The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support

Abstract

We consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. Although it is the final value which determines which attractors eventually exist, the sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In particular, we investigate numerically how dissipation monotonically varying in time changes the sizes of the basins of attraction. It turns out that, in order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters where the systems can be considered as a perturbation of the simple pendulum, which is integrable, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. Finally, we pass to the case of an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, and we numerically observe the resulting effects on the sizes of the basins of attraction: when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. Furthermore, if during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, it can happen that, in the limit of very large variation time, a fixed point asymptotically attracts the entire phase space, up to a zero-measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values.

Appendices

Appendix 1: Global attraction to the two fixed points

To compute the conditions for attraction to the origin, we use the method outlined in [4]; see also [2]. We define $f(\tau )$ as in (1.3) and require $f(\tau ) > 0$: the consequences of this restriction are that the method can only be applied to the downward pointing pendulum when $\alpha > \beta$. Then, we apply the Liouville transformation

$$\begin{aligned} \tilde{\tau } = \int \limits _0^\tau \sqrt{f(s)}\mathrm{d}s \end{aligned}$$

(7.1)

and write our Eq. (1.3) in terms of the new time $\tilde{\tau }$ as

$$\begin{aligned} \theta _{\tilde{\tau }\tilde{\tau }} + \left( \frac{\tilde{f}(\tilde{\tau })_{\tilde{\tau }}}{2\tilde{f}(\tilde{\tau })}+ \frac{\gamma }{\sqrt{\tilde{f}(\tilde{\tau })}}\right) \theta _{\tilde{\tau }} + \sin \theta = 0, \end{aligned}$$

(7.2)

where the subscript $\tilde{\tau }$ represents derivative with respect to the new time $\tilde{\tau }$ and $\tilde{f}(\tilde{\tau }) := f(\tau )$. This can be represented as the two-dimensional system on ${\mathbb {T}}\times {\mathbb {R}}$, by setting $x(\tilde{\tau })=\theta (\tilde{\tau })$ and writing

$$\begin{aligned} x_{\tilde{\tau }} \!=\! y, \qquad y_{\tilde{\tau }} \!=\! \!-\!\frac{y}{\sqrt{\tilde{f}}} \left( \frac{\tilde{f}_{\tilde{\tau }}}{2\sqrt{\tilde{f}}} \!+\! \gamma \right) \!-\! \sin {x},\qquad \end{aligned}$$

(7.3)

for which we have the energy $E(x,y) = 1 -\cos {x} + y^2/2$. By setting $H(\tilde{\tau })=E(x(\tilde{\tau }),y(\tilde{\tau }))$, one finds

$$\begin{aligned} H_{\tilde{\tau }} = -\frac{y^2}{\sqrt{\tilde{f}}}\left( \frac{\tilde{f}_{\tilde{\tau }}}{2\sqrt{\tilde{f}}} + \gamma \right) , \end{aligned}$$

(7.4)

thus $H_{\tilde{\tau }} \le 0$, i.e. $x,\,y$ are bounded given that $\gamma$ satisfies

$$\begin{aligned} \gamma > - \min _{\tilde{\tau } \ge 0} \frac{\tilde{f}_{\tilde{\tau }}}{2\sqrt{\tilde{f}}} = - \min _{\tau \ge 0} \frac{f'}{2f}. \end{aligned}$$

(7.5)

Moreover, we have that for all $\tilde{\tau } > 0$

$$\begin{aligned} H(\tilde{\tau }) + \int \limits _0^{\tilde{\tau }} \frac{y^2}{\sqrt{\tilde{f}}} \left( \frac{\tilde{f}'}{2\sqrt{\tilde{f}}} + \gamma \right) \mathrm{d}s = H(0), \end{aligned}$$

(7.6)

so that, as $\tilde{\tau } \rightarrow \infty$, using the properties above we can arrive at

$$\begin{aligned} \min _{s \ge 0} \left[ \frac{1}{\sqrt{\tilde{f}}}\left( \frac{\tilde{f}_{\tilde{\tau }}}{2\sqrt{\tilde{f}}} + \gamma \right) \right] \int \limits _0^\infty y^2(s) \mathrm{d}s < \infty . \end{aligned}$$

(7.7)

Hence, $y \rightarrow 0$ as time tends to infinity. There are two regions of phase space to consider. Any level curve of $H$ strictly inside the separatrix of the unperturbed pendulum is the boundary of a positively invariant set $D$ containing the origin: since $S = \{(x(\tilde{\tau }),y(\tilde{\tau })) : H_{\tilde{\tau }} = 0\} \cap D$ consists purely of the origin, we can apply the local Barbashin–Krasovsky–La Salle theorem [24] to conclude that every trajectory that begins strictly inside the separatrix will converge to the origin as $\tilde{\tau }\rightarrow +\infty$.

