Journal of Vibration Testing and System Dynamics On

Journal of Vibration Testing and System Dynamics 3(1) (2019) 55-69

Journal of Vibration Testing and System Dynamics Journal homepage: https://lhscientificpublishing.com/Journals/JVTSD-Default.aspx

On Experimental Periodic Motions in a Duffing Oscillatory Circuit Yu Guo1 , Albert C.J. Luo2†, Zeltzin Reyes1 , Abigail Reyes1 , Rojitha Goonesekere1 1 2

McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, IL 62026-1805, USA Submission Info Communicated by Steve Suh Received 24 September 2018 Accepted 25 December 2018 Available online 1 April 2019 Keywords Twin-well Duffing oscillator Duffing oscillatory circuits Periodic motions Experimental harmonic amplitudes

Abstract In this paper, the dynamics of a Duffing oscillatory system are studied through both an experimental method and analytical predictions. A Duffing oscillatory circuit is built and experimental data of various periodic motions are obtained. The semi-analytical method using discrete implicit maps is adopted to determine such periodic trajectories analytically using the same parameters as in the experiments. Finally, illustrations of various periodic motions on both trajectories and harmonic amplitudes are demonstrated between the experimental data and analytical predictions. From the current experimental technology, experimental results may not provide real results in nonlinear dynamical systems. ©2019 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Study of periodic solutions in nonlinear dynamical systems have been of great interest for a long time. Such an issue started in 1788 when Lagrange [1] investigated the three-body problem through a perturbation of the two-body problem using the method of averaging. In 1899, Poincare [2] further developed the perturbation theory, and applied such a method to the motions of celestial bodies. In early 19th century, van der Pol [3] adopted the method of averaging for the periodic solutions of oscillatory circuits. However, the asymptotic validity of the method of averaging was not proved until 1928, when Fatou [4] provided proof of the asymptotic validity using the solution existence theorems of differential equations. Such a method of averaging was further developed by Krylov and Bogoliubov [5] for nonlinear vibration systems in 1935. Since then, researchers applied extensively the perturbation method on approximate periodic solutions of nonlinear dynamical systems. The search for analytical solutions to nonlinear dynamical systems has always continued until the accurate solutions of periodic motions can be obtained. In 2012, Luo [6] developed an analytical method for analytical solutions of periodic motions in nonlinear dynamical systems through a generalized harmonic balance method. This approach has changed the traditional analysis, and the more accurate solutions of periodic motions can be obtained. † Corresponding

author. Email address: [email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2019 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2019.03.005

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Yu Guo et al. / Journal of Vibration Testing and System Dynamics 3(1) (2019) 55–69

Luo and Huang [7] applied such a method to a Duffing oscillator for approximate solutions of periodic motions, and the analytical bifurcation trees of period-m motions to chaos were presented in the Duffing oscillator [8]. Since then, in Luo and Huang [9, 10], the analytical bifurcation trees of period-m motion to chaos in the Duffing oscillator with twin-well potentials were presented. However, the generalized harmonic balance method is computationally expensive and only for polynomial nonlinear functions. Thus, in 2005, Luo [11,12] presented the mapping dynamics of discontinuous dynamical systems. Then in Luo [13], the implicit mapping dynamics for discrete dynamical systems were developed. Such discrete maps can be either implicit or explicit functions, which makes it very convenient to be applied for the analytical prediction of periodic motions in discrete dynamical systems. Stability and bifurcation analysis of periodic motions can also be completed easily through eigenvalue analysis. Based on such discrete mapping structures, in 2015, Luo [14] developed a semi-analytical method to determine periodic motions in nonlinear dynamical systems. Luo and Guo [15, 16] applied such a method to investigate the nonlinear dynamics of a Duffing oscillator. Later on, The methodology was also adopted to study the dynamics of pendulums in Luo and Guo [17, 18]. On the other hand, in 1993, Chua et al [19] introduced the Chua’s oscillatory circuit for generating chaos and bifurcation phenomena. For the first time, experimental observation of chaos was achieved. Since then, one has started to investigate the dynamics in experimental nonlinear oscillatory circuits. In 1997, Kapitaniak et al [20] studied the synchronization between two coupled Chua’s circuits initially operating on different co-existing attractors. In 2009, Tafo Wembe and Yamapi [21] studied the synchronization of two resistively coupled Duffing oscillatory circuits. In 2013, Trejo-Guerra et al [22] discussed the integrated design of oscillatory circuits and the electronic implementation of nonlinear differential equations. In 2014, Buscarino et al [23] developed detailed circuit implementations of various nonlinear oscillators using modern electronics. Therefore, such electronic implementations of nonlinear oscillatory circuits are adopted in this paper for experimental investigations of periodic motions in comparison to analytical predictions. In this paper, the Duffing oscillatory system will be studied both experimentally and analytically. Experimental data of various periodic motions will be obtained through a Duffing oscillatory circuit. Analytical prediction of the same motion will be obtained through discrete implicit maps. Comparison between the experimental data and analytical predictions will be carried out on both trajectories and harmonic amplitudes.

