BIFURCATIONS OF PERIODIC ORBITS IN THREE-WELL DUFFING SYSTEM WITH A PHASE SHIFT
|
|
- Gordon McCormick
- 6 years ago
- Views:
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.
COMPLEX DYNAMICS AND CHAOS CONTROL IN DUFFING-VAN DER POL EQUATION WITH TWO EXTERNAL PERIODIC FORCING TERMS
International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 9 No. III (September, 2015), pp. 197-210 COMPLEX DYNAMICS AND CHAOS CONTROL IN DUFFING-VAN DER POL EQUATION WITH TWO EXTERNAL
More informationDifference Resonances in a controlled van der Pol-Duffing oscillator involving time. delay
Difference Resonances in a controlled van der Pol-Duffing oscillator involving time delay This paper was published in the journal Chaos, Solitions & Fractals, vol.4, no., pp.975-98, Oct 9 J.C. Ji, N. Zhang,
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More informationCitation Acta Mechanica Sinica/Lixue Xuebao, 2009, v. 25 n. 5, p The original publication is available at
Title A hyperbolic Lindstedt-poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators Author(s) Chen, YY; Chen, SH; Sze, KY Citation Acta Mechanica Sinica/Lixue Xuebao,
More informationDYNAMICS OF ASYMMETRIC NONLINEAR VIBRATION ABSORBER
8 Journal of Marine Science and Technology, Vol. 11, No. 1, pp. 8-19 (2003) DYNAMICS OF ASYMMETRIC NONLINEAR VIBRATION ABSORBER Chiou-Fong Chung and Chiang-Nan Chang* Key words: bifurcation, absorber,
More informationCALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISE-LINEAR SYSTEMS USING THE SHOOTING METHOD
Vietnam Journal of Mechanics, VAST, Vol. 34, No. 3 (2012), pp. 157 167 CALCULATION OF NONLINEAR VIBRATIONS OF PIECEWISE-LINEAR SYSTEMS USING THE SHOOTING METHOD Nguyen Van Khang, Hoang Manh Cuong, Nguyen
More information1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More informationEffect of various periodic forces on Duffing oscillator
PRAMANA c Indian Academy of Sciences Vol. 67, No. 2 journal of August 2006 physics pp. 351 356 Effect of various periodic forces on Duffing oscillator V RAVICHANDRAN 1, V CHINNATHAMBI 1, and S RAJASEKAR
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationZERO-HOPF BIFURCATION FOR A CLASS OF LORENZ-TYPE SYSTEMS
This is a preprint of: Zero-Hopf bifurcation for a class of Lorenz-type systems, Jaume Llibre, Ernesto Pérez-Chavela, Discrete Contin. Dyn. Syst. Ser. B, vol. 19(6), 1731 1736, 214. DOI: [doi:1.3934/dcdsb.214.19.1731]
More informationHopf Bifurcation Analysis and Approximation of Limit Cycle in Coupled Van Der Pol and Duffing Oscillators
The Open Acoustics Journal 8 9-3 9 Open Access Hopf ifurcation Analysis and Approximation of Limit Cycle in Coupled Van Der Pol and Duffing Oscillators Jianping Cai *a and Jianhe Shen b a Department of
More informationIn-Plane and Out-of-Plane Dynamic Responses of Elastic Cables under External and Parametric Excitations
Applied Mathematics 5, 5(6): -4 DOI:.59/j.am.556. In-Plane and Out-of-Plane Dynamic Responses of Elastic Cables under External and Parametric Excitations Usama H. Hegazy Department of Mathematics, Faculty
More informationHORSESHOES CHAOS AND STABILITY OF A DELAYED VAN DER POL-DUFFING OSCILLATOR UNDER A BOUNDED DOUBLE WELL POTENTIAL
Available at: http://publications.ictp.it IC/2009/040 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL
More informationWHAT IS A CHAOTIC ATTRACTOR?
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties
More informationInvariant manifolds of the Bonhoeffer-van der Pol oscillator
Invariant manifolds of the Bonhoeffer-van der Pol oscillator R. Benítez 1, V. J. Bolós 2 1 Dpto. Matemáticas, Centro Universitario de Plasencia, Universidad de Extremadura. Avda. Virgen del Puerto 2. 10600,
More informationBifurcation and Chaos in Coupled Periodically Forced Non-identical Duffing Oscillators
APS/13-QED Bifurcation and Chaos in Coupled Periodically Forced Non-identical Duffing Oscillators U. E. Vincent 1 and A. Kenfack, 1 Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria.
