Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 24, No. 5 (2014) (28 pages) c World Scientific Publishing Company DOI: /S Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Albert C. J. Luo and Bo Yu Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL , USA aluo@siue.edu Received September 4, 2013 In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well. Keywords: Parametric quadratic nonlinear oscillator; period-1 motions to chaos; period-2 motions to chaos; nonlinear dynamical systems. 1. Introduction The stability of solutions in parametric linear systems has been understood through Hill equation and Mathieu equation. Parametric nonlinear systems extensively exist in engineering. The parametric nonlinear systems possess dynamical behaviors distinguishing from periodically forced nonlinear systems. In the traditional perturbation analysis, only symmetric periodic motions in parametrically forced nonlinear systems can be determined. In this paper, one of the simplest parametric nonlinear oscillators will be investigated, and the nonlinear behaviors of such possible periodic motions will be discussed through the frequency amplitude curves. For parametric oscillators, one should go back to periodic solutions in the Mathieu equations. Mathieu [1868] investigated the linear Mathieu equation (also see [Mathieu, 1873; McLachlan, 1947]). The corresponding stability of periodic solutions in the linear Mathieu equation was discussed. Whittaker [1913] presented a method to find the unstable solutions for very weak excitation (also see [Whittaker & Watson, 1935]). In engineering, Sevin [1961] used the Mathieu equation to investigate the vibration-absorber with parametric excitation. Hsu [1963] discussed the first approximation analysis and stability criteria for a multipledegree of freedom dynamical system (also see [Hsu, 1965]). Hayashi [1964] discussed the approximate periodic solutions in parametric systems and the corresponding stability by the averaging method and harmonic balance method. Tso and Caughey [1965] discussed the stability of parametric, nonlinear systems. Mond et al. [1993] did the stability

2 A. C. J. Luo & B. Yu analysis of nonlinear Mathieu equation. Zounes and Rand [2000] discussed the transient response for the quasi-periodic Mathieu equation. Luo [2004] discussed chaotic motions in the resonant separatrix bands of the Mathieu Duffing oscillator with a twin-well potential. Shen et al. [2008] used the incremental harmonic balance method to investigate the bifurcation route to chaos in the Mathieu Duffing oscillator. Luo [2012] developed a generalized harmonic balance method to determine periodic motions in nonlinear dynamical systems. Luo and Huang [2012a] applied the generalized harmonic balance method to determine period-m solutions in the Duffing oscillator (also see [Luo & Huang, 2012b, 2012c]). In [Luo & O Connor, 2013], periodic motions in hardening Mathieu Duffing oscillator were presented. In the traditional analysis, asymmetric periodic motions cannot be obtained. In other words, period-1 motions in parametric linear oscillator cannot be obtained. In [Luo & Yu, 2013a, 2013b, 2013c], periodically forced, quadratic nonlinear oscillators were investigated, and bifurcation trees of period-1 motion to chaos were presented. Herein, parametric quadratic nonlinear oscillators will be investigated. In this paper, analytical solutions of parametric nonlinear oscillators will be developed through the generalized harmonic balance. Analytical bifurcation trees of period-2 motion to chaos in such a parametric oscillator will be determined first and the bifurcation trees of period-1 to chaos will be also discussed. The corresponding stability and bifurcation of periodic motions should be presented through eigenvalue analysis. Numerical illustrations will be completed for a better understanding of periodic motions in the simplest parametric nonlinear oscillators. 2. Approximate Solutions Consider a parametric, quadratic nonlinear oscillator as ẍ + δẋ +(α + Q 0 cos Ωt)x + βx 2 =0 (1) where parameters α and β are positive constants. The damping coefficient δ is positive. A parametric excitation xq 0 cos Ωt has excitation amplitude Q 0 and frequency Ω. The standard form of Eq. (1) is written as ẍ + f(x, ẋ, t) =0 (2) where f(x, ẋ, t) =δẋ +(α + Q 0 cos Ωt)x + βx z. (3) As in [Luo, 2012], the analytical solution of periodm motion is assumed as x (t) =a (m) 0 (t)+ k=1 ( ) k b k/m (t)cos m Ωt ( ) k + c k/m (t)sin m Ωt. (4) From the foregoing approximate expression, the first and second order derivatives of x (t) are ẋ (t) =ȧ (m) 0 + ẍ (t) =ä (m) 0 + ( ḃ k/m + kω ) ( ) kωt m c k/m cos m ) ( ) kωt sin, (5) m ( bk/m +2 kω m ċk/m k=1 ( + ċ k/m kω m b k/m k=1 ( ) kω 2 ( ) kω b m k/m) cos m t ( + c k/m 2 kω ( ) kω 2 m ḃk/m c m k/m) ( ) kω sin m t. (6) Substituting Eqs. (4) (6) intoeq.(2) and averaging all terms of cos(kωt/m) and sin(kωt/m) gives ä (m) 0 + F (m) 0 (a (m) 0, b (m), c (m), ȧ (m) 0, ḃ (m), ċ (m) )=0 bk/m +2 kω ( ) kω 2 m ċk/m b m k/m + F (m) 1k (a(m) 0, b (m), c (m), ȧ (m) 0, ḃ(m), ċ (m) )=0 c k/m 2 kω ( ) kω 2 m ḃk/m c m k/m + F (m) 2k (a(m) 0, b (m), c (m), ȧ (m) 0, ḃ(m), ċ (m) )=0 for k =1, 2,...,N. (7)

