Further analytic solutions for periodic motions in the Duffing oscillator

Abstract

Harmonic balance method is employed for analytic periodic motions in the Duffing oscillator. Traditionally, one used the perturbation method to discuss the nonlinearity of periodic motions in the Duffing oscillator. For decades, one cannot achieve the appropriate analytical solutions of periodic motions. The harmonic balance method with prescribed-computational accuracy will be used, for a better understanding of nonlinear characteristics of periodic motions in the Duffing oscillator. In this method, the finite Fourier series with coefficient varying with time is adopted to approximately describe periodic motions in nonlinear dynamical system. The dynamical system of the coefficients of the Fourier series is obtained. The equilibrium of such a system gives the constant coefficients in the Fourier series. Thus, the periodic solution is obtained and the stability and bifurcation of the periodic motions can be determined. The analytic periodic motions should be further revisited, since the Duffing oscillator is extensively applied in engineering, and the comprehensive investigation of such periodic motions in the Duffing oscillator is necessary and significant. In this paper, the further analytic solutions of periodic motion in the Duffing oscillator will be investigated, and frequency amplitude characteristics will be discussed, and numerical simulation of periodic motions will be presented. Some new roads from periodic motions to chaos based on generalized harmonic balance method are found, such as independent period-2 motions to chaos and independent period-4 motions to chaos.

Appendix

The $F_0^{\left( m \right) } ,F_{k/m}^{\left( c \right) } ,F_{k/m}^{\left( s \right) }$ are expressed as follows:

$$\begin{aligned} F_0^{\left( m \right) }= & {} \frac{1}{mT}\int _0^{mT} {f\left( {\mathbf{x}^{*},\dot{\mathbf{x}}^{*},t,\mathbf{p}} \right) dt} =\frac{1}{mT}\int _0^{mT} {Fdt} \nonumber \\= & {} -\delta \dot{a}_0^{(m)} -\alpha a_0^{(m)} -\beta \left( {\left( {a_0^{(m)} } \right) ^3 +\sum _{i=1}^N {\left[ {\frac{3}{2}a_0^{(m)} } \right. } } \right. \nonumber \\&\times \left( {b_{i/m} b_{i/m} +c_{i/m} c_{i/m} } \right) +\sum _{l=1}^N {\sum _{i=1}^N {\sum _{j=1}^N {\left[ {\frac{1}{4}b_{i/m} } \right. } } } \nonumber \\&\times b_{j/m} b_{l/m} \delta _{301} +\left. {\left. {\frac{3}{4}b_{i/m} c_{j/m} c_{l/m} \delta _{302} } \right] } \right) \end{aligned}$$

(18)

$$\begin{aligned} F_{k/m}^{\left( c \right) }= & {} \frac{2}{mT}\int _0^{mT} {f\left( {\mathbf{x}^{*},\dot{\mathbf{x}}^{*},t,\mathbf{p}} \right) \cos \left( \frac{k}{m}\Omega t \right) dt} \nonumber \\= & {} \frac{2}{mT}\int _0^{mT} {F\cos \left( \frac{k}{m}\Omega t \right) dt} \nonumber \\= & {} Q_0 \delta _m^k -\delta {b}'_{k/m} -\alpha b_{k/m} -\beta \left( 3 \left( {a_0^{(m)} } \right) \right. ^{2}b_{k/m} \nonumber \\&+\sum _{i=1}^N {\sum _{j=1}^N {\left[ {\frac{3a_0^{(m)} }{2}\left( {b_{i/m} b_{j/m} \delta _{2c1} \hbox {+}c_{i/m} c_{j/m} \delta _{2c2} } \right) } \right] } } \nonumber \\&+\sum _{l=1}^N {\sum _{i=1}^N {\sum _{j=1}^N {\left[ {\frac{1}{4}b_{i/m} b_{j/m} b_{l/m} \delta _{3c1} } \right. } } } \nonumber \\&+\left. \left. {\frac{3}{4}b_{i/m} c_{j/m} c_{l/m} \delta _{3c2} } \right] \right) \end{aligned}$$

(19)

$$\begin{aligned} F_{k/m}^{\left( s \right) }= & {} \frac{2}{mT}\int _0^{mT} {f\left( {\mathbf{x}^{*},\dot{\mathbf{x}}^{*},t,\mathbf{p}} \right) \sin \left( \frac{k}{m}\Omega t \right) dt}\nonumber \\= & {} \frac{2}{mT}\int _0^{mT} {F\sin \left( \frac{k}{m}\Omega t \right) dt} \nonumber \\= & {} -\delta {c}'_{k/m} -\alpha c_{k/m} -\beta \left( 3\left( {a_0^{(m)} } \right) ^{2}c_{k/m}\right. \nonumber \\&+\sum _{i=1}^N {\sum _{j=1}^N {\left( {3a_0^{(m)} b_{i/m} c_{j/m} \delta _{2s1} } \right) } } \nonumber \\&+\sum _{i=1}^N {\sum _{j=1}^N {\sum _{l=1}^N {\left[ {\frac{1}{4}} \right. } } } c_{i/m} c_{j/m} c_{l/m} \delta _{3s1} \nonumber \\&+\left. \left. {\frac{3}{4}c_{i/m} b_{j/m} b_{l/m} \delta _{3s2} } \right] \right) \end{aligned}$$