2 Analytical prediction As discussed in Luo and Guo [15, 16]. The equation of motion for the Duffing oscillatory system is given as x¨ + δ x˙ + α x + β x3 = Q0 cos Ωt.

(1)

The corresponding equation in state space is x˙ = y, y˙ = Q0 cos Ωt − δ x˙ − α x − β x3 .

(2)

Thus, the differential equation in Eq. (2) can be discretized by a midpoint scheme for t ∈ [tk ,tk+1 ] to form an implicit map Pk (k = 0, 1, 2, ...) as Pk : (xk−1 , yk−1 ) → (xk , yk ) ⇒ (xk , yk ) = Pk (xk−1 , yk−1 )

(3)


57

with 1 xk =xk−1 + h(yk−1 + yk ), 2

1 1 yk =yk−1 + h[Q0 cos Ω(tk−1 + h) − δ (yk−1 + yk ) 2 2 1 1 − α (xk−1 + xk ) − β (xk−1 + xk )3 ]. 2 8

(4)

The above discretization experiences an accuracy of O(h3 ) for each step. To keep computational accuracy less than 10−8 , h < 10−3 needs to be maintained. In general, a period-m periodic motion in the Duffing oscillatory system is represented by a discrete mapping structure as (m)

(m)

(m)

(m)

(m)

(m)

P = PmN ◦ PmN−1 ◦ · · · ◦ P2 ◦ P1 : (x0 , y0 ) → (xmN , ymN ) mN−actions

(5)

with (k = 1, 2, · · · , mN). (m)

(m)

Pk : (xk−1 , yk−1 ) → (xk , yk ) (m)

(m)

(m)

(6)

(m)

⇒ (xk , yk ) = Pk (xk−1 , yk−1 ).

From Eq.(4), the corresponding algebraic equations for the mapping structure in Eq.(5) are developed for Pk in Eq. (7), which are then used for analytical predictions of periodic motions. (m)

xk

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

1 (m) (m) (m) =xk−1 + h(yk−1 + yk ), 2

1 (m) =yk−1 + h[Q0 cos Ω(tk−1 + h) 2 1 1 (m) (m) (m) (m) for Pk . − δ (yk−1 + yk ) − α (xk−1 + xk )⎪ ⎪ 2 2 ⎪ ⎪ ⎪ 1 ⎪ (m) (m) ⎪ ⎪ − β (xk−1 + xk )3 ] ⎪ ⎪ 8 ⎪ ⎭ (k =1, 2, · · · , mN)

(m)

yk

(7)

The corresponding periodicity condition is given as (m)

(m)

(m)

(m)

(xmN , ymN ) = (x0 , y0 ).

(8)

From Eqs. (7) and (8), nodes at the discretized Duffing oscillatory system can be determined by (m)∗ 2(mN + 1) equations. For nodes xk (k = 1, 2, · · · , mN), the stability of period-m motion can be discussed (m)∗ (m) (m)∗ (m) by the corresponding Jacobian matrix. For a small perturbation in vicinity of xk , xk = xk + Δxk (k = 0, 1, 2, · · · , mN), (m)

(m)

ΔxmN = DPΔx0 = DPmN · DPmN−1 · . . . · DP2 · DP1 Δx0 mN -muplication

(9)

with (k = 1, 2, · · · , mN). (m)

(m)

Δxk

(m)

= DPk Δxk−1 ≡ [

∂ xk

]

(m)

(m)∗

(m) (x ∂ xk−1 k

(m)∗

,xk−1 )

Δxk−1 ,

(10)