More informationDynamical behaviour of a controlled vibro-impact system
Vol 17 No 7, July 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(07)/2446-05 Chinese Physics B and IOP Publishing Ltd Dynamical behaviour of a controlled vibro-impact system Wang Liang( ), Xu Wei( ), and
More informationBIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs
BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.
More information(8.51) ẋ = A(λ)x + F(x, λ), where λ lr, the matrix A(λ) and function F(x, λ) are C k -functions with k 1,
2.8.7. Poincaré-Andronov-Hopf Bifurcation. In the previous section, we have given a rather detailed method for determining the periodic orbits of a two dimensional system which is the perturbation of a
More information2.034: Nonlinear Dynamics and Waves. Term Project: Nonlinear dynamics of piece-wise linear oscillators Mostafa Momen
2.034: Nonlinear Dynamics and Waves Term Project: Nonlinear dynamics of piece-wise linear oscillators Mostafa Momen May 2015 Massachusetts Institute of Technology 1 Nonlinear dynamics of piece-wise linear
More informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
More informationRational Energy Balance Method to Nonlinear Oscillators with Cubic Term
From the SelectedWorks of Hassan Askari 2013 Rational Energy Balance Method to Nonlinear Oscillators with Cubic Term Hassan Askari Available at: https://works.bepress.com/hassan_askari/4/ Asian-European
More informationNew Homoclinic and Heteroclinic Solutions for Zakharov System
Commun. Theor. Phys. 58 (2012) 749 753 Vol. 58, No. 5, November 15, 2012 New Homoclinic and Heteroclinic Solutions for Zakharov System WANG Chuan-Jian ( ), 1 DAI Zheng-De (à ), 2, and MU Gui (½ ) 3 1 Department
More informationNonlinear Stability and Bifurcation of Multi-D.O.F. Chatter System in Grinding Process
Key Engineering Materials Vols. -5 (6) pp. -5 online at http://www.scientific.net (6) Trans Tech Publications Switzerland Online available since 6//5 Nonlinear Stability and Bifurcation of Multi-D.O.F.
More information2:1 Resonance in the delayed nonlinear Mathieu equation
Nonlinear Dyn 007) 50:31 35 DOI 10.1007/s11071-006-916-5 ORIGINAL ARTICLE :1 Resonance in the delayed nonlinear Mathieu equation Tina M. Morrison Richard H. Rand Received: 9 March 006 / Accepted: 19 September
More informationTwo models for the parametric forcing of a nonlinear oscillator
Nonlinear Dyn (007) 50:147 160 DOI 10.1007/s11071-006-9148-3 ORIGINAL ARTICLE Two models for the parametric forcing of a nonlinear oscillator Nazha Abouhazim Mohamed Belhaq Richard H. Rand Received: 3
More informationAnti-synchronization of a new hyperchaotic system via small-gain theorem
Anti-synchronization of a new hyperchaotic system via small-gain theorem Xiao Jian( ) College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China (Received 8 February 2010; revised
More informationarxiv: v1 [nlin.cd] 20 Jul 2010
Invariant manifolds of the Bonhoeffer-van der Pol oscillator arxiv:1007.3375v1 [nlin.cd] 20 Jul 2010 R. Benítez 1, V. J. Bolós 2 1 Departamento de Matemáticas, Centro Universitario de Plasencia, Universidad
More informationGlobal analysis of the nonlinear Duffing-van der Pol type equation by a bifurcation theory and complete bifurcation groups method
Global analysis of the nonlinear Duffing-van der Pol type equation by a bifurcation theory and complete bifurcation groups method Raisa Smirnova 1, Mikhail Zakrzhevsky 2, Igor Schukin 3 1, 3 Riga Technical
More informationResearch Article Chaotic Behavior of the Biharmonic Dynamics System
Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 9, Article ID 979, 8 pages doi:.55/9/979 Research Article Chaotic Behavior of the Biharmonic Dynamics
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
More informationSolutions of Nonlinear Oscillators by Iteration Perturbation Method
Inf. Sci. Lett. 3, No. 3, 91-95 2014 91 Information Sciences Letters An International Journal http://dx.doi.org/10.12785/isl/030301 Solutions of Nonlinear Oscillators by Iteration Perturbation Method A.