3 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator The coefficients of constant, cos(kωt/m) and sin(kωt/m) for the function of f(x, ẋ, t) can be obtained in the form of + β [(b i/m b j/m + c i/m c j/m )δ k j i i=1 j=1 F (m) 0 (a (m) 0, b (m), c (m), ȧ (m) 0, ḃ(m), ċ (m) ) = 1 mt mt 0 F (x, ẋ, t)dt = δȧ (m) 0 + αa (m) 0 + β(a (m) 0 ) Q 0a k/m δ k m + β 2 (b 2 i/m + c2 i/m ) i=1 F (m) 1k (a(m) 0, b (m), c (m), ȧ (m) 0, ḃ(m), ċ (m) ) = 2 mt ( ) k F (x, ẋ, t)cos mt 0 m Ωt dt ( ) kω = δ ḃ k/m + c k/m + αb m k/m +2βa (m) 0 b k/m + f k/m F (m) 2k (a(m) 0, b (m), c (m), ȧ (m) 0, ḃ(m), ċ (m) ) = 2 mt ( ) k F (x, ẋ, t)sin mt 0 m Ωt dt ( ) kω = δ ċ k/m b k/m + αc m k/m where +2βa (m) 0 c k/m + f (s) k/m f k/m = a(m) 0 Q 0 δ k m Q 0 a i/m (δ k i+m + δ k m i + δ k i m) i=1 g (m) (z (m), z (m) 1 )= F (m) F (m) (8) f (s) k/m = 1 2 Q 0 Define (b i/mb j/m c i/m c j/m )δ k i+j] + β b i/m (δ k i+m + δk i m δk m i ) i=1 i=1 j=1 z (m) (a (m) 0, b (m), c (m) ) T b i/m c j/m (δ k i+j + δ k j i δ k i j). =(a (m) 0,b 1/m,...,b N/m,c 1/m,...,c N/m ) T (z (m) 0,z (m) 1,...,z (m) 2N )T, z (m) 1 = ż (m) =(ȧ (m) 0, ḃ(m), ċ (m) ) T where =(ȧ (m) 0, ḃ1/m,...,ḃn/m, ċ 1/m,...,ċ N/m ) T (9) (ż (m) 0, ż (m) 1,...,ż (m) 2N )T (10) b (m) =(b 1/m,b 2/m,...,b N/m ) T, c (m) =(c 1/m,c 2/m,...,c N/m ) T. Equation (7) becomes where ż (m) = z (m) 1 and ż (m) 1 = g (m) (z (m), z (m) 1 ) F (m) 0 (z (m), z (m) 1 ) ( ) 1 (z (m), z (m) Ω Ω 2 1 ) 2k mċ(m) 1 + k 2 b (m) m ( ) 2 (z (m), z (m) Ω Ω 2 1 )+2k mḃ(m) 1 + k 2 c (m) m (11) (12) (13)

4 A. C. J. Luo & B. Yu with and k 1 =diag(1, 2,...,N), k 2 =diag(1, 2 2,...,N 2 ); F (m) 1 =(F (m) 11,F(m) 12,...,F(m) 1N )T, F (m) 2 =(F (m) 21,F(m) 22,...,F(m) 2N )T for N =1, 2,..., y (m) (z (m), z (m) 1 ) and (14) f (m) =(z (m) 1, g (m) ) T (15) Thus, equation (12) becomes ẏ (m) = f (m) (y (m) ). (16) The steady-state solutions for periodic motions in the parametric quadratic nonlinear oscillator can be obtained by setting ẏ (m) = 0, i.e. F (m) 0 (a (m) 0, b (m), c (m), 0, 0, 0) =0, F (m) 1 (a (m) 0, b (m), c (m), 0, 0, 0) Ω2 m 2 k 2b (m) = 0, F (m) 2 (a (m) 0, b (m), c (m), 0, 0, 0) Ω2 m 2 k 2c (m) = 0. (17) The (2N + 1) nonlinear equations in Eq. (17) are solved by the Newton Raphson method. In [Luo, 2012], the linearized equation at y (m) =(z (m), 0) T is where ẏ (m) = Df (m) (y (m) ) y (m) (18) Df (m) (y (m) )= f (m) (y (m) ) y (m) The Jacobian matrix is and Df (m) (y (m) ) = 0 (2N+1) (2N+1) G (2N+1) (2N+1) y (m) I (2N+1) (2N+1) H (2N+1) (2N+1). (19) (20) G = g(m) z (m) =(G(0), G, G (s) ) T (21) with G (0) =(G (0) 0,G(0) 1,...,G(0) 2N ), G =(G 1, G 2,...,G N )T, G (s) =(G (s) 1, G(s) 2,...,G(s) N )T for N =1, 2,..., with G k G (s) k =(G k0,g k1,...,g k(2n) ), (22) =(G (s) k0,g(s) k1,...,g(s) k(2n) ) (23) for k =1, 2,...,N. The corresponding components are G (0) r = αδ r Q 0δ r m βg(0) r, ( ) G kω 2 kr = δ r k m αδr k δ kω m δr k+n 1 2 Q 0δ k m δr Q 0 i=1 (δ k i+m + δ k m i + δ k i m)δ r i βg kr, ( ) G (s) kω 2 kr = δ r k+n m + δ kω m δr k αδr k+n where g (0) r 1 4 Q 0 βg (s) kr (δ k i+m + δ k i m δ k m i)δ r i+n i=1 (24) =2a (m) 0 δ r 0 + b k/mδ r k + c k/mδ r k+n, (25) g kr =2b k/mδ 0 r +2a(m) 0 δ r k + i=1 j=1 [b j/m (δ k j i + δ k i j + δ k i+j)δ r i + c j/m (δ k j i + δk i j δk i+j )δr i+n ], (26) g (s) kr =2c k/mδ r 0 +2a(m) 0 δ r k+n + i=1 j=1 c j/m (δ k i+j + δ k j i δ k i j)δ r i + b i/m (δ k i+j + δk j i δk i j )δr j+n (27) for r =0, 1,...,2N.