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Fig. 1 Electronic implementation of the Duffing oscillatory circuit. (R1 = 10MΩ, R2 = 1kΩ, R3 = 1kΩ, R4 = 10kΩ, R5 = 100Ω, R6 = 1kΩ, R7 = 1kΩ, R8 = 4kΩ, R9 = 1kΩ, R10 = 1kΩ, R11 = 10MΩ, R12 = 1kΩ, R13 = 1kΩ, R14 = 75Ω, C1 = 0.1 μ F, C2 = 0.1 μ F).

where

DPk =

(m) ∂ xk (m) ∂ xk−1

⎡

(m)∗ (m)∗ (xk ,xk−1 )

⎢ =⎢ ⎣

(m)

⎤

(m)

∂ xk

(m)

∂ xk

∂ xk−1

∂ yk−1

(m) ∂ yk (m) ∂ xk−1

(m) ∂ yk (m) ∂ yk−1

(m)

⎥ ⎥ ⎦

. (m)∗

(xk

(11)

(m)∗

,xk−1 )

The stability and bifurcation of period-m motion are determined by eigenvalues of DP. That is, |DP − λ I| = 0 where DP =

1

∏

k=mN

(12)

(m)

[

∂ xk

]

(m)∗

(m) (xk

∂ xk

(m)∗

,xk−1 )

.

(13)

The stability of periodic motion can be determined by the following conditions: 1. If all |λi | < 1 for (i = 1, 2), the periodic motion is stable. 2. If one of |λi | > 1 for (i ∈ {1, 2}), the periodic motion is unstable. And the bifurcation points can be determined as follows: 1. If one of λi = −1 and |λ j | < 1 for (i, j ∈ {1, 2} and j = i), the period-doubling bifurcation of periodic motion occurs. 2. If one of λi = 1 and |λ j | < 1 for (i, j ∈ {1, 2} and j = i), the saddle-node bifurcation of the periodic motion occurs. 3. If |λ1,2 | = 1 is a pair of complex eigenvalues, the Neimark bifurcation of the periodic motion occurs.


59

3 Experimental implementation From Buscarino et al [23], the implementation of a Duffing oscillatory circuit is shown in Fig. 1 with slight modification of component values to fit the experimental need. The Duffing oscillatory system is associated to such electronic implementation as following: C1 R2 x˙ = y, R9 R9 R9 R9 3 x . C2 R10 y˙ = Vsin − x˙ + x − R7 R8 R6 10R5

(14)

where Vsin is the excitation signal from a function generator. The corresponding parameters of the Duffing oscillatory system are

α =−

R9 R9 R9 = −1, β = = 1, δ = = 0.25. R6 10R5 R8

(15)

with such implementation, a temporal rescaling factor κ is introduced:

κ=

1 1 = = 10000. C1 R2 C2 R10

(16)

The corresponding excitation amplitude and frequency are Q0 =

R9 2π f |Vsin | and Ω = R7 κ

(17)

where f is the frequency of the excitation signal Vsin from the function generator. 4 Result illustration In this section, pairs of asymmetric periodic motions will be presented through both experiments and analytical predictions. The experimental data is recorded with a sampling rate of at least 100kHz. The analytical predictions are achieved with N = 1024 nodes per excitation period. In order to match the experimental parameters, the following parameters are used for the analytical illustrations:

α = −1, β = 1, δ = 0.25.

(18)

Asymmetric periodic motions are observed in the twin-well Duffing oscillator as in Luo and Guo [10, 11]. In such a nonlinear system, the co-existing asymmetric periodic motions are related to the potential wells. Thus, to demonstrate the co-existing asymmetric motions analytically and experimentally, a relationship of the initial conditions for pairs of asymmetric motions is 1 (19) ti = ti + T, xi = −xi , x˙i = −x˙i . 2 with i ∈ {1, 2} representing the two co-existing branches of asymmetric motions. By controlling the initial conditions according to Eq.(19), one is able to demonstrate the co-existence of asymmetric motions on the Duffing oscillatory circuit. Furthermore, two different types of motions will be illustrated herein: (i) the local motions possesses trajectories mainly existing in one of two potential wells of the Duffing oscillatory system, and (ii) the global motions possess trajectories existing in the two potential wells. In all illustrations, the trajectories of the motions are presented with black curves representing the experimental data, and the red curves with circular symbols indicating the analytical prediction. The acronyms ‘B1’ and ‘B2’ are for the two different co-existing asymmetric motions. The corresponding harmonic amplitudes are also presented by black and red dots, which represent the experimental data and analytical prediction, respectively. Finally, the gray dots are for the FFT responses of the corresponding experimental data.