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationLecture 5. Outline: Limit Cycles. Definition and examples How to rule out limit cycles. Poincare-Bendixson theorem Hopf bifurcations Poincare maps
Lecture 5 Outline: Limit Cycles Definition and examples How to rule out limit cycles Gradient systems Liapunov functions Dulacs criterion Poincare-Bendixson theorem Hopf bifurcations Poincare maps Limit
More informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
More informationRICH VARIETY OF BIFURCATIONS AND CHAOS IN A VARIANT OF MURALI LAKSHMANAN CHUA CIRCUIT
International Journal of Bifurcation and Chaos, Vol. 1, No. 7 (2) 1781 1785 c World Scientific Publishing Company RICH VARIETY O BIURCATIONS AND CHAOS IN A VARIANT O MURALI LAKSHMANAN CHUA CIRCUIT K. THAMILMARAN
More informationPeriod-One Rotating Solutions of Horizontally Excited Pendulum Based on Iterative Harmonic Balance
Advances in Pure Mathematics, 015, 5, 413-47 Published Online June 015 in Scies. http://www.scirp.org/journal/apm http://dx.doi.org/10.436/apm.015.58041 Period-One otating Solutions of Horizontally Excited
More informationEE222 - Spring 16 - Lecture 2 Notes 1
EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued
More informationPractice Problems for Final Exam
Math 1280 Spring 2016 Practice Problems for Final Exam Part 2 (Sections 6.6, 6.7, 6.8, and chapter 7) S o l u t i o n s 1. Show that the given system has a nonlinear center at the origin. ẋ = 9y 5y 5,
More informationA Model of Evolutionary Dynamics with Quasiperiodic Forcing
paper presented at Society for Experimental Mechanics (SEM) IMAC XXXIII Conference on Structural Dynamics February 2-5, 205, Orlando FL A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth
More informationGeneralized projective synchronization between two chaotic gyros with nonlinear damping
Generalized projective synchronization between two chaotic gyros with nonlinear damping Min Fu-Hong( ) Department of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China
More informationLYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR
LYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR Peter H. Baxendale Department of Mathematics University of Southern California Los Angeles, CA 90089-3 USA baxendal@math.usc.edu
More informationDynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited
Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited arxiv:1705.03100v1 [math.ds] 8 May 017 Mark Gluzman Center for Applied Mathematics Cornell University and Richard Rand Dept.
More informationStochastic response of fractional-order van der Pol oscillator
HEOREICAL & APPLIED MECHANICS LEERS 4, 3 4 Stochastic response of fractional-order van der Pol oscillator Lincong Chen,, a, Weiqiu Zhu b College of Civil Engineering, Huaqiao University, Xiamen 36, China
More informationComplicated behavior of dynamical systems. Mathematical methods and computer experiments.