Outside of the separatrix, we may use Eq. (7.4), which shows the energy to be strictly decreasing while $y \ne 0$, provided $\gamma$ is chosen large enough, coupled with $y \rightarrow 0$ as time tends to infinity. The result is that all trajectories tend to the invariant points on the $x$-axis as time tends to infinity. One of two cases must occur: either the trajectory moves inside the separatrix or it does not. In the first instance, we have already shown that the limiting solution is the origin. In the latter, there is only one possibility. As all points on the $x$-axis are contained within the separatrix other than the unstable fixed point, the trajectory must move onto such a fixed point and hence belongs to its stable manifold, which is a zero-measure set. Therefore, we conclude that a full-measure set of initial conditions are attracted by the origin. Reverting back to the original system with time $\tau$, we conclude that for that system too the basin of attraction of the origin has full measure, provided $\beta <\alpha$ and $\gamma$ satisfies (7.5).

Appendix 2: Action-angle variables

In this section, we detail the calculation of the action-angle variables for the simple pendulum in time $\tau$. More details on calculating action-angle variables can be found in [9, 14, 32]. The simple pendulum has equation of motion given by

$$\begin{aligned} \theta '' + \alpha \sin {\theta } = 0, \end{aligned}$$

(8.1)

where the dashes represent derivative with respect to the scaled time $\tau$. The Hamiltonian for the simple pendulum in this notation is

$$\begin{aligned} E = H(\theta ,\theta ') = \frac{1}{2} (\theta ')^2 - \alpha \cos {\theta } \end{aligned}$$

(8.2)

or, in terms of the usual notation for Hamiltonian dynamics,

$$\begin{aligned} E = H(p,q) = \frac{1}{2} p^2 - \alpha \cos {q}, \end{aligned}$$

(8.3)

where $q = \theta$ and $p= q'=\theta '$. Rearranging this for $p$ we obtain $p=\pm p(E,q)$, with

$$\begin{aligned} p(E,q) \!=\! \sqrt{2(E \!+\! \alpha \cos {q})} \!=\! \sqrt{2\alpha (E_{0} \!+\! \cos {q})},\nonumber \\ \end{aligned}$$

(8.4)

where $E_{0} = E/\alpha$. It is clear that there are two types of dynamics, oscillatory dynamics when $E_{0} < 1$ and rotational dynamics when $E_{0} > 1$, separated at a separatrix when $E_{0} = 1$, for which no action-angle variables exist.

1.1 Librations

We first consider the case $E_{0} < 1$. The action variable is

$$\begin{aligned} I&= \frac{1}{2\pi } \oint p \mathrm{d}q = \frac{2}{\pi }\sqrt{2\alpha } \int \limits _{0}^{q_1} \sqrt{E_{0} + \cos {q}} \; \mathrm{d}q\nonumber \\&= \frac{8}{\pi }\sqrt{\alpha }\Bigl [(k_1^2 - 1)\mathbf{K}(k_1) + \mathbf{E}(k_1)\Bigr ], \end{aligned}$$

(8.5)

where $k_1^2 = (E_{0}+1)/2$ and $q_{1}=\arccos (-E_{0})$. The functions $\mathbf{K}(k)$ and $\mathbf{E}(k)$ are the complete elliptic integrals of the first and second kinds, respectively.

The angle variable $\varphi$ can be found as follows

$$\begin{aligned} \varphi ' = \frac{\partial H}{\partial I } = \frac{\mathrm{d}E}{\mathrm{d}I} = \left( \frac{\mathrm{d}I}{\mathrm{d}E}\right) ^{-1}, \end{aligned}$$

(8.6)

so that

$$\begin{aligned} \frac{\mathrm{d}I}{\mathrm{d}E}&= \frac{\mathrm{d}}{\mathrm{d}E} \frac{2}{\pi } \int \limits _{0}^{q_1} \sqrt{E + \alpha \cos {q}} \; \mathrm{d}q \nonumber \\&= \frac{2}{\pi \sqrt{\alpha }} \mathbf{K}(k_1). \end{aligned}$$