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Fig. 2 Asymmetric period-1 motions with Q0 = 0.415 and Ω = 1.649: (a) Trajectory (B1), (b) trajectory (B2), (c) harmonic amplitudes (B1), (d) harmonic amplitudes (B2). (α = −1.0, β = 1.0, δ = 0.25).

4.1

Local periodic motions

In Fig. 2, a pair of asymmetric local period-1 motions is presented for Ω = 1.649 and Q0 = 0.415. The initial conditions given as: t0 = 0.0, x0 ≈ 0.5924, x˙0 ≈ 0.1612 for B1, and t0 = 1.9050, x0 ≈ −0.5924, x˙0 ≈ −0.1612 for B2. In Figs. 2 (a) and (b), presented are the trajectories of the two co-existing asymmetric motions. The pair of asymmetric motions are asymmetric to each other around (0, 0) in the phase portrait. The experimental trajectories of motions are not as clear as the analytical trajectories. However, the experimental and analytical trajectories match each other. The corresponding harmonic amplitudes are presented in Figs.2 (c) and (d) for B1 and B2, respectively. Due to the limitation on experimental observation, leakage can be observed in the FFT responses of experimental motion. Such leakage could cause mismatch between the experimental and analytical amplitudes. For both B1 and B2, the harmonic amplitudes a0 and Ak , k = 1, 2, 3, 4 match very well between the experiment and analytical prediction. However, as the harmonic order k increases, the harmonic amplitudes for analytical prediction continue decreasing ,while the experimental harmonic amplitudes maintain a level of about 10−4 . This is due to the leakage in FFT and limited resolution in the experimental data during sampling. As the excitation frequency Ω decreases with the same amplitudes Q0 , the asymmetric period-1 motions possess period-doubling bifurcations. After such period-doubling bifurcations, a pair of co-existing local asymmetric period-2 motions are presented in Fig. 3 at Ω = 1.484. corresponding initial conditions are given as: t0 = 0.0, x0 ≈ −1.1884, x˙0 ≈ 0.5200 (B1), and t0 = 2.1170, x0 ≈ 1.1884, x˙0 ≈ −0.5200 (B2). The trajectories of the two period-2 motions are illustrated in Figs. 3 (a) and (b) for branches



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B1 and B2, respectively. The trajectories in phase plane demonstrate two cycles to form a complete period-2 motion. The trajectories of the period-2 motions on B1 and B2 are skew symmetric to the origin (0, 0). The analytical trajectories holds same pattern with the experimental trajectories. However, the experimental and analytical trajectories do not match as well as the period-1 motions. This is due to the rapidly shrinking stable ranges as the period-doubling bifurcation occurs. The corresponding harmonic amplitudes of the period-2 motions on B1 and B2 are presented in Figs. 3 (c) and (d), respectively. For the period-2 motions, there exist half order terms (i.e., A1/2 , A3/2 , A5/2 ...). Leakage is also observed from the FFT responses for both period-2 motions on B1 and B2 due to the limitation on experimental observations. Such leakage causes slight mismatch on harmonic amplitudes between the experimental and analytical results. The harmonic amplitudes a0 and Ak/2 (k = 1, 2, 3, ..., 9) match between the experiment and analytical predictions. With the harmonic order increases, the analytical harmonic amplitudes continuously decrease to A10 ∼ 10−10 . However, the experimental harmonic amplitudes maintain a quantity level of about 10−4 . Again, this is caused by the leakage effect in the FFT response and the limited resolution in the experimental data during sampling. The period-4, period-8. . . motions after further cascaded period-doubling bifurcations have extremely small stable ranges, therefore will not be demonstrated herein. At a lower excitation frequency of Ω = 0.451, there is a pair of complex asymmetric local period-1 motions, as presented in Fig. 4 with Q0 = 0.415. The corresponding initial conditions are: t0 = 0.0, x0 ≈ −0.7156, x˙0 ≈ 0.1047 (B1), and t0 = 6.9658, x0 ≈ 0.7156, x˙0 ≈ −0.1047 (B2). The trajectories of such asymmetric motions are illustrated in Figs.4 (a) and (b). The trajectories of asymmetric motions are asymmetric to (0, 0) in phase plane. The experimental trajectories match with the analytical