5 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator where H = g(m) z (m) 1 =(H (0), H, H (s) ) T H (0) =(H (0) 0,H(0) 1,...,H(0) 2N ), H =(H 1, H 2,...,H N )T, H (s) =(H (s) 1, H(s) 2,...,H(s) N )T for N =1, 2,...,, with H k H (s) k (28) (29) =(H k0,h k1,...,h k(2n) ), (30) =(H (s) k0,h(s) k1,...,h(s) k(2n) ) for k =1, 2,...,N. The corresponding components are H (0) r = δδ r 0, H kr = 2kΩ m δr k+n δδr k, (31) H (s) kr =2kΩ m δr k δδr k+n for r =0, 1,...,2N. The corresponding eigenvalues are determined by Df (m) (y (m) ) λi 2(2N+1) 2(2N+1) =0. (32) From Luo [2012], the eigenvalues of Df (m) (y (m) ) are classified as (n 1,n 2,n 3 n 4,n 5,n 6 ). (33) The corresponding boundary between the stable and unstable solutions is given by the saddle-node bifurcation and Hopf bifurcation. 3. Frequency Amplitude Curves The curves of harmonic amplitude varying with excitation frequency Ω for each harmonic term are illustrated. The corresponding solution in Eq. (4) can be rewritten as ( ) x (t) =a (m) k 0 + A k/m cos m Ωt ϕ k/m, k=1 (34) where the harmonic amplitude and phase are defined by A k/m b 2 k/m + c2 k/m, The system parameters are ϕ k/m =arctan c k/m b k/m (35) δ =0.5, α =5, β =20. (36) In all frequency amplitude curves, the acronyms and represent the saddle-node and Hopf bifurcations, respectively. Solid curves represent stable period-m motions. Long dashed, short dashed and chain curves represent unstable period-1, period-2 and period-4 motions, respectively. Consider a bifurcation tree of period-2 motion to chaos through period-2 to period-4 motion in parametrically excited, quadratic nonlinear oscillator. Using the parameters in Eq. (36), the frequency amplitude curves based on 40 harmonic terms of period-2 to period-4 motion are presented in Figs. 1 3 for Q 0 = 10. In Fig. 1, a global view of frequency amplitude curve for period-2 to period-4 motion is presented. In Fig. 1, constant a (m) 0 versus excitation frequency Ω is presented, and all the values of a (m) 0 are less than zero. a (m) 0 ( 0.17, ) is for period-2 and period-4 motion. The saddle-node bifurcations of period-2 motions are their onset points at Ω cr 2.6, 6.4. The Hopf bifurcations of period-2 motions yield the onset of period-4 motions at Ω cr 2.59, 5.95 for the large branch and Ω cr 2.92, 3.0 for the small branch. In Fig. 1, the harmonic amplitude A 1/4 versus excitation frequency Ω is presented for period-4 motion. We have A 1/4 (, ). The Hopf bifurcation points of period-4 motions are the saddlenode bifurcation points for period-8 motions. Since the stable period-8 motion exists in the short range, period-8 motions will not be presented herein. In Fig. 1, the harmonic amplitudes A 1/2 versus excitation frequency Ω are presented for period-2 and period-4 motions, as for constant a (m) 0 in Fig. 1. We have A 1/2 (, 0.3). In Fig. 1, the harmonic amplitude A 3/4 versus excitation frequency Ω is presented for period-4 motion in the range of A 3/4 (, 0.12). The frequency amplitude curves in (Ω,A 1 ) are illustrated in Fig. 1(e) for period-2 and period-4 motions in the range of

6 A. C. J. Luo & B. Yu 0 Constant Term, a 0 (m) Harmonic Amplitude, A 1/ Harmonic Amplitude, A 1/2 Harmonic Amplitude, A Harmonic Amplitude, A 3/4 Harmonic Amplitude, A e e e e (e) 1e Fig. 1. Frequency amplitude curves (Q 0 = 10) based on 40 harmonic terms (40) of period-2 motion to period-4 motion in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (h) harmonic amplitude A k/m (k =1, 2,...,4, 8, 12, 40, m=4).(δ =0.5, α=5,β= 20). (f)

7 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Harmonic Amplitude, A 3 1e-2 1e-3 1e-4 1e e (g) Fig. 1. A 1 (, 0.3). To save space, the harmonic amplitude A k/4 (mod(k, 4) 0) will not be presented. To further show effects of harmonic terms, the harmonic amplitudes A 2,A 3,A 10 are presented in Figs. 1(f) 1(h). A 2 < 10 1, A 3 < 10 2 and A 10 < 10 9 are observed. Thus, effects of the higher order harmonic terms on period-2 and period-4 motions can be ignored. To clearly illustrate the bifurcation trees of period-2 motions, the bifurcation tree-1 of period-2 motion to period-4 motion based on 40 harmonic terms is presented in Fig. 2 for Q 0 = 10 within the range of Ω (5.70, 6.0). In Fig. 2, the constant versus excitation frequency is presented. The bifurcation tree is very clearly presented. In -24 Harmonic Amplitude, A 10 1e-11 1e-12 1e-13 1e-14 1e-15 1e-16 (Continued) 1e-12 1e-13 1e-17 1e-18 1e-19 1e (h) Fig. 2, the amplitude versus excitation frequency for period-4 motion in the bifurcation tree is presented. We have A 1/4 = 0 for period-2 motion. The bifurcation tree-1 for harmonic amplitude A 1/2 is presented in Fig. 2. For the zoomed range, the quantity level of the harmonic amplitudes is A 1/ Similar to A 1/4, the harmonic amplitude A 3/4 for period-4 motion in bifurcation tree-1 is presented in Fig. 2. The harmonic amplitude A 1 for the bifurcation tree-1 of period-2 to period- 4 motion is given in Fig. 2(e). For the zoomed range, the quantity level of the harmonic amplitude is A For the period-2 motion, the harmonic amplitude A k/2 (k =2l 1, l =1, 2,...)is very important. Thus, the harmonic amplitude A 3/2 6 Constant Term, a 0 (m) Harmonic Amplitude, A 1/ Fig. 2. Bifurcation tree-1 of period-2 motion (Q 0 = 10) to period-4 motion based on 40 harmonic terms (40) in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (h) harmonic amplitude A k/m (k =1, 2,...,4, 6, 8, 24, m =4).(δ =0.5,α=5,β= 20)

8 A. C. J. Luo & B. Yu Harmonic Amplitude, A 1/ Harmonic Amplitude, A 3/ Harmonic Amplitude, A (e) 1.6e-4 Harmonic Amplitude, A 3/ (f) 3e-11 Harmonic Amplitude, A 2 1.4e-4 1.2e-4 1.0e-4 Harmonic Amplitude, A 6 2e-11 2e-11 2e e (g) Fig. 2. 1e (h) (Continued)

9 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Constant Term, a 0 (m) Harmonic Amplitude, A 1/ Harmonic Amplitude, A 1/2 Harmonic Amplitude, A (e) Harmonic Amplitude, A 3/4 Harmonic Amplitude, A 3/ Fig. 3. Bifurcation tree-2 of period-2 motion (Q 0 = 10) to period-4 motion based on 40 harmonic terms (40) in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (h) harmonic amplitude A k/m (k =1, 2,...,4, 6, 8, 24, m =4).(δ =0.5, α=5,β= 20). (f)