(20)

The $\delta$ function and sign function are

$\delta _k^l =\left\{ {\begin{array}{ll} 1&{} {l=k} \\ 0&{} {l\ne k} \\ \end{array} } \right.$ and $\hbox {sgn}\left( k \right) =\left\{ {\begin{array}{ll} 1&{} k\ge 0 \\ -1&{}k<0 \\ \end{array}} \right.$

In Eqs. (18), (19), (20), the $\delta$ functions are expressed as follows.

$$\begin{aligned} \delta _{301}= & {} \delta _{i-j+l}^0 +\delta _{i+j-l}^0 +\delta _{i-j-l}^0\\ \delta _{302}= & {} \delta _{i-j+l}^0 +\delta _{i+j-l}^0 -\delta _{i-j-l}^0\\ \delta _{2c1}= & {} \delta _{i+j}^k +\delta _{\left| {i-j} \right| }^k ,\quad {\delta _{2c2} } =-\delta _{i+j}^k +\delta _{\left| {i-j} \right| }^k\\ \delta _{2s1}= & {} \delta _{i+j}^k -\hbox {sgn}(i-j)\delta _{\left| {i-j} \right| }^k\\ \delta _{3c1}= & {} \delta _{i+j+l}^k +\delta _{\left| {i+j-l} \right| }^k +\delta _{\left| {i-j+l} \right| }^k +\delta _{\left| {i-j-l} \right| }^k \\ \delta _{3c2}= & {} -\delta _{i+j+l}^k +\delta _{\left| {i+j-l} \right| }^k +\delta _{\left| {i-j+l} \right| }^k -\delta _{\left| {i-j-l} \right| }^k \\ \delta _{3s1}= & {} -\delta _{i+j+l}^k +\hbox {sgn}\left( {i+j-l} \right) \delta _{\left| {i+j-l} \right| }^k \\&+\hbox {sgn}\left( {i-j+l} \right) \delta _{\left| {i-j+l} \right| }^k -\hbox {sgn}\left( {i-j-l} \right) \delta _{\left| {i-j-l} \right| }^k \\ \delta _{3s2}= & {} \delta _{i+j+l}^k +\hbox {sgn}\left( {i+j-l} \right) \delta _{\left| {i+j-l} \right| }^k \\&+\hbox {sgn}\left( {i-j+l} \right) \delta _{\left| {i-j+l} \right| }^k +\hbox {sgn}\left( {i-j-l} \right) \delta _{\left| {i-j-l} \right| }^k \end{aligned}$$

In the Jacobian matrix, G and H are

$$\begin{aligned} G_r^{\left( 0 \right) }= & {} \frac{\partial F_0^{(m)} }{\partial z\left( r \right) }=-\alpha \delta _r^1 -3\beta \left( {a_0^{(m)} } \right) ^{2}\delta _r^1 \nonumber \\&-\beta \sum _{i=1}^N {\left[ {\frac{3}{2}\left( {b_{i/m}^2 +c_{i/m}^2 } \right) \delta _r^1 } \right. } \nonumber \\&+\left. {\frac{3a_0^{(m)} }{2}\left( {2b_{i/m} \delta _{i+1}^r +2c_{i/m} \delta _{N+i+1}^r } \right) } \right] \nonumber \\&-\beta \sum _{l=1}^N {\sum _{i=1}^N {\sum _{j=1}^N {\left[ {\frac{3}{4}b_{j/m} b_{l/m} \delta _{301} \delta _{i+1}^r } \right. } } } \nonumber \\&+\left. {\left( {\frac{3}{4}c_{j/m} c_{l/m} \delta _{i+1}^r +\frac{3}{2}b_{i/m} c_{l/m} \delta _{N+j+1}^r } \right) \delta _{302} } \right] \end{aligned}$$