(8.7)

Hence, we have

$$\begin{aligned} \varphi (\tau ) = \frac{\pi }{2\mathbf{K}(k_1)} \sqrt{\alpha }(\tau - \tau _0). \end{aligned}$$

(8.8)

Take $s = \sin {(q/2)}$; then, using Eq. (8.2), it is easy to show that

$$\begin{aligned} (s')^2 = \frac{g}{l}(1-s^2)\left( k_1^2 - s^2\right) . \end{aligned}$$

(8.9)

Integrating using the Jacobi elliptic functions

$$\begin{aligned} s(\tau ) = k_1 {{\mathrm{sn}}}{\left( \sqrt{\frac{g}{l}}(\tau - \tau _0),k_1\right) }, \end{aligned}$$

(8.10)

the expression can then be rearranged to achieve the following result:

$$\begin{aligned} q&= 2\arcsin {\left[ k_1 {{\mathrm{sn}}}{\left( \frac{2\mathbf{K}(k_1)}{\pi }\varphi ,k_1\right) }\right] },\nonumber \\ p&= 2 k_1 \sqrt{\alpha } {{\mathrm{cn}}}{\left( \frac{2\mathbf{K}(k_1)}{\pi }\varphi ,k_1\right) }, \end{aligned}$$

(8.11)

which coincide with Eqs. (2.4). Using (11.3) in Appendix 3, one obtains from (8.5)

$$\begin{aligned} \frac{\partial I}{\partial k_{1}} = \frac{8}{\pi } k_{1} \mathbf{K}(k_{1}) \, \sqrt{\alpha } , \end{aligned}$$

(8.12)

a relation which has been used to derive (2.12).

1.2 Rotations

In the case of rotational dynamics, we have

$$\begin{aligned}&I = \frac{1}{2\pi } \int \limits _{0}^{2\pi } p \, q = \frac{1}{2\pi }\sqrt{\alpha } \int \limits _{0}^{2\pi } \sqrt{E_{0} + \cos {q}} \ \mathrm{d}q\nonumber \\&\quad = \frac{4}{k_2\pi }\sqrt{\alpha } \, \mathbf{E}(k_2), \end{aligned}$$

(8.13)

where this time we let $k_2^2 = 2/(E_{0} + 1) = 1/k_1^2$. The angle variable $\varphi$ can similarly be found using (8.6), where $\mathrm{d}I/\mathrm{d}E$ can be similarly calculated as

$$\begin{aligned} \frac{\mathrm{d}I}{\mathrm{d}E}&= \frac{\mathrm{d}}{\mathrm{d}E} \frac{1}{2\pi } \int \limits _0^{2\pi } \sqrt{E +\alpha \cos {q}} \;\mathrm{d}q \nonumber \\&= \frac{k_2}{\pi \sqrt{\alpha }} \mathbf{K}(k_2), \end{aligned}$$

(8.14)

which hence gives

$$\begin{aligned} \varphi (\tau ) = \frac{\pi }{\mathbf{K}(k_2)}\sqrt{\alpha }\frac{(\tau - \tau _0)}{k_2}. \end{aligned}$$

(8.15)

Using (8.9) and the definition of $k_2$, for the rotating solutions we find that

$$\begin{aligned} s(\tau ) = {{\mathrm{sn}}}{\left( \sqrt{\alpha } \frac{(\tau - \tau _0)}{k_2},k_2\right) } , \end{aligned}$$

(8.16)

and similarly, by simple rearrangement, we find that

$$\begin{aligned} q&= 2 \arcsin {\left[ {{\mathrm{sn}}}{\left( \frac{\mathbf{K}(k_2)}{\pi }\varphi ,k_2\right) }\right] },\nonumber \\ p&= \frac{2}{k_2} \sqrt{\alpha } \, {{\mathrm{dn}}}{\left( \frac{\mathbf{K}(k_2)}{\pi }\varphi ,k_2\right) }, \end{aligned}$$

(8.17)

which again yields Eqs. (2.6). Using (11.3) in Appendix 3, one obtains from (8.13)

$$\begin{aligned} \frac{\partial I}{\partial k_{2}} = - \frac{4}{\pi k_{2}^{2}} \mathbf{K}(k_{2}) \, \sqrt{\alpha } \end{aligned}$$

(8.18)

which has been used to derive (2.23).