10 A. C. J. Luo & B. Yu e-7 Harmonic Amplitude, A Harmonic Amplitude, A 6 1.2e-7 8.0e-8 4.0e (g) (h) Fig. 3. for the bifurcation tree-1 is presented in Fig. 2(f). For the zoomed range, the quantity level of the harmonic amplitudes is A 3/ The harmonic amplitude with the range of A is presented in Fig. 2(g). For reduction of abundant illustrations, the harmonic amplitude A for the bifurcation is presented in Fig. 2(h). Other harmonic amplitudes can be similarly presented. The second bifurcation tree of period-2 motion to period-4 motion is in the narrow range of Ω (2.90, 3.02), as shown in Fig. 3. In this bifurcation tree, the period-4 motion has four parts of stable solutions and three parts of unstable solution on the same curve. In Fig. 3, constant a (m) is presented. The four segments of stable solutions Constant Term, a 0 (m) (Continued) and three segments of unstable solutions of period-4 motions are clearly observed. The harmonic amplitude A 1/ is presented in Fig. 3 and the two segments of stable solutions in the middle of frequency ranges are very tiny, which is difficult to be observed. The harmonic amplitude A 1/2 0.3 is presented in Fig. 3 for the bifurcation of period- 2 to period-4 motion. The harmonic amplitude A 3/ in Fig. 3 is similar to the harmonic amplitude A 1/4. The harmonic amplitude A 1 4 is similar to A 1/2, as shown in Fig. 3(e). To avoid abundant illustrations, the harmonic amplitudes A k/4 (k = 4l +1, l = 1, 2,...) will not be presented, which are similar to A 1/4. The harmonic amplitude A 3/ is presented in Fig. 3(f), Harmonic Amplitude, A 1/ Fig. 4. Global view for frequency amplitude curves (Q 0 = 15) based on 80 harmonic terms (80) of period-1 motion to period-4 motion in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (h) harmonic amplitude A k/m (k =1, 2,...,4, 8, 12, 80,m=4).(δ =0.5, α=5,β= 20)

11 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Harmonic Amplitude, A 1/ Harmonic Amplitude, A 3/ Harmonic Amplitude, A 1 Harmonic Amplitude, A (e) Harmonic Amplitude, A 2 Harmonic Amplitude, A (f) 1e+0 1e-1 1e-2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 1e-11 1e-12 1e-13 1e-14 1e-15 1e-16 1e-17 1e-18 1e-19 1e (g) (h) Fig. 4. (Continued)

12 A. C. J. Luo & B. Yu which is different from A 1/2. The harmonic amplitude A is similar to A 1,asshown in Fig. 3(g). For this bifurcation tree, the harmonic amplitude A is presented in Fig. 3(h). For a linear parametric system, one can find period-2 motions instead of period-1 motion. However, for a nonlinear parametric system, period-1 motions can be found. The global view of the frequency amplitude for bifurcation trees of period- 1 motion to period-4 motion is presented for Q 0 = 15. Such illustrations of constants and harmonic amplitude (a (m) 0, A 1/4,A 1/2,A 3/4, A 1,A 2,A 3,A 20 ) are presented for the frequency range of Ω (0, 3.0) in Figs. 4 4(h), respectively. In Fig. 4, the constant a (m) 0 ( 0.4, ) versus excitation frequency is presented. There are a few branches of bifurcation trees and the stable and unstable solutions of period-1 to period-4 motions are crowded together. The saddle-node bifurcation of period-1 motion occurs between the stable and unstable period-1 motions without the onset of a new periodic motion. In addition to unstable period-1 motion, the Hopf bifurcation of stable period-1 motion will generate the onset of period-2 motion. Continually, the saddle-node bifurcation of period-2 motion is the Hopf bifurcation of period-1 motion, and the Hopf bifurcation of period-2 motion is the onset of period-4 motion with a saddlenode bifurcation. For period-4 motion, the harmonic amplitude A 1/4 5 is presented in Fig. 4. Most of the solutions are unstable, and the stable solutions are in a few short ranges. In Constant Term, a 0 (m) addition, independent unstable period-4 motions are observed. The harmonic amplitude A 1/2 0.4 is presented in Fig. 4. The onset of period-2 motion is at the Hopf bifurcation of the period-1 motion. Nonindependent period-4 motions are generated by the Hopf bifurcations of period-2 motions, but the independent period-4 motion is separate from period-2 motions. For the period-4 motion, the harmonic amplitude A 3/4 0.4 ispresentedin Fig. 4, which is similar to A 1/4. The harmonic amplitude A is presented in Fig. 4(e). To show the quantity levels of harmonic amplitudes, the harmonic amplitude A 2 5 is presented in Fig. 4(f). The harmonic amplitude A 3 5 is presented in Fig. 4(g), which is different from A 2.For lower frequency, the quantity levels of A 1 and A 2 are the same, but for higher frequency, the quantity level of A 1 is much higher than the quantity level of A 2. To illustrate such change, the harmonic amplitude A 20 is presented with a common logarithmic scale. The quantity level of the harmonic amplitude A 20 decreases with excitation frequency with a power law. For stable solutions, the harmonic amplitude A is observed. Since the global view of the bifurcation trees from period-1 motion to period-4 motion is very crowded, it is very difficult to observe the nonlinear characteristics. The local view of frequency amplitude curves is presented in Fig. 5 with all possible solutions together. In Figs. 5 and 5, two local views of constant term a (m) 0 versus excitation frequency are presented for Ω (1.6, 2.1) and Constant Term, a 0 (m) Fig. 5. Local views for frequency amplitude curves (Q 0 = 15) based on 80 harmonic terms (80) of period-1 motion to period-4 motion in the parametric, nonlinear quadratic oscillator: and constant term a (m) 0, (n) harmonic amplitude A k/m (k =1, 2,...,4, 6, 8, 12, m=4).(δ =0.5, α=5,β= 20)

13 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Harmonic Amplitude, A 1/ Harmonic Amplitude, A 1/ Harmonic Amplitude, A 1/2 Harmonic Amplitude, A (e) Harmonic Amplitude, A 3/4 Harmonic Amplitude, A (f) (g) (h) Fig. 5. (Continued)

14 A. C. J. Luo & B. Yu Harmonic Amplitude, A 3/ Harmonic Amplitude, A 3/2 5 Harmonic Amplitude, A 3 Harmonic Amplitude, A (i) (k) (m) Fig. 5. Harmonic Amplitude, A 3 Harmonic Amplitude, A (j) (l) (n) (Continued)

15 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator 4 Constant Term, a 0 (m) Harmonic Amplitude, A Harmonic Amplitude, A 2 Harmonic Amplitude, A 4 8 P1 P4 6 P Harmonic Amplitude, A 3 Harmonic Amplitude, A e-8 1.0e-8 5.0e (e) Fig. 6. Bifurcation tree-1 of period-1 motion to period-4 motion (Q 0 = 15) based on 80 harmonic terms (80) in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (f) harmonic amplitude A k/m (k =1, 4,...,16, 36, m =4).(δ =0.5, α=5,β= 20). (f)