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Fig. 4 Asymmetric period-1 motions with Q0 = 0.415 and Ω = 0.451: (a) Trajectory (B1), (b) trajectory (B2), (c) harmonic amplitudes (B1), (d) harmonic amplitude (B2). (α = −1.0, β = 1.0, δ = 0.25).

prediction in a very good manner. The corresponding harmonic amplitudes are presented in Figs. 4 (c) and (d) for the paired period-1 motions for B1 and B2, respectively. Due to the limitation on experiment, the leakage can be observed in the FFT responses of experimental motion. Such leakage could cause mismatch between the experimental and analytical harmonic amplitudes. For the period-1 motions on B1 and B2, the harmonic amplitudes a0 and Ak (k = 1, 2, 3, ..., 10) match very well between the experimental and analytical results. However, as the harmonic order k increases, the analytical harmonic amplitudes continuously decrease, while the experiment harmonic amplitudes maintain a level of about 10−4 . This is due to limited resolution in the experimental data collection during sampling. Overall the experimental and analytical results are in a good agreement. With decreasing excitation frequency Ω, the paired asymmetric period-1 motion have a perioddoubling bifurcation to generate a pair of asymmetric period-2 motions, as presented in Fig. 5 at Ω = 0.431. The corresponding initial conditions are t0 = 0.0, x0 ≈ −0.6452, x˙0 ≈ 0.1668 for B1, and t0 = 7.2891, x0 ≈ 0.6452, x˙0 ≈ −0.1668 for B2. The trajectories of such period-2 motions are illustrated in Figs. 5 (a) and (b) on the two branches of B1 and B2, respectively. The experimental and analytical results do not match very well but they have similar patterns. However, some harmonic amplitudes match each other , as shown in Figs. 5 (c) and (d), respectively. Leakage is observed from the FFT responses for the two branches B1 and B2 due to the limitation on experiments. Overall, the primary harmonic amplitudes a0 and Ak (k = 1, 2, 3, · · · , 10) match quite well between the experimental and analytical results. On the other hand, the half order harmonic terms Ak/2 (k = 1, 3, 5, · · · ) show a significant mismatching. As the harmonic order increases, the analytically predicted harmonic amplitudes decrease to A10 ∼ 10−6 . However, the experimental harmonic amplitudes maintain a quantity




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level of about 10−4 . Such a problem is caused by the leakage effect in the FFT response and the limited resolution in the experimental data during sampling. The corresponding period-4, period-8. . . motions after further cascaded period-doubling bifurcations will not be demonstrated herein because the stable ranges of such periodic motions are very small. 4.2

Global periodic motions

The global periodic motions exist in the two potential domains in the Duffing oscillatory system. A pair of co-existing asymmetric global period-1 motions is presented in Fig. 6 for Q0 = 0.415, Ω = 0.4228. The trajectories of the paired asymmetric period-1 motions on the two branches of B1 and B2 are presented in Figs.6 (a) and (b), respectively. The symmetry of the two paired asymmetric co-existing motions can be observed. Such trajectories demonstrate one local cycle and one global cycle to complete a periodic orbit. In Figs. 6 (a) and (b), the analytical and experimental trajectories agree with each other quit well. The corresponding harmonic amplitudes are illustrated in Figs. 6 (c) and (d) for the paired period-1 motion on the branches of B1 and B2, respectively. For the two period-1 motions, the constant a0 and harmonic amplitudes Ak (k = 1, 2, · · · , 10) closely match between the experimental and analytical results. As the harmonic order increases, the harmonic amplitudes Ak (k = 11, 12, · · · , 19) do not match as well. But the experimental and analytical harmonic amplitudes continue to follow a similar trend. After the 20th order harmonics, the trend of convergence for such harmonic amplitudes can be clearly observed in Figs. 6 (c) and (d). The analytical harmonic amplitudes continuously drop to a quantity level of about 10−7 , However, experimental harmonic amplitudes maintain a quantity level of about 10−4 . Such difference primarily results from the limited resolution of data acquisition,


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Fig. 6 Asymmetric period-1 motions at Q0 = 0.415 and Ω = 0.4228: (a) Trajectory (B1), (b) trajectory (B2), (c) harmonic amplitudes (B1), (d) harmonic amplitudes (B2). (α = −1.0, β = 1.0, δ = 0.25).