Complicated behavior of dynamical systems. Mathematical methods and computer experiments. Kuznetsov N.V. 1, Leonov G.A. 1, and Seledzhi S.M. 1 St.Petersburg State University Universitetsky pr. 28 198504
More informationWIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY
WIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY J.M. TUWANKOTTA Abstract. In this paper we present an analysis of a system of coupled oscillators suggested
More informationChaos suppression of uncertain gyros in a given finite time
Chin. Phys. B Vol. 1, No. 11 1 1155 Chaos suppression of uncertain gyros in a given finite time Mohammad Pourmahmood Aghababa a and Hasan Pourmahmood Aghababa bc a Electrical Engineering Department, Urmia
More informationSolution of a Quadratic Non-Linear Oscillator by Elliptic Homotopy Averaging Method
Math. Sci. Lett. 4, No. 3, 313-317 (215) 313 Mathematical Sciences Letters An International Journal http://dx.doi.org/1.12785/msl/4315 Solution of a Quadratic Non-Linear Oscillator by Elliptic Homotopy
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 14, 2019, at 08 30 12 30 Johanneberg Kristian
More informationInvariant manifolds in dissipative dynamical systems
Invariant manifolds in dissipative dynamical systems Ferdinand Verhulst Mathematisch Instituut University of Utrecht PO Box 80.010, 3508 TA Utrecht The Netherlands Abstract Invariant manifolds like tori,
More informationResponse of A Hard Duffing Oscillator to Harmonic Excitation An Overview
INDIN INSTITUTE OF TECHNOLOGY, KHRGPUR 710, DECEMBER 8-0, 00 1 Response of Hard Duffing Oscillator to Harmonic Excitation n Overview.K. Mallik Department of Mechanical Engineering Indian Institute of Technology
More informationAnalysis of Dynamical Systems
2 YFX1520 Nonlinear phase portrait Mathematical Modelling and Nonlinear Dynamics Coursework Assignments 1 ϕ (t) 0-1 -2-6 -4-2 0 2 4 6 ϕ(t) Dmitri Kartofelev, PhD 2018 Variant 1 Part 1: Liénard type equation
More informationSystem Control Engineering 0
System Control Engineering 0 Koichi Hashimoto Graduate School of Information Sciences Text: Nonlinear Control Systems Analysis and Design, Wiley Author: Horacio J. Marquez Web: http://www.ic.is.tohoku.ac.jp/~koichi/system_control/
More informationPerturbation Theory of Dynamical Systems
Perturbation Theory of Dynamical Systems arxiv:math/0111178v1 [math.ho] 15 Nov 2001 Nils Berglund Department of Mathematics ETH Zürich 8092 Zürich Switzerland Lecture Notes Summer Semester 2001 Version:
More informationThree ways of treating a linear delay differential equation
Proceedings of the 5th International Conference on Nonlinear Dynamics ND-KhPI2016 September 27-30, 2016, Kharkov, Ukraine Three ways of treating a linear delay differential equation Si Mohamed Sah 1 *,
More informationAvailable online at ScienceDirect. Procedia IUTAM 19 (2016 ) IUTAM Symposium Analytical Methods in Nonlinear Dynamics
Available online at www.sciencedirect.com ScienceDirect Procedia IUTAM 19 (2016 ) 11 18 IUTAM Symposium Analytical Methods in Nonlinear Dynamics A model of evolutionary dynamics with quasiperiodic forcing
More informationTECHNICAL NOTE: PREDICTION OF THE THRESHOLD OF GLOBAL SURF-RIDING BY AN EXTENDED MELNIKOV METHOD
10 th International Conference 441 TECHNICAL NOTE: PREDICTION OF THE THRESHOLD OF GLOBAL SURF-RIDING BY AN EXTENDED MELNIKOV METHOD Wan Wu, Virginia Polytechnic Institute and State University, wanwu@vt.edu
More informationModeling the Duffing Equation with an Analog Computer
Modeling the Duffing Equation with an Analog Computer Matt Schmitthenner Physics Department, The College of Wooster, Wooster, Ohio 44691, USA (Dated: December 13, 2011) The goal was to model the Duffing
More informationRecent new examples of hidden attractors
Eur. Phys. J. Special Topics 224, 1469 1476 (2015) EDP Sciences, Springer-Verlag 2015 DOI: 10.1140/epjst/e2015-02472-1 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Review Recent new examples of hidden
More informationGeneralized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters
Vol 16 No 5, May 2007 c 2007 Chin. Phys. Soc. 1009-1963/2007/16(05)/1246-06 Chinese Physics and IOP Publishing Ltd Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with
More informationPhysics 235 Chapter 4. Chapter 4 Non-Linear Oscillations and Chaos
Chapter 4 Non-Linear Oscillations and Chaos Non-Linear Differential Equations Up to now we have considered differential equations with terms that are proportional to the acceleration, the velocity, and
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More informationEXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 9 EXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION JIBIN LI ABSTRACT.
More informationDETERMINATION OF THE FREQUENCY-AMPLITUDE RELATION FOR NONLINEAR OSCILLATORS WITH FRACTIONAL POTENTIAL USING HE S ENERGY BALANCE METHOD
Progress In Electromagnetics Research C, Vol. 5, 21 33, 2008 DETERMINATION OF THE FREQUENCY-AMPLITUDE RELATION FOR NONLINEAR OSCILLATORS WITH FRACTIONAL POTENTIAL USING HE S ENERGY BALANCE METHOD S. S.