16 A. C. J. Luo & B. Yu Constant Term, a 0 (m) Harmonic Amplitude, A Harmonic Amplitude, A 2 Harmonic Amplitude, A (e) Harmonic Amplitude, A 3 Harmonic Amplitude, A e-7 1.6e-7 8.0e Fig. 7. Bifurcation tree-2 of period-1 motion to period-2 motion (Q 0 = 15) based on 80 harmonic terms (80) in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (h) harmonic amplitude A k/m (k =1, 2,...,8, 18, m =2).(δ =0.5, α=5,β= 20). (f)

17 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Constant Term, a 0 (m) Harmonic Amplitude, A Harmonic Amplitude, A 2 Harmonic Amplitude, A Harmonic Amplitude, A Harmonic Amplitude, A e-7 8e-7 7e-7 6e-7 5e e (e) (f) Fig. 8. Bifurcation tree-3 of period-1 motion to period-4 motion (Q 0 = 15) based on 80 harmonic terms (80) in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (f) harmonic amplitude A k/m (k =1, 4,...,16, 36, m =4).(δ =0.5, α=5,β= 20)

18 A. C. J. Luo & B. Yu Constant Term, a 0 (m) Harmonic Amplitude, A Harmonic Amplitude, A 2 Harmonic Amplitude, A (e) Harmonic Amplitude, A 9 Harmonic Amplitude, A e-8 1.4e-8 1.2e-8 1.0e-8 8.0e-9 6.0e (f) Fig. 9. Bifurcation tree-4 of period-1 motion to period-4 motion (Q 0 = 15) based on 80 harmonic terms (80) in the parametric, nonlinear quadratic oscillator: constant term a (m) 0, (f) harmonic amplitude A k/m (k =1, 4,...,16, 36, m =4).(δ =0.5, α=5,β= 20)

19 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Ω (2.4, 2.75), respectively. From the local view, the relation of frequency amplitude is very clearly presented. The frequency amplitude curves are complicated, which means multiple coexisting solutions exist. In Fig. 5, the lower part relative to stable solutions are presented, and the solution varying with excitation frequency is observed. In Fig. 5, the upper part relative to stable solutions are presented. In Fig. 5, the harmonic amplitude A 1/4 relative to the stable solution of period-4 motion is zoomed. To observe period-2 motion and period-4 motion, two local views of harmonic amplitude A 1/2 are presented for Ω (1.6, 2.1) and Ω (2.4, 2.7) in Figs. 5 and 5(e), respectively. The stability and bifurcations of period-2 and period-4 motions are observed clearly. The harmonic amplitude A 3/4 relative to the stable solution of period-4 motion is zoomed in Fig. 5(f), which is similar to the harmonic amplitude A 1/4 for period-4 motion only herein. The two zoomed areas for harmonic amplitude A 1 are presented in Figs. 5(g) and 5(h) for Ω (1.6, 2.1) and Ω (2.4, 2.75). Again, the frequency amplitude relations of coexisting solutions of period-1, period-2 and period-4 motions are very crowded in the two zoomed local views. Similarly, the two zoomed areas for the harmonic amplitude A 3/2 are presented for Ω (1.6, 2.1) and Ω (2.4, 2.7), in Figs. 5(i) and 5(j). The frequency amplitude curves for period-2 and period-4 motions are still fully occupied. To look into higher order harmonic terms effects, the harmonic amplitudes A 2 and A 3 are presented in Figs. 5(k) 5(n). In Figs. 5(k) and 5(l), two local views for harmonic amplitude A 2 are presented for Ω (2.4, 2.75) and Ω (1.6, 2.1), respectively. The more detailed view is observed. Similarly, two local views for harmonic amplitude A 3 are presented in Figs. 5(m) and 5(n) for Ω (2.4, 2.75) and Ω (1.6, 2.1), respectively. In similar fashion, other higher order harmonic terms can be illustrated. To avoid abundant illustrations, they will not be presented. From the local views, the bifurcation trees of period-1 to period-4 motions are not clear because of too many coexisting stable and unstable solutions. The further refined local views should be presented to demonstrate the bifurcation trees of period-1 to period-4 motion. To show bifurcation trees of period-1 to period- 4 motion, the harmonic amplitudes (a (m) 0 and A k/m, m = 4, k = 4l, l = 1, 2, 3, 4, 9) are presented for the bifurcation tree-1 with Ω (2.55, 2.71) in Figs. 6 6(f), respectively. The bifurcation tree-2 for Ω (2.40, 2.7)isbasedontheperiod-1motion to period-2 motion, and the harmonic amplitudes (a (m) 0 and A k/2, k =2l, l =1, 2, 3, 4, 9) are illustrated in Figs. 7 7(f). In Figs. 8 8(f), the harmonic amplitudes (a (m) 0 and A k/m, m =4,k =4l, l =1, 2, 3, 4, 9) for the bifurcation tree-3 are presented for Ω (1.98, 2.06). The bifurcation tree-4 for Ω (1.665, 1.685) is presented in Figs. 9 9(f) through the harmonic amplitudes (a (m) 0, m =4, k =4l, l =1, 2, 3, 4, 9). 4. Numerical Simulations In this section, numerical simulations of periodic motions in parametric quadratic nonlinear oscillator will be completed for illustrations of periodic motions. The initial conditions for numerical simulations are computed from approximate analytical solutions of periodic solutions. The numerical and analytical results can be compared. In all plots, circular symbols give approximate analytical solutions, and solid curves give numerical results. The numerical solutions of periodic motions are generated via the midpoint scheme. For system parameters (δ = 0.5,α = 5,β = 20), a period-2 motion (Ω = 6.8) is presented in Fig. 10 for Q 0 = 15 and the initial condition (x , y ) is computed from the analytical solution with ten harmonic terms (10). Displacement and velocity responses are presented in Figs. 10 and 10, respectively. The displacement and velocity curves for such a periodic motion are not simple sinusoidal curves. In Fig. 10, the trajectory of period-2 motion is presented. The asymmetry of periodic motion with one cycle is observed but not a simple cycle. The amplitude spectrum based on five harmonic terms is presented in Fig. 10. The main harmonic amplitudes are a (2) , A 1/ , A E 3, A 3/ E 3. The other harmonic amplitudes are A , A 5/ , A , A 7/ , A ,A 9/ , and A For this periodic motion, the harmonic amplitude A 1/ plays an important role in period- 2motion.Fork 6, the harmonic amplitudes A k/2 can be ignored. For the aforementioned period-2 motion, once the Hopf bifurcation occurs, the period-4 motion