causing the experimental results to have a lower precision than the analytical prediction. As excitation frequency Ω decreases, the pair of two global asymmetric period-1 motions have a period-doubling bifurcation to generate a pair of two paired global asymmetric period-2 motions. The two paired asymmetric period-2 motions are presented in Fig. 7 at Ω = 0.3962. The trajectories of such two asymmetric period-2 motions for the two branches of B1 and B2 are illustrated in Figs. 7 (a) and (b), respectively. Such trajectories demonstrate two local cycles and two global cycles to complete the period-2 motion. The analytical trajectories match with the experimental motion in a very good manner. The corresponding harmonic amplitudes for the two asymmetric period-2 motions are presented in Figs. 7 (c) and (d). The zoomed views of such harmonic amplitudes are illustrated in Fig. 7 (e) and (f) for the harmonic orders (k = 1, 2, · · · , 10). For period-2 motions, half order harmonic amplitudes exist (i.e., A1/2 , A3/2 , A5/2 , · · · ). For the harmonic amplitudes Ak/2 (k = 1, 2, · · · , 20), the experimental and analytical results match very well. The limited differences due to leakage from the FFT responses is caused by the limitation on experimental observations. Such leakage effect could result in slightly lower or higher experimental magnitudes than the actual harmonic amplitudes under different circumferences. As the harmonic order increases, for the harmonic amplitudes Ak (k = 11, 12, · · · , 20), observed are some mismatches between the experimental and analytical results. However, the overall maximum quantity level for these harmonic amplitudes is about 10−2 , which is much lower than the lower order harmonics. As the harmonic order further increases, the convergence of the harmonic amplitudes can be clearly observed in Figs. 7 (c) and (d). The analytically harmonic amplitudes continuously drop to a quantity level of A30 ∼ 10−7 . However, the experimental harmonic amplitudes




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Fig. 7 Asymmetric period-2 motions at Q0 = 0.415 and Ω = 0.3962: (a) Trajectory (B1), (b) trajectory (B2), (c) harmonic amplitudes (B1), (d) harmonic amplitudes (B2), (e) zoomed harmonic amplitudes (B1), (f) zoomed harmonic amplitudes (B2). (α = −1.0, β = 1.0, δ = 0.25).

maintain a quantity level of about 10−4 . The period-4, period-8. . . motions after further cascaded period-doubling bifurcations also possess extremely small stable ranges, which will not be demonstrated herein. A symmetric global period-1 motion is presented in Fig. 8 for Q0 = 2.080, Ω = 0.5009. The trajectory of the motion is symmetric to the origin, as shown in Fig. 8 (a). Such trajectory demonstrate one local cycle on both ends and one global cycle to connect with both ends. In Fig. 8 (a), the analytical and experimental trajectories agree with each other quit well. The corresponding harmonic amplitudes


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Fig. 8 Symmetric period-1 motions at Q0 = 2.080 and Ω = 0.5009: (a) Trajectory, (b) harmonic amplitude. (α = −1.0, β = 1.0, δ = 0.25).