Appendix 3: Jacobian determinant

Here, we compute the entries of the Jacobian matrix $J$ of the transformation to action-angle variables, which will be used in the next Appendix. As a by-product, we check that $J$ determinant equal to 1, that is

$$\begin{aligned} \frac{\partial q}{\partial \varphi }\frac{\partial p}{\partial I} - \frac{\partial q}{\partial I}\frac{\partial p}{\partial \varphi } =1. \end{aligned}$$

(9.1)

For further details on the proof of (9.1), we refer the reader to [9], where the calculations are given in great detail. The derivative with respect to $\varphi$ is straightforward in both the libration and rotation case, however, the dependence of $p$ and $q$ on the action $I$ is less obvious. That said, the dependence of $p$ and $q$ on $k_1$ in the oscillating case and $k_2$ in the rotating case is clear and we know the relationship between $I$ and $k$ in both cases; hence, the derivative of the Jacobi elliptic functions can be calculated using that

$$\begin{aligned}&\frac{\partial }{\partial I} \!= \!\frac{\partial k}{\partial I}\frac{\partial }{\partial k} \!+\! \frac{\partial u}{\partial I}\frac{\partial }{\partial u} \!=\! \frac{\partial k}{\partial I}\left( \frac{\partial }{\partial k} \!+\! \frac{\partial u}{\partial k}\frac{\partial }{\partial u}\right) ,\qquad \end{aligned}$$

(9.2)

where $u$ is the first argument of the functions, i.e. ${{\mathrm{sn}}}(u,k)$, etc. Then, for the oscillations, we have

$$\begin{aligned} \frac{\partial q}{\partial I}&= \frac{\pi }{4k_1\mathbf{K}(k_1)\sqrt{\alpha }}\left[ \frac{{{\mathrm{sn}}}(\cdot )}{{{\mathrm{dn}}}(\cdot )}+ \frac{2\mathbf{E}(k_1)\varphi {{\mathrm{cn}}}(\cdot )}{\pi k_{1}'^2}\right. \nonumber \\&\left. + \frac{k_1^2{{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}^2(\cdot )}{k_1'^2{{\mathrm{dn}}}(\cdot )} - \frac{\mathbf{E}(\cdot ){{\mathrm{cn}}}(\cdot )}{k_1'^2} \right] , \nonumber \\ \frac{\partial p}{\partial I}&= \frac{\pi }{4k_1\mathbf{K}(k_1)}\left[ {{\mathrm{cn}}}(\cdot ) - \frac{2\mathbf{E}(k_1)\varphi {{\mathrm{sn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{\pi k_{1}'^2}\right. \nonumber \\&\left. - \frac{k_1^2{{\mathrm{sn}}}^2(\cdot ){{\mathrm{cn}}}(\cdot )}{k_1'^2} + \frac{\mathbf{E}(\cdot ){{\mathrm{sn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{k_1'^2}\right] ,\nonumber \\ \frac{\partial q}{\partial \varphi }&= \frac{4k_1\mathbf{K}(k_1){{\mathrm{cn}}}(\cdot )}{\pi } ,\nonumber \\ \frac{\partial p}{\partial \varphi }&= -\sqrt{\alpha } \, \frac{4k_1\mathbf{K}(k_1){{\mathrm{sn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{\pi }, \end{aligned}$$

(9.3)

where $(\cdot ) = \left( \frac{2\mathbf{K}(k_1)\varphi }{\pi },k_1\right)$ and $k_1' = \sqrt{1 - k_1^2}$. From the above, it is easy to check that Eq. (9.1) is satisfied. Similarly, for the rotations, we have

$$\begin{aligned} \frac{\partial q}{\partial I}&= - \frac{\pi k_2^2}{2 \mathbf{K}(k_2) \, \sqrt{\alpha }} \left[ \frac{\varphi \, \mathbf{E}(k_{2})\,{{\mathrm{dn}}}(\cdot )}{\pi k_{2}k_{2}'^2}\right. \nonumber \\&\left. + \frac{k_2{{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot )}{k_2'^2} - \frac{\mathbf{E}(\cdot ){{\mathrm{dn}}}(\cdot )}{k_2k_2'^2}\right] , \nonumber \\ \frac{\partial p}{\partial I}&= \frac{\pi k_2^2}{2 \mathbf{K}(k_2)} \left[ \frac{{{\mathrm{dn}}}(\cdot )}{k_2^2} + \frac{\varphi \, \mathbf{E}(k_{2})\,{{\mathrm{sn}}}(\cdot ) \, {{\mathrm{cn}}}(\cdot )}{\pi k_{2}'^2}\right. \nonumber \\&\left. + \frac{{{\mathrm{sn}}}^2(\cdot ){{\mathrm{dn}}}(\cdot )}{k_2'^2} - \frac{\mathbf{E}(\cdot ){{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot )}{k_2'^2} \right] , \nonumber \\ \frac{\partial q}{\partial \varphi }&= \frac{2\mathbf{K}(k_2){{\mathrm{dn}}}(\cdot )}{\pi },\nonumber \\ \frac{\partial p}{\partial \varphi }&= -\sqrt{\alpha } \, \frac{2k_2\mathbf{K}(k_2){{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot )}{\pi } , \end{aligned}$$