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationResearch Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System
Abstract and Applied Analysis Volume, Article ID 3487, 6 pages doi:.55//3487 Research Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System Ranchao Wu and Xiang Li
More informationTime-delay feedback control in a delayed dynamical chaos system and its applications
Time-delay feedback control in a delayed dynamical chaos system and its applications Ye Zhi-Yong( ), Yang Guang( ), and Deng Cun-Bing( ) School of Mathematics and Physics, Chongqing University of Technology,
More informationShilnikov bifurcations in the Hopf-zero singularity
Shilnikov bifurcations in the Hopf-zero singularity Geometry and Dynamics in interaction Inma Baldomá, Oriol Castejón, Santiago Ibáñez, Tere M-Seara Observatoire de Paris, 15-17 December 2017, Paris Tere
More informationRen-He s method for solving dropping shock response of nonlinear packaging system
Chen Advances in Difference Equations 216 216:279 DOI 1.1186/s1662-16-17-z R E S E A R C H Open Access Ren-He s method for solving dropping shock response of nonlinear packaging system An-Jun Chen * *
More informationMath 4200, Problem set 3
Math, Problem set 3 Solutions September, 13 Problem 1. ẍ = ω x. Solution. Following the general theory of conservative systems with one degree of freedom let us define the kinetic energy T and potential
More informationAnalysis of the Takens-Bogdanov bifurcation on m parameterized vector fields
Analysis of the Takens-Bogdanov bifurcation on m parameterized vector fields Francisco Armando Carrillo Navarro, Fernando Verduzco G., Joaquín Delgado F. Programa de Doctorado en Ciencias (Matemáticas),
More informationA Delay-Duhem Model for Jump-Resonance Hysteresis*
A Delay-Duhem Model for Jump-Resonance Hysteresis* Ashwani K. Padthe and Dennis S. Bernstein Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 489-4, USA, {akpadthe,dsbaero}@umich.edu
More informationBifurcation control and chaos in a linear impulsive system
Vol 8 No 2, December 2009 c 2009 Chin. Phys. Soc. 674-056/2009/82)/5235-07 Chinese Physics B and IOP Publishing Ltd Bifurcation control and chaos in a linear impulsive system Jiang Gui-Rong 蒋贵荣 ) a)b),
More informationJournal of Applied Nonlinear Dynamics
Journal of Applied Nonlinear Dynamics 4(2) (2015) 131 140 Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/journals/jand-default.aspx A Model of Evolutionary Dynamics with Quasiperiodic
More informationRotational Number Approach to a Damped Pendulum under Parametric Forcing
Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004, pp. 518 522 Rotational Number Approach to a Damped Pendulum under Parametric Forcing Eun-Ah Kim and K.-C. Lee Department of Physics,
More informationChapter 14 Three Ways of Treating a Linear Delay Differential Equation
Chapter 14 Three Ways of Treating a Linear Delay Differential Equation Si Mohamed Sah and Richard H. Rand Abstract This work concerns the occurrence of Hopf bifurcations in delay differential equations
More informationANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM. China
Mathematical and Computational Applications, Vol. 9, No., pp. 84-9, 4 ANALYSIS AND CONTROLLING OF HOPF BIFURCATION FOR CHAOTIC VAN DER POL-DUFFING SYSTEM Ping Cai,, Jia-Shi Tang, Zhen-Bo Li College of
More informationThe SD oscillator and its attractors
The SD oscillator and its attractors Qingjie Cao, Marian Wiercigroch, Ekaterina Pavlovskaia Celso Grebogi and J Michael T Thompson Centre for Applied Dynamics Research, Department of Engineering, University
More informationStudies on Rayleigh Equation
Johan Matheus Tuwankotta Studies on Rayleigh Equation Intisari Dalam makalah ini dipelajari persamaan oscilator tak linear yaitu persamaan Rayleigh : x x = µ ( ( x ) ) x. Masalah kestabilan lokal di sekitar
More informationStrange dynamics of bilinear oscillator close to grazing
Strange dynamics of bilinear oscillator close to grazing Ekaterina Pavlovskaia, James Ing, Soumitro Banerjee and Marian Wiercigroch Centre for Applied Dynamics Research, School of Engineering, King s College,
More informationnonlinear oscillators. method of averaging