are illustrated in Fig. 8 (b). The harmonic amplitudes Ak (k = 1, 3, 5, · · · , 25) closely match between the experimental data and analytical prediction. a0 = 0 and A2l = 0(l = 1, 2, 3, · · · ). On the other hand, a0 ≈ 7.1080 × 10−4 . For the experimental harmonic amplitudes of even orders, A2 ≈ 2.0593 × 10−3 , A4 ≈ 9.3870×10−3 , A6 ≈ 2.8127×10−3 , A8 ≈ 1.3037×10−3 , and A10 ≈ 6.3112×10−4 . Such a difference between the experimental and analytical results are mainly caused from the noise in the signal and leakage effect in the experimental FFT. As the harmonic order increases, the analytical harmonic amplitudes Ak (k > 25) continuously drop towards 10−9 . However, the experimental harmonic amplitudes maintain a quantity level of 10−5 . For a constant Ω = 0.5009, consider the variation of excitation amplitude Q0 . A pair of two asymmetric global period-1 motions are presented in Fig. 9 for Q0 = 2.525. The trajectories of the two paired period-1 motions on thw two branches of B1 and B2 are presented in Figs. 9 (a) and (b), respectively. Similar to the symmetric period-1 motions at Q0 = 2.080, the two asymmetric motions have one local cycle on both ends and and one global cycle connecting with two local cycles on the both sides to complete a period-1 motion. The analytical trajectories agree with the experimental trajectories on the two branches of B1 and B2 in a very good manner. The analytical and experimental harmonic amplitudes corresponding to B1 and B2 are then presented in Figs. 9 (c) and (d), respectively. The harmonic amplitudes Ak (k = 1, 2, 3, · · · , 10) are illustrated in Figs. 9 (e) and (f) for a zoomed view. The main mismatches can be observed for the harmonic amplitudes of A9 , A11 , A19 . However, until the twentieth harmonic order, the overall agreements of harmonic amplitudes are very good for most of the harmonic terms. The existence of noise in the experimental signals and the effect of leakage from the FFT responses from experimental data are the main reasons to the mismatches. As the harmonic order increases, for the harmonic amplitudes Ak (k = 20, 21, 22, ...), the analytically harmonic amplitudes continues to converge towards A30 ∼ 10−6 . However, the experimental harmonic amplitudes maintain a quantity level of about 10−4 for the two branches of B1 and B2 owing to the limited resolution of data acquisition. Such limited resolution causes the experimental data to have a lower precision than the analytical prediction. Finally, a symmetric global period-3 motion is presented in Fig.10 for Q0 = 0.940 and Ω = 0.7381. The trajectory of such a symmetric period-3 motion has one local cycle on both sides and three global cycles connecting with the two local cycles to form a period-3 motion. The trajectory is symmetric in phase plane, as shown in Fig. 10 (a). Again, observed is a good agreement between the experimental and analytical trajectories. The corresponding harmonic amplitudes of the motion for analytical and experimental results are presented in Fig. 10 (b) for harmonic order up to twenty. A zoomed view of harmonic amplitudes for harmonic orders less than ten is the illustrated in Fig.10 (c). For such a period-




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Fig. 9 Asymmetric period-1 motions at Q0 = 2.525 and Ω = 0.5009: (a) Trajectory (B1), (b) trajectory (B2), (c) harmonic amplitudes (B1), (d) harmonic amplitudes (B2), (e) zoomed harmonic amplitudes (B1), (f) zoomed harmonic amplitudes (B2). (α = −1.0, β = 1.0, δ = 0.25).

3 motion, there exist harmonic amplitudes of A(2l−1)/3 (l = 1, 2, · · · ). a0 = 0 and A2l/3 = 0 (l = 1, 2, · · · ) for the analytical results. On the other hand, for experimental results, the overall quantity levels of a0 and A2l/3 (l = 1, 2, · · · ) are 10−3 , ( i.e., a0 ≈ 6.5299 × 10−4 , A2/3 ≈ 1.6915 × 10−3 , A4/3 ≈ 3.5432 × 10−3 , A2 ≈ 3.5255 × 10−3 , A8/3 ≈ 2.3394 × 10−3 , etc.). Such experimental harmonic amplitudes of A2l/3 = 0 (l = 1, 2, · · · ) continuously decrease with the harmonic order increase, which are eventually converge to a quantity level of 10−4 . Such a difference is mainly from the leakage effects of the experimental FFT and the limited resolution of data acquisition. For the harmonic amplitudes A(2l−1)/3 (l = 1, 2, · · · ), a


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Fig. 10 Symmetric period-1 motions at Q0 = 0.940 and Ω = 0.7381: (a) Trajectory, (b) harmonic amplitude, (c) zoomed view of the harmonic amplitudes. (α = −1.0, β = 1.0, δ = 0.25).

very good agreement between the experimental and analytical results as illustrated in Figs. 10 (b) and (c). Such agreement maintains until the harmonic order goes beyond fourteen. For harmonic amplitude A(2l−1)/3 (l > 22), the quantity level for analytical harmonic amplitudes continue to show a good convergence towards 10−8 . However, the experimental harmonic amplitudes maintain a quantity level of about 10−4 as presented in Fig. 10 (b). 5 Conclusion In this paper, the dynamics of a Duffing oscillatory system was studied experimentally and analytically. For experiments, a Duffing oscillatory circuit was built, and experimental results of coexisting asymmetric periodic motions was obtained. The semi-analytical method based on discrete implicit maps was adopted for analytical predictions. From the same parameters, analytical results of corresponding periodic motions were obtained. Good agreements between the experimental and analytical results were achieved for local period-1 and period-2 motions. Finally, global period-1, period-2, and period-3 motions were also presented with good agreement between experimental and analytical results. The difference between the analytical and experimental results were also observed, and the reasons causing the experimental results not being accurate are the data leakage and the low resolutions of data acquisition in experiments. Thus, experimental results may not be real results in nonlinear dynamical systems. Such an issue on experimental measurements should be careful.