(9.4)

where $(\cdot ) = \left( \frac{\mathbf{K}(k_2)}{\pi }\varphi ,k_2\right)$ and $k_2' = \sqrt{1 - k_2^2}$. It is once again easily checked from the above that (9.1) is satisfied.

Appendix 4: Equations of motion for the perturbed system

By (9.1), one has

$$\begin{aligned} \begin{pmatrix} \partial \varphi /\partial q &{} \partial \varphi / \partial p \\ \partial I /\partial q &{} \partial I / \partial p \end{pmatrix} = \begin{pmatrix} \partial p / \partial I &{} - \partial q / \partial I \\ - \partial p/ \partial \varphi &{} \partial q / \partial \varphi . \end{pmatrix} \end{aligned}$$

(10.1)

We rewrite the Eq. (1.3) in the action-angle coordinates introduced in Appendix 2 as follows.

1.1 Librations

In this section, we want to write (1.3) in terms of the action-angle introduced in Appendix 2. By taking into account the forcing term $-\beta \cos \tau \cos \theta$ in (1.3), one finds

$$\begin{aligned}&I' \!=\! \frac{\partial I}{\partial q}q' \!+\! \frac{\partial I}{\partial p}p' \!=\! - \frac{\partial p}{\partial \varphi }q' \!+\! \frac{\partial q}{\partial \varphi }p' \nonumber \\&\quad = \frac{8 \beta k_1^2 \mathbf{K}(k_1)}{\pi }\cos (\tau \!-\! \tau _0){{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot ), \nonumber \\&\varphi ' \!=\! \frac{\partial \varphi }{\partial q}q' \!+\! \frac{\partial \varphi }{\partial p}p' \!=\! \frac{\partial p}{\partial I}q' \!-\! \frac{\partial q}{\partial I}p' \nonumber \\&\quad = \frac{\pi \sqrt{\alpha }}{2\mathbf{K}(k_1)} \!-\! \frac{\pi \beta }{2\mathbf{K}(k_1)\,\sqrt{\alpha }}\nonumber \\&\left[ {{\mathrm{sn}}}^2(\cdot ) \!+\! \frac{2\mathbf{E}(k_1)\varphi {{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{\pi k_{1}'^2}+ \left. \frac{k_1^2{{\mathrm{sn}}}^2(\cdot ){{\mathrm{cn}}}^2(\cdot )}{1\!-\!k_1^2} \right. \right. \nonumber \\&\quad \left. - \frac{\mathbf{E}(\cdot ) {{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{1 \!-\! k_1^2}\right] \cos (\tau \!-\! \tau _0) , \end{aligned}$$

(10.2)

where we have used the properties of the Jacobi elliptic functions in Appendix 5. As in Appendix 3, we are shortening $(\cdot ) = \left( \frac{2\mathbf{K}(k_1)\varphi }{\pi },k_1\right)$.

We then wish to add the dissipative term $\gamma \theta '$. This results in the following equations:

$$\begin{aligned} I'&= \frac{8 \beta k_1^2 \mathbf{K}(k_1)}{\pi } \cos (\tau - \tau _0){{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )\nonumber \\&-\frac{8 \gamma k_1^2 \sqrt{\alpha } \, \mathbf{K}(k_1)}{\pi }{{\mathrm{cn}}}^2(\cdot ), \nonumber \\ \varphi '&= \frac{\pi \sqrt{\alpha }}{2\mathbf{K}(k_1)}- \frac{\pi \beta }{2\mathbf{K}(k_1)\,\sqrt{\alpha }}\left[ {{\mathrm{sn}}}^2(\cdot )\right. \nonumber \\&\left. +\, \frac{2\mathbf{E}(k_1)\varphi {{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{\pi (1-k_{1}^2)}+ \frac{k_1^2{{\mathrm{sn}}}^2(\cdot ){{\mathrm{cn}}}^2(\cdot )}{1-k_1^2}\right. \nonumber \\&\left. -\, \frac{\mathbf{E}(\cdot ) {{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{1 - k_1^2}\right] \cos (\tau - \tau _0) \nonumber \\&+\, \frac{\gamma \pi {{\mathrm{cn}}}(\cdot )}{2\mathbf{K}(k_1)}\left[ \frac{{{\mathrm{sn}}}(\cdot )}{{{\mathrm{dn}}}(\cdot )} \right. \left. + \frac{2 \mathbf{E}(k_1)\varphi {{\mathrm{cn}}}(\cdot )}{\pi (1-k_{1}^2)}\right. \nonumber \\&\left. +\, \frac{k_1^2{{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}^2(\cdot )}{(1-k_1^2){{\mathrm{dn}}}(\cdot )} - \frac{ \mathbf{E}(\cdot ){{\mathrm{cn}}}(\cdot )}{1 - k_1^2}\right] . \end{aligned}$$

(10.3)

Using the property that, see [26], $\mathbf{E}(u,k) \!=\! \mathbf{E}(k)u/\mathbf{K}(k)$ $+ \mathbf{Z}(u,k)$, we arrive at equations (2.9). The function $\mathbf{Z}(u,k)$ is the Jacobi zeta function, which is periodic with period $2\mathbf{K}(k)$ in $u$.

1.2 Rotations

The presence of the forcing term leads to the Equations.

$$\begin{aligned} I'&= - \frac{\partial p}{\partial \varphi }q' + \frac{\partial q}{\partial \varphi }p'\nonumber \\&= \frac{4 \beta \mathbf{K}(k_2)}{\pi }\cos (\tau - \tau _0) {{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot ), \nonumber \\ \varphi '&= \frac{\partial p}{\partial I}\dot{q} - \frac{\partial q}{\partial I}\dot{p}= \frac{\pi \sqrt{\alpha }}{k_2\mathbf{K}(k_2)}+\, \frac{\pi k_2 \beta }{\sqrt{\alpha }\mathbf{K}(k_2)}\nonumber \\&\times \left[ \frac{ \mathbf{E}(k_{2}) \, \varphi \,{{\mathrm{sn}}}(\cdot )\,{{\mathrm{cn}}}(\cdot )\,{{\mathrm{dn}}}(\cdot )}{\pi (1-k_{2}^2)} \right. +\,\frac{k^2_2 {{\mathrm{sn}}}^2(\cdot ){{\mathrm{cn}}}^2(\cdot )}{1 - k_2^2}\nonumber \\&\left. -\frac{\mathbf{E}(\cdot ){{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{1 - k_2^2} \right] \cos (\tau - \tau _0). \end{aligned}$$

(10.4)

Again, if we wish to add a dissipative term, we arrive at the equations

$$\begin{aligned} I'&= \frac{4 \beta \mathbf{K}(k_2)}{\pi }\cos (\tau \!-\! \tau _0) {{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )\nonumber \\&-\frac{4\gamma \sqrt{\alpha } \, \mathbf{K}(k_2)}{\pi k_2}{{\mathrm{dn}}}^2(\cdot ), \nonumber \\ \varphi '&= \frac{\pi \sqrt{\alpha }}{k_2\mathbf{K}(k_2)} \!+\! \frac{\pi k_2 \beta }{\sqrt{\alpha }\mathbf{K}(k_2)} \left[ \frac{ \mathbf{E}(k_{2}) \, \varphi \,{{\mathrm{sn}}}(\cdot )\,{{\mathrm{cn}}}(\cdot )\,{{\mathrm{dn}}}(\cdot )}{\pi (1\!-\!k_{2}^2)} \right. \nonumber \\&\left. + \frac{k^2_2 {{\mathrm{sn}}}^2(\cdot ){{\mathrm{cn}}}^2(\cdot )}{1 \!-\! k_2^2}\!-\! \frac{\mathbf{E}(\cdot ){{\mathrm{sn}}}(\cdot ){{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{1 \!-\! k_2^2} \right] \nonumber \\&\times \cos (\tau \!-\! \tau _0) \!-\! \frac{\gamma \pi }{\mathbf{K}(k_2)} \left[ \frac{ \mathbf{E}(k_{2}) \, \varphi \,{{\mathrm{dn}}}^{2} (\cdot )}{\pi (1\!-\!k_{2}^2)}\right. \nonumber \\&\left. + \frac{k_2^2{{\mathrm{sn}}}(\cdot ) {{\mathrm{cn}}}(\cdot ){{\mathrm{dn}}}(\cdot )}{1 \!-\! k_2^2} \!-\! \frac{ \mathbf{E}(\cdot ){{\mathrm{dn}}}^2(\cdot )}{1\!-\!k_2^2} \right] . \end{aligned}$$