20 A. C. J. Luo & B. Yu 2T 0.8 2T Time, t Time, t I.C Amplitude, A k/ A 1/2 1e-3 1e-4 1e-5 1e-6 1e-7 1e A 1 A 3/2 A 2 A 5/2 A 3 A 7/2 A 4 A 9/2 A Harmonics Order, k/2 A 2 A 3 A 4 Fig. 10. Period-2 motion (Ω = 6.8): displacement, velocity, trajectory and amplitude. Initial condition (x , y ). (δ =0.5, α=5,β= 20, Q 0 = 15). canbeobservedinsuchabifurcationtree.thus, with system parameters (δ =0.5, α=5,β= 20), a period-4 motion (Ω = 6.78) is illustrated in Fig. 11 for Q 0 = 15 and the initial condition (x , y ) is computed from the analytical solution with 24 harmonic terms (24). The time-histories of displacement and velocity are presented in Figs. 11 and 11, respectively. The displacement and velocity curves for such a periodic motion are periodic with 4T. In Fig. 11, the trajectory of period-4 motion is presented, and there are two cycles instead of one cycle in the period-2 motion. The amplitude spectrum based on 24 harmonic terms is presented in Fig. 11. The main harmonic amplitudes are a (4) , A 1/ , A 1/ , A 3/ E 3, A E 3, A 5/ E 3, A 3/ E 3, A 5/ E 4. The other harmonic amplitudes are A , A 9/ ,A 5/ ,A 11/ , A ,A 13/ ,A 7/ , A 15/ , A , A 17/ ,A 9/ ,A 19/ , A A 21/ ,A 11/ ,A 19/ , and A For this periodic motion, the harmonic amplitudes of A 1/ and A 1/ make important contributions on the period-4 motion. For k 12, the harmonic amplitudes A k/4 can be ignored. For another branch of period-2 motion to period-4 motion, to save space, only phase A

21 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator 4T 0.8 4T Time, t Time, t I.C Amplitude, A k/ A 1/2 1e-3 1e-4 A 1e-5 3 1e-6 A 4 1e-7 1e-8 A 5 A 6 1e-11 1e A 1/4 A A 1 3/2 A 3/4 A A 2 A 5/2 A 3 A A 4 5/4 7/2 A 9/2 A 5 A A 6 11/ Harmonics Order, k/4 Fig. 11. Period-4 motion (Ω = 6.78): displacement, velocity, trajectory and amplitude. Initial condition (x , y ). (δ =0.5, α=5,β= 20, Q 0 = 15). trajectories and spectrums for period-2 motions will be presented in Fig. 12 for Ω = with (x , y ) and Ω = 2.89 with (x , y ). In Fig. 12, the trajectory of period-2 motion with Ω = is presented. The analytical solutions possess 20 harmonic terms (20). The harmonic amplitude in spectrum is shown in Fig. 12. The main harmonic amplitudes are a (2) , A 1/ , A , A 3/ and A The other harmonic amplitudes are A 5/ ,A ,A 7/ ,A , A 9/ , and A A k/2 (10 10, 10 5 )fork =11, 12,...,20. For this periodic motion, the harmonic amplitudes A 1/2 to A 2 make significant contributions on period-2 motion rather than majorly from A 1/2. A 1/2 and A 1 are two most important terms. For k 10, the harmonic amplitudes A k/2 can be ignored. In Fig. 12, the phase trajectory of period-2 motion for Ω = 2.89 is illustrated, which is different from the one in Fig. 12. The corresponding spectrum of period- 2 motion is presented in Fig. 12. The main harmonic amplitudes are a (2) E 3, A 1/ , A , A 3/ , A and A 5/ The other harmonic amplitudes are A ,A 7/ ,A ,A 9/ , and A A k/2 (10 11, 10 5 )fork = 11, 12,...,20. For this period-2 motion, the harmonic amplitudes (A 1/2 to A 5/2 ) make significant

22 A. C. J. Luo & B. Yu I.C. Amplitude, A k/ A 1/2 A 1 A 1e-4 1e-5 A 6 1e-6 1e-7 A 7 1e-8 8 A 9 1e A 3/2 A A 5/2 A 3 A 7/2 A 4 A 9/2 A Harmonics Order, k/ Amplitude, A k/ A 1/2 A 1 A 3/2 A 1e-4 1e-5 A 6 1e-6 1e-7 A 7 1e-8 8 A 9 1e A 2 A 5/2 A 3 A 7/2 A 4 A9/2 A Harmonics Order, k/2 Fig. 12. Period-2 motion (Ω = ) with initial condition (x , y ): trajectory and amplitude. Period-4 motion (Ω = 2.89) with initial condition (x , y ): trajectory and amplitude. (δ =0.5, α=5,β= 20, Q 0 = 15). contributions on the period-2 motion. The harmonic amplitude A 3/2 has more contribution than A 1 on period-2 motion (i.e. A 3/2 2A 1 ). Next illustrations are trajectories for period- 1 motion to period-4 motions. In linear parametric oscillators, no such period-1 motions can be observed. However, the nonlinear parametric oscillator possesses bifurcation tree of period-1 motion to chaos. For different branches, nonlinear dynamical behaviors are different, as shown in Figs To save space, only phase trajectories and spectrums are illustrated, and the initial conditions are listed in Table 1. On the bifurcation tree relative to period-1 motion with Ω = 1.665, the trajectories and amplitude spectrums of the period-1 motion (Ω = 1.665), period-2 motion (Ω = 1.67) and period-4 motion (Ω = 1.673) are presented in Figs (f), respectively. In Fig. 13, the trajectory of period-1 motion for Ω = possesses two cycles because the second harmonic term (A 2 ) plays an important role on the period-1 motion. The main harmonic amplitudes are a 0 = E 3, A 1 = E 3, A 2 = E 3, A 3 = E 3, A 4 = E 4, A 5 = E 5. The other harmonic amplitudes are A ,A ,A ,A ,A With increasing excitation frequency, period-2 motion can be observed. In Fig. 13, the trajectory of period-2 motion for Ω = 1.67 possesses four cycles. The distribution of harmonic amplitudes