Physics 4 Spring 7 nonlinear oscillators. method of averaging lecture notes, spring semester 7 http://www.phys.uconn.edu/ rozman/courses/p4_7s/ Last modified: April, 7 Oscillator with nonlinear friction
More informationThe Big, Big Picture (Bifurcations II)
The Big, Big Picture (Bifurcations II) Reading for this lecture: NDAC, Chapter 8 and Sec. 10.0-10.4. 1 Beyond fixed points: Bifurcation: Qualitative change in behavior as a control parameter is (slowly)
More informationControl of Chaos in Strongly Nonlinear Dynamic Systems
Control of Chaos in Strongly Nonlinear Dynamic Systems Lev F. Petrov Plekhanov Russian University of Economics Stremianniy per., 36, 115998, Moscow, Russia lfp@mail.ru Abstract We consider the dynamic
More informationModels Involving Interactions between Predator and Prey Populations
Models Involving Interactions between Predator and Prey Populations Matthew Mitchell Georgia College and State University December 30, 2015 Abstract Predator-prey models are used to show the intricate
More informationConnecting orbits in perturbed systems
Noname manuscript No. (will be inserted by the editor) Connecting orbits in perturbed systems Fritz Colonius Thorsten Hüls Martin Rasmussen December 3, 8 Abstract We apply Newton s method in perturbed
More informationImproving convergence of incremental harmonic balance method using homotopy analysis method
Acta Mech Sin (2009) 25:707 712 DOI 10.1007/s10409-009-0256-4 RESEARCH PAPER Improving convergence of incremental harmonic balance method using homotopy analysis method Yanmao Chen Jike Liu Received: 10
More informationUNKNOWN BIFURCATION GROUPS WITH CHAOTIC AND RARE ATTRACTORS IN THE DUFFING-UEDA AND DUFFING - VAN DER POL ARCHETYPAL OSCILLATORS
11 th International Conference on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 9-12 September 2013 UNKNOWN BIFURCATION GROUPS WITH CHAOTIC AND RARE ATTRACTORS IN THE DUFFING-UEDA AND
More informationPerturbation theory for anharmonic oscillations
Perturbation theory for anharmonic oscillations Lecture notes by Sergei Winitzki June 12, 2006 Contents 1 Introduction 1 2 What is perturbation theory 1 21 A first example 1 22 Limits of applicability
More informationDynamicsofTwoCoupledVanderPolOscillatorswithDelayCouplingRevisited
Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 7 Issue 5 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals
More information8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)
More informationChapter 23. Predicting Chaos The Shift Map and Symbolic Dynamics
Chapter 23 Predicting Chaos We have discussed methods for diagnosing chaos, but what about predicting the existence of chaos in a dynamical system. This is a much harder problem, and it seems that the
More informationPeriod-doubling cascades of a Silnikov equation
Period-doubling cascades of a Silnikov equation Keying Guan and Beiye Feng Science College, Beijing Jiaotong University, Email: keying.guan@gmail.com Institute of Applied Mathematics, Academia Sinica,
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY P. Yu 1,2
More information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationUNIFORM SUBHARMONIC ORBITS FOR SITNIKOV PROBLEM
Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 UNIFORM SUBHARMONIC ORBITS FOR SITNIKOV PROBLEM CLARK ROBINSON Abstract. We highlight the
More informationProblem Set Number 5, j/2.036j MIT (Fall 2014)
Problem Set Number 5, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Fri., October 24, 2014. October 17, 2014 1 Large µ limit for Liénard system #03 Statement:
More informationPeriod-One Rotating Solution of Parametric Pendulums by Iterative Harmonic Balance
Period-One Rotating Solution of Parametric Pendulums by Iterative Harmonic Balance A THESIS SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI I AT MĀNOA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
More informationOn a Codimension Three Bifurcation Arising in a Simple Dynamo Model
On a Codimension Three Bifurcation Arising in a Simple Dynamo Model Anne C. Skeldon a,1 and Irene M. Moroz b a Department of Mathematics, City University, Northampton Square, London EC1V 0HB, England b
More information