The Duffing Oscillator with Damping for a Softening Potential

Abstract

The solution to the softening Duffing differential equation ${\ddot{x}}+2\beta {\dot{x}}+\alpha x-\varepsilon x^{3}=0$, for an unforced system with damping is described in terms of the Jacobi elliptic functions. The parameter of the Jacobi elliptic functions is allowed to be time-dependent and the analytical solution is found to be accurate compared to the numerical one. An explicit expression for the the period of oscillation is derived and found to be longer initially, however gradually decreasing to that of a linearized model.

Appendix: The Derivation of the Function \(\xi (u)\)

In this section the scaling function $\xi (u)$ is derived such that the relationsship:

$$\begin{aligned} {\xi }^{\prime }-\frac{{m}^{\prime }}{m} f= 1 \end{aligned}$$

(21)

is fulfilled for any value of the variable u [24, 25]. The function f(u) is for small values of the elliptic parameter approximately given by $f(u)\cong (\frac{1}{4}m(u)+\frac{9}{32}m(u)^{2}) u$ [24, 25], and the elliptic parameter m is in the present case given by Eq. (12):

$$\begin{aligned} m(u)=\frac{a e^{-2 \gamma u}}{1-a e^{-2 \gamma u}} \end{aligned}$$

It is seen, that with increasing values of u the elliptic parameter varies between the maximum value of $m(0)=a/{(1-a)}$ towards zero value.

From Eq. (21) one now finds that:

$$\begin{aligned}&{\xi }^{\prime }\cong 1+\frac{{m}^{\prime }}{m} f\\&\quad \Updownarrow \\&{\xi }^{\prime }\cong 1+\left( \frac{1}{4}+\frac{9}{32}m\right) {m}^{\prime } u\\&\quad \Updownarrow \\&\xi \cong u+\int {\left( \frac{1}{4}+\frac{9}{32}m\right) \frac{dm}{du} u}\, du\, + const. \end{aligned}$$

Using integration by parts this gives:

$$\begin{aligned}&\xi \cong u+\left( \frac{1}{4}m+\frac{9}{64}m^{2}\right) u-\int {\left( \frac{1}{4}m+\frac{9}{64}m^{2}\right) } du\, + const.\\&\quad \Updownarrow \\&\xi \cong \left( 1+\frac{1}{4}m+\frac{9}{64}m^{2}\right) u-\frac{1}{4} \int {\frac{a e^{-2 \gamma u}}{1-a e^{-2 \gamma u}}} du-\frac{9}{64} \int {\frac{a^{2} e^{-4 \gamma u}}{\left( 1-a e^{-2 \gamma u}\right) ^{2}}} du\, +const. \end{aligned}$$

By use of the substitution, $v=1-a e^{-2 \gamma u}$, both of these integrals are easily evaluated and one finds that:

$$\begin{aligned} \xi\cong & {} \left( 1+\frac{1}{4}m+\frac{9}{64}m^{2}\right) u-\frac{1}{8 \gamma } \ln \left( 1-a e^{-2 \gamma u}\right) \\&+\frac{9}{128 \gamma } \left( {\ln (1-a e^{-2 \gamma u})+\frac{1}{1-a e^{-2 \gamma u}}} \right) \, + const. \end{aligned}$$

Finally, the constant of integration is found from the criteria, that $\xi (0)=0$, and thus an expression for the parameter scaling expression $\xi (u)$ is:

$$\begin{aligned} \xi\cong & {} \left( 1+\frac{1}{4}m+\frac{9}{64}m^{2}\right) u-\frac{1}{8 \gamma } \ln \left( {\frac{1-a e^{-2 \gamma u}}{1-a}} \right) \nonumber \\&+\frac{9}{128 \gamma } \left( {\ln \left( {\frac{1-a e^{-2 \gamma u}}{1-a}} \right) +\frac{1}{1-a e^{-2 \gamma u}}-\frac{1}{1-a}} \right) \end{aligned}$$

(22)

In the limit of $\gamma \rightarrow 0$ we have $\xi (u)\rightarrow u$ as expected, which can readily be verified by use of the theorem by L’Hospital.