23 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator I.C. Amplitude, A k 12 8 a 0 4 A 1 A 2 1e-3 1e-4 1e-5 A 6 A 7 1e-6 1e-7 A8 A 9 1e A 3 A 4 A Harmonics Order, k 4 I.C Amplitude, A k/ a 0 (2) 4 A 1/2 A 1 A 3/2 A 2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 1e A 5/2 A 7/2 A 4 A 9/ A 2 A 3 Harmonics Order, k/2 1e-4 1e-5 1e-6 1e-7 1e-8 A 5 A 7 A 6 A 8 A9 A 7 A 8 A 9 A 10 A 11 A 10 A I.C. Amplitude, A k/4 8 a 0 (m) 4 A 1/2 A 1 A 3/2 A 5/2 A 3 1e-11 1e-12 1e (e) A 1/4 A A 5/4 A 3/4 7/4 A 9/4 A 11/4 A 7/2 A Harmonics Order, k/4 Fig. 13. Period-1 motion (Ω = 1.665) with (x , y E 3): trajectory and amplitude. Period-2 motion (Ω = 1.67) with (x , y E 3): trajectory and amplitude. Period-4 motion (Ω = 1.673) with (x , y E 3): (e) trajectory and (f) amplitude. (δ =0.5, α=5,β= 20, Q 0 = 15). (f) A 9/2 A

24 A. C. J. Luo & B. Yu I.C. Amplitude, A k a 0 A 1 A 2 1e-3 1e-4 1e-5 1e-6 1e-7 A 6 A 7 A 8 A9 A 10 A 11 1e A 3 A 4 A 5 A Harmonics Order, k I.C I.C. Amplitude, A k/2 Amplitude, A k/ a 0 (2) A 1 A 3/2 A 2 1e-3 1e-4 A 7 A 1e-5 8 A 9 1e-6 A10 1e-7 A 11 1e A 6 A 1/2 5/2 A 3 A 7/2 A 4 A 9/2 A 5 A 11/2 A a 0 (4) A 1 A 3/2 A 2 Harmonics Order, k/2 1e-4 A 7 1e-5 A 8 A 1e-6 9 A 1e-7 10 A 11 1e A 1/2 A 5/ (e) A 1/4 A 3/4 A 5/4 A 7/4 A 9/4 A 3 A A 7/2 11/4 A 4 A 9/2 A Harmonics Order, k/4 Fig. 14. Period-1 motion (Ω = 2.06) with (x , y ): trajectory and amplitude. Period-2 motion (Ω = 2.02) with (x , y ): trajectory and amplitude. Period-4 motion (Ω = 2.0) with (x , y ): (e) trajectory and (f) amplitude. (δ =0.5, α=5,β= 20, Q 0 = 15). (f)

25 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator I.C. Amplitude, A k A 1 1e-4 1e-5 A 7 1e-6 A 8 1e-7 A 9 1e-8 A 10 1e A A 3 A4 A 5 A Harmonics Order, k I.C I.C. 0.6 Amplitude, A k/2 Amplitude, A k/ A 1/2 A 1 A 3/2 A 2 1e-4 1e-5 1e-6 A 7 A 8 1e-7 1e-8 A 9 A 10 A 11 1e-11 1e A 5/2 A 3 A 7/2 A 4 A 9/2 A 5 A 11/2 A A 1 Harmonics Order, k/2 A 2 1e-4 1e-5 A 1e-6 7 A 8 1e-7 A 9 1e-8 A 10 A 11 1e-11 1e-12 1e A 1/2 A 3/ (e) A 5/2 A A 1/4 3/4 A 5/4 A A 7/4 A A 3 9/4 11/ Harmonics Order, k/4 (f) Fig. 15. Period-1 motion (Ω = 2.625) with (x , y ): trajectory and amplitude. Period-2 motion (Ω = 2.613) with (x , y ): trajectory and amplitude. Period-4 motion (Ω = 2.608) with (x , y ): (e) trajectory and (f) amplitude. (δ =0.5, α=5,β= 20, Q 0 = 15)

26 A. C. J. Luo & B. Yu Table 1. Input data for numerical simulations (δ =0.5,α=5,β = 20). Frequency Ω Initial Conditions (x 0,y 0 ) Type and Stability Figs. 13, ( 12807, E 3) Period-1 motion (stable) Figs. 13, ( 19510, E 3) Period-2 motion (stable) Figs. 13(e), 13(f) ( 21881, E 3) Period-4 motion (stable) Figs. 14, ( 98897, ) Period-1 motion (stable) Figs. 14, ( , ) Period-2 motion (stable) Figs. 14(e), 14(f) ( , 67075) Period-4 motion (stable) Figs. 15, ( 50625, ) Period-1 motion (stable) Figs. 15, ( 40570, ) Period-2 motion (stable) Figs. 15(e), 15(f) ( 98209, ) Period-4 motion (stable) presented in Fig. 13 can give us an important clue. The main harmonic amplitudes for this period-2 motion are a (2) E 3, A 1/ E 3, A E 3, A 3/ E 3, A , A 5/ E 3, A E 3, A 7/ E 4, A E 4, A 9/ E 5, A E 5. The other harmonic amplitudes are A k/2 (10 10, 10 5 ) (k = 11, 12,...,20). For period-4 motion (Ω = 1.673), the corresponding trajectory with eight cycles is presented in Fig. 13(e), and the harmonic amplitude distribution in spectrum is presented in Fig. 13(f). The main harmonic amplitudes for the period-4 motion are a (4) E 3, A 1/ E 4, A 1/ E 3, A 3/ E 4, A E 3, A 5/ E 4, A 3/ E 3, A 7/ E 4, A , A 9/ E 4, A 5/ E 3, A 11/ E 4, A E 3, A 13/ E 4, A 7/ E 4, A 15/ E 5, A E 4, A 17/ E 5, A 9/ E 5, A 19/ E 6, A E 5. The other harmonic amplitudes are A k/4 (10 11, 10 5 )(k =11, 12,...,48). For the bifurcation tree relative to period-1 motion with Ω = 2.06, the trajectories and amplitude spectrums of the period-1 motion (Ω = 2.06), period-2 motion (Ω = 2.02) and period-4 motion (Ω = 2.0) are presented in Figs (f), respectively. In Fig. 14, the phase trajectory of period-1 motion for Ω = 2.06 possesses two cycles because the second harmonic terms (A 1 and A 2 ) play an important role on the period-1 motion. In Fig. 14, the main harmonic amplitudes are a 0 = , A 1 = 42535, A 2 = , A 3 = 16871, A 4 = E 3, A 5 = E 4, and A E 5. The other harmonic amplitudes are A ,A ,A ,A ,A , A With decreasing excitation frequency, period-2 motion can be observed. In Fig. 14, the trajectory of period-2 motion for Ω=2.02 possesses three cycles. The distribution of harmonic amplitudes is presented in Fig. 14. The main harmonic amplitudes for the period-2 motion are a (2) , A 1/ , A , A 3/ , A , A 5/ , A , A 7/ E 3, A 9/ E 3, A E 3, A E 4, A 11/ E 4, and A E 4. The other harmonic amplitudes are A k/2 (10 9, 10 4 ) (k = 13, 14,...,24). For period-4 motion (Ω = 2.0), the corresponding trajectory with six cycles is presented in Fig. 14(e), and the harmonic amplitude distribution in spectrum is presented in