69

References [1] Lagrange, J.L. (1788), Mecanique Analytique, (2 vol.) (edition Albert Balnchard: Paris, 1965). [2] Poincare, H. (1899), Methodes Nouvelles de la Mecanique Celeste, Vol.3, Gauthier-Villars: Paris. [3] van der Pol, B. (1920), A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, pp.701-710, pp.754-762. [4] Fatou, P. (1928), Sur le mouvement d’un systeme soumis ‘a des forces a courte periode, Bull. Soc. Math., 56, 98-139. [5] Krylov, N.M. and Bogolyubov, N.N. (1935), Methodes approchees de la mecanique non-lineaire dans leurs application a l’Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s’y rapportant, Academie des Sciences d’Ukraine:Kiev. (in French). [6] Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L&H Scientific: Beijing/Glen Carbon. [7] Luo, A.C.J. and Huang, J.Z. (2012), Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance, Journal of Vibration and Control, 18, 1661-1871. [8] Luo, A.C.J. and Huang, J.Z. (2012), Analytical dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No. 1250093 (29 pages). [9] Luo, A.C.J. and Huang, J.Z. (2012), Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential, Journal of Applied Nonlinear Dynamics, 1, 73-108. [10] Luo, A.C.J. and Huang, J.Z. (2012), Unstable and stable period-m motions in a twin-well potential Duffing oscillator, Discontinuity, Nonlinearity and Complexity, 1, 113-145. [11] Luo, A.C.J. (2005), The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation, Journal of Sound and Vibration, 283, 723-748. [12] Luo, A.C.J. (2005), A theory for non-smooth dynamic systems on the connectable domains, Communications in Nonlinear Science and Numerical Simulation, 10, 1-55. [13] Luo, A.C.J. (2012), Regularity and Complexity in Dynamical Systems, Springer: New York. [14] Luo, A.C.J. (2015), Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems, International Journal of Bifurcation and Chaos, 25(3), Article No. 1550044 (62 pages). [15] Luo, A.C.J. and Guo, Y. (2015), A semi-analytical prediction of periodic motions in Duffing oscillator through mapping structures, Discontinuity, Nonlinearity, and Complexity, 4(2), pp.13-44. [16] Guo, Y. and Luo, A.C.J. (2015), Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings, International Journal of Dynamics and Control, 5(2), 223-238. [17] Luo, A.C.J. and Guo, Y. (2016), Periodic motions to chaos in pendulum, International Journal of Bifurcation and Chaos, 26(9), 1650159. [18] Guo, Y. and Luo, A.C.J. (2017), Routes of periodic motions to chaos in a periodically forced pendulum, International Journal of Dynamics and Control, 5(3), 551-569. [19] Chua, L.O., Wu, C.W., Huang, A., and Zhong, G.Q. (1993), A universal circuit for studying and generating chaos, Part I, II, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40(10), 732-761. [20] Kapitaniak T., Chua, L.O., and Zhong, G.Q. (1997), Experimental evidence of locally intermingled basins of attraction in coupled Chua’s circuits, Chaos, Solitons & Fractals, 8(9), 1517-1522 [21] Tafo Wembe, E. and Yamapi, R. (2009), Chaos synchronization of resistively coupled Duffing systems: numerical and experimental investigations, Communications in Nonlinear Science and Numerical Simulation, 14(2009), 1439-1453. [22] Trejo-Guerra, R., Tlelo-Cuautle, E., Carbajal-Gomez, V.H., and Rodriguez-Gomez, G. (2013), A survey on the integrated design of chaotic oscillators, Applied Mathematics and Computation, 219, 5113-5122. [23] Buscarino, A., Fortuna, L., Frasca, M., and Scuito, G. (2014), A concise guide to chaotic electronic circuits, Springer Briefs in Applied Sciences and Technology.

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