References [1] F. C. Moon, Chaotic Fractal Dynamics, Wiley, New York, 199. [] C. Holmes P. Holmes, -order averaging bifurcations to subharmonics in Duffing s equation, Journal of Sound Vibration, 1981, 78(: [3] P. Holmes D. Whitley, On the attracting set for Duffing s equation, Physicia D, 1983, 7(1 3:

13 BIFURCATIONS OF PERIODIC ORBITS IN A DUFFING SYSTEM 531 [4] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems Chaos, Springer-Verlag, New York, 199. [5] M. Lakshnianan K. Murali, Chaos in Nonlinear Oscillations, World Scientific Publishing Co. ltd, [6] K. Yagasaki, -order averaging Melnikov analysis for forced non-linear oscillators, Journal of Sound Vibration, 1996, 19(4: [7] K. Yagasaki, A simple feedback control system: Bifurcations of periodic orbits chaos, Nonlinear Dynamics, 1996, 9: [8] H. Bunz, H. Ohno, H. Haken, Subcritical period doubling in Duffing equation-type III intermittency, attractor crisis, Z. Phys B, 1984, 56: [9] V. Parlitz W. Lauterborn, Supersturcture in the bifurcation set of the Duffing equation, Physics Letters A, 1985, 17: [1] E. D. Rio, M. G. Velarde, A. R. Lozanno, Long time date series difficulties with characterization of chaotic attractors: A case with intermittency III, Chaos, Solitons Fractals, 1994, 4(1: [11] K. Yagasaki, Detecting of bifurcation structures by higher-order averaging for Duffing s equation, Nonlinear Dynamics, 1999, 18: [1] K. Yagasaki, Degenerate resonances in forced oscillators, Discrete Continuous Dynamical Systems Series B, 3, 3(3: [13] G. X. Li F. C. Moon, Criteria for chaos of a three-well potential oscillator with homoclinic heteroclinic orbits, Journal of Sound Vibration, 199, 136(1: [14] R. Chacón J. D. Bejarano, Homoclinic heteroclinic chaos in a triple-well oscillator, Journal of Sound Vibration, 1995, 186(: [15] A. H. Nayfeh B. Balachran, Applied Nonlinear Dynamics-Analytical, Computational, Experimental Methods, John Wiley & Sons, [16] H. J. Cao Z. J. Jing, Chaotic dynamics for Josephson equation with a phase shift, Chaos, Solitons Fractals, 1, 14: [17] Z. J. Jing H. J. Cao, Bifurcations of periodic orbits in a Josephson equation with a phase shift, Inter. J. of Bifurcation Chaos,, 1(7: [18] J. C. Huang Z. J. Jing, Bifurcations chaos in three-well Duffing system with one external forcing, Chaos, Solitons Fractals, 9, 4: [19] Z. J. Jing, J. C. Huang, J. Deng, Complex dynamics in three-well duffing system with two external forcings, Chaos, Solitons Fractals, 7, 33(3: [] J. Guckenheimer P. Holmes, Nonlinear Oscillations,Dynamical Systems Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.