(21)

$$\begin{aligned} G_{kr}^{\left( c \right) }= & {} \frac{\partial \left( {F_{k/m}^{(c)} -2\frac{\Omega }{m}k\dot{c}_{k/m} +\left( {\frac{k\Omega }{m}} \right) ^{2}b_{k/m} } \right) }{\partial z\left( r \right) } \nonumber \\&=-\delta \frac{k\Omega }{m}\delta _{N+k+1}^r -\alpha \delta _{k+1}^r -\beta \left( {6a_0^{(m)} b_{k/m} \delta _r^1 } \right. +3\left( {a_0^{(m)} } \right) ^{2}\nonumber \\&\times \delta _{k+1}^r -\beta \sum _{i=1}^N {\sum _{j=1}^N {\left[ {\frac{3}{2}} \right. \left( {b_{i/m} b_{j/m} \delta _{2c1} } \right. } } \left. {+c_{i/m} c_{j/m} \delta _{2c2} } \right) \delta _r^1 \nonumber \\&+\left. {\frac{3a_0^{(m)} }{2}\left( {2b_{j/m} \delta _{2c1} \delta _{i+1}^r } \right. +\left. {2c_{j/m} \delta _{2c2} \delta _{N+i+1}^r } \right) } \right] \nonumber \\&- \beta \sum \limits _{{l = 1}}^{N} {\sum \limits _{{i = 1}}^{N} {\sum \limits _{{j = 1}}^{N} {\left[ {\frac{3}{4}} \right. b_{{j/m}} } } } b_{{l/m}} \delta _{{3c1}} \delta _{{i + 1}}^{r} + \left( {\frac{3}{4}c_{{j/m}} c_{{l/m}} \delta _{{i + 1}}^{r} } \right. \nonumber \\&\left. {\left. {+\frac{3}{2}b_{i/m} c_{l/m} \delta _{N+j+1}^r } \right) \delta _{3c2} } \right] +\left( {\frac{k\Omega }{m}} \right) ^{2}\delta _{k+1}^r \end{aligned}$$

(22)

$$\begin{aligned} G_{kr}^{\left( s \right) }= & {} \frac{\partial \left( {F_{k/m}^{(s)} +2\frac{\Omega }{m}k\dot{b}_{k/m} +\left( {\frac{k\Omega }{m}} \right) ^{2}c_{k/m} } \right) }{\partial z\left( r \right) } \nonumber \\&=\delta \frac{k\Omega }{m}\delta _{k+1}^r -\alpha \delta _{N+k+1}^r -\beta \left( {6a_0^{(m)} c_{k/m} \delta _r^1 } \right. \nonumber \\&+3\left( {a_0^{(m)} } \right) ^{2}\left. {\delta _{N+k+1}^r } \right) -\beta \sum _{i=1}^N {\sum _{j=1}^N {\left[ {3\left( {b_{i/m} c_{j/m} \delta _r^1 } \right. } \right. } } \nonumber \\&\left. {\left. {+a_0^{(m)} c_{j/m} \delta _{i+1}^r +a_0^{(m)} b_{i/m} \delta _{N+j+1}^r } \right) \delta _{2s1} } \right] \nonumber \\&-\beta \sum _{l=1}^N {\sum _{i=1}^N {\sum _{j=1}^N {\left[ \frac{3}{4}c_{j/m} c_{l/m} \delta _{N+i+1}^r \delta _{3s1} +\right. } } } \frac{3}{4}\left( {b_{j/m} b_{l/m} } \right. \nonumber \\&\left. \left. {\times \delta _{N+i+1}^r +2c_{i/m} b_{l/m} \delta _{j+1}^r } \right) \delta _{3s2} \right] +\left( {\frac{k\Omega }{m}} \right) ^{2}\delta _{N+k+1}^r \end{aligned}$$

(23)

$$\begin{aligned} H_r^{\left( 0 \right) }= & {} \frac{\partial F_0^{(m)} }{\partial \dot{z}\left( r \right) }=-\delta \delta _r^1 \end{aligned}$$

(24)

$$\begin{aligned} H_{kr}^{\left( c \right) }= & {} \frac{\partial \left( {F_{k/m}^{(c)} -2\frac{\Omega }{m}k\dot{c}_{k/m} +\left( {\frac{k\Omega }{m}} \right) ^{2}b_{k/m} } \right) }{\partial \dot{z}\left( r \right) } \nonumber \\= & {} -\delta \delta _{k+1}^r -2\frac{\Omega }{m}k\delta _{N+k+1}^r \end{aligned}$$

(25)

$$\begin{aligned} H_{kr}^{\left( s \right) }= & {} \frac{\partial \left( {F_{k/m}^{(s)} +2\frac{\Omega }{m}k\dot{b}_{k/m} +\left( {\frac{k\Omega }{m}} \right) ^{2}c_{k/m} } \right) }{\partial \dot{z}\left( r \right) } \nonumber \\= & {} -\delta \delta _{N+k+1}^r +2\frac{\Omega }{m}k\delta _{k+1}^r \end{aligned}$$

(26)