27 Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator Fig. 14(f). The main harmonic amplitudes for the period-4 motion are a (4) , A 1/ , A 1/ , A 3/ E 3, A , A 5/ , A 3/ , A 7/ E 3, A , A 9/ E 3, A 5/ , A 11/ E 3, A , A 13/ E 4, A 7/ E 3, A 15/ E 4, A E 3, A 17/ E 4, A 9/ E 3, A 19/ E 4, A E 4, A 21/ E 5, A 11/ E 5, A 23/ E 5, A E 4. The other harmonic amplitudes are A k/4 (10 11, 10 5 )(k =13, 14,...,48). For the bifurcation tree relative to period-1 motion with Ω = 2.625, the trajectories and amplitude spectrums of the period-1 motion (Ω = 2.625), period-2 motion (Ω = 2.613) and period-4 motion (Ω = 2.608) are presented in Figs (f), respectively. In Fig. 15, the phase trajectory of period-1 motion for Ω = possesses one cycle because the first harmonic term (A 1 ) plays an important role on the period-1 motion. In Fig. 15, the main harmonic amplitudes are a 0 = 20284, A 1 = , A 2 =47170, A 3 = E 3, A 4 = E 4, A 5 = E 5, and A E 6. The other harmonic amplitudes are A ,A ,A ,A , A With decreasing excitation frequency, period-2 motion can be observed. In Fig. 15, the trajectory of period-2 motion for Ω=2.613 possesses two cycles. The distribution of harmonic amplitudes is presented in Fig. 15. The main harmonic amplitudes for the period-2 motion are a (2) , A 1/ , A , A 3/ , A , A 5/ E 3, A E 3, A 7/ E 4, A E 4, A 9/ E 5, A E 5, A 11/ E 6, and A E 6. The other harmonic amplitudes are A k/2 (10 9, 10 5 ) (k = 13, 14,...,22). For period-4 motion (Ω = 2.608), the corresponding trajectory with four cycles is presented in Fig. 15(e), and the harmonic amplitude distribution in spectrum is presented in Fig. 15(f). The main harmonic amplitudes for the period-4 motion are a (4) , A 1/ E 3, A 1/ , A 3/ E 3, A , A 5/ E 3, A 3/ , A 7/ E 3, A , A 9/ E 4, A 5/ E 3, A 11/ E 4, A E 3, A 13/ E 4, A 7/ E 4, A 15/ E 5, A E 4, A 17/ E 5, A 9/ E 4, A 19/ E 6, A E 5, A 21/ E 6, A 11/ E 5, A 23/ E 6, A E 6. The other harmonic amplitudes are A k/4 (10 11, 10 5 )(k =13, 14,...,48). 5. Conclusions In this paper, the analytical solutions of periodic motions in a parametric quadratic nonlinear oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcations of periodic motions are discussed by eigenvalue analysis. The analytical bifurcation trees of periodic motions to chaos in such a parametric oscillator were obtained. Nonlinear behaviors of such periodic motions are characterized through frequency amplitude curves of each harmonic term in the finite Fourier series solution. Numerical illustrations of periodic motions were carried out through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees of period-1 and period-2 motions to chaos exist as well

28 A. C. J. Luo & B. Yu References Hayashi, C. [1964] Nonlinear Oscillations in Physical Systems (McGraw-Hill Book Company, NY). Hsu, C. S. [1963] On the parametric excitation of a dynamics system having multiple degrees of freedom, ASME J. Appl. Mech. 30, Hsu, C. S. [1965] Further results on parametric excitation of a dynamics system, ASME J. Appl. Mech. 32, Luo, A. C. J. [2004] Chaotic motion in the resonant separatrix bands of a Mathieu Duffing oscillator with twin-well potential, J. Sound Vibr. 273, Luo, A. C. J. [2012] Continuous Dynamical Systems (Higher Education Press/L&H Scientific, Beijing/ Glen Carbon). Luo, A. C. J. & Huang, J. Z. [2012a] Analytical dynamics of period-m flows and chaos in nonlinear systems, Int. J. Bifurcation and Chaos 22, Luo, A. C. J. & Huang, J. [2012b] Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential, J. Appl. Nonlin. Dyn. 1, Luo, A. C. J. & Huang, J. [2012c] Unstable and stable period-m motions in a twin-well potential Duffing oscillator, Discontin. Nonlin. Compl. 1, Luo, A. C. J. & O Connor, D. [2013] On periodic motions in Mathieu Duffing oscillator, Int. J. Bifurcation and Chaos, in press. Luo, A. C. J. & Yu, B. [2013a] Analytical solutions for stable and unstable period-1 motions in a periodically forced oscillator with quadratic nonlinearity, ASME J. Vibr. Acoust. 135, Luo, A. C. J. & Yu, B. [2013b] Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator, J. Vibr. Contr., doi: / Luo, A. C. J. & Yu, B. [2013c] Period-m motions in a periodically forced oscillator with quadratic nonlinearity, Discontin. Nonlin. Compl. 2, Mathieu, E. [1868] Memoire sur le mouvement vibratoire d une membrance deforme elliptique, J. Math. 2, Mathieu, E. [1873] Cours de Physique Methematique (Gauthier-Villars, Paris). McLachlan, N. W. [1947] Theory and Applications of Mathieu Equations (Oxford University Press, London). Mond, M., Cederbaum, G., Khan, P. B. & Zarmi, Y. [1993] Stability analysis of non-linear Mathieu equation, J. Sound Vibr. 167, Sevin, E. [1961] On the parametric excitation of pendulum-type vibration absorber, ASME J. Appl. Mech. 28, Shen, J. H., Lin, K. C., Chen, S. H. & Sze, K.Y. [2008] Bifurcation and route-to-chaos analysis for Mathieu Duffing oscillator by the incremental harmonic balance method, Nonlin. Dyn. 52, Tso, W. K. & Caughey, T. K. [1965] Parametric excitation of a nonlinear system, ASME J. Appl. Mech. 32, Whittaker, E. T. [1913] General solution of Mathieu s equation, Proc.EdinburghMath.Soc.32, Whittaker, E. T. & Watson, G. N. [1935] A Course of Modern Analysis (Cambridge University Press, London). Zounes, R. S. & Rand, R. H. [2000] Transition curves for the quasi-periodic Mathieu equations, SIAM J. Appl. Math. 58,