(10.5)

Again, using that $\mathbf{E}(u,k) = \mathbf{E}(k)u/\mathbf{K}(k) + \mathbf{Z}(u,k)$, we arrive at the equations (2.22).

Appendix 5: Useful properties of the elliptic functions

The complete integrals of the first and second kind are, respectively,

$$\begin{aligned} \mathbf{K}(k)&= \int \limits _{0}^{\pi /2} \frac{\mathrm{d}\psi }{\sqrt{1-k^2 \sin ^2 \psi }} ,\nonumber \\ \mathbf{E}(k)&= \int \limits _{0}^{\pi /2} \mathrm{d}\psi \, \sqrt{1-k^2 \sin ^2 \psi } , \end{aligned}$$

(11.1)

whereas the incomplete elliptic integral of the second kind is

$$\begin{aligned} \mathbf{E}(u,k) = \int \limits _{0}^{{{\mathrm{sn}}}(u,k)} \mathrm{d}x \frac{\sqrt{1-k^2 x^{2}}}{\sqrt{1-x^2}}. \end{aligned}$$

(11.2)

One has

$$\begin{aligned}&\!\!\!\frac{\partial \mathbf{K}(k)}{\partial k} = \frac{1}{k} \left( \frac{\mathbf{E}(k)}{1-k^2} - \mathbf{K}(k) \right) ,\nonumber \\&\!\!\!\frac{\partial \mathbf{E}(k)}{\partial k} = \frac{1}{k} \left( \mathbf{E}(k) - \mathbf{K}(k) \right) . \end{aligned}$$

(11.3)

The following properties of the Jacobi elliptic functions have been used in the previous sections. The derivatives with respect to the first arguments are

$$\begin{aligned} \frac{\partial }{\partial u} {{\mathrm{sn}}}(u,k)&= {{\mathrm{cn}}}(u,k) \, {{\mathrm{dn}}}(u,k) , \nonumber \\ \frac{\partial }{\partial u} {{\mathrm{cn}}}(u,k)&= - {{\mathrm{sn}}}(u,k) \, {{\mathrm{dn}}}(u,k) , \\ \frac{\partial }{\partial u} {{\mathrm{dn}}}(u,k)&= - k^2 {{\mathrm{sn}}}(u,k) \, {{\mathrm{cn}}}(u,k) ,\nonumber \end{aligned}$$

(11.4)

while the derivatives with respect to the elliptic modulus are

$$\begin{aligned} \frac{\partial }{\partial k} {{\mathrm{sn}}}(u,k)&= \frac{u}{k} {{\mathrm{cn}}}(u,k) \, {{\mathrm{dn}}}(u,k)\nonumber \\&+\, \frac{k}{k'^2} \, {{\mathrm{sn}}}(u,k) \, {{\mathrm{cn}}}^{2} (u,k)\nonumber \\&-\,\frac{1}{k k'^2} \mathbf{E}(u,k)\,{{\mathrm{cn}}}(u,k) \, {{\mathrm{dn}}}(u,k) ,\nonumber \\ \frac{\partial }{\partial k} {{\mathrm{cn}}}(u,k)&= - \frac{u}{k} {{\mathrm{sn}}}(u,k) \, {{\mathrm{dn}}}(u,k)\nonumber \\&-\, \frac{k}{k'^2} \, {{\mathrm{sn}}}^2(u,k) \, {{\mathrm{cn}}}(u,k)\nonumber \\&+\,\frac{1}{k k'^2} \mathbf{E}(u,k)\,{{\mathrm{sn}}}(u,k) \, {{\mathrm{dn}}}(u,k) ,\nonumber \\ \frac{\partial }{\partial k} {{\mathrm{dn}}}(u,k)&= -ku \, {{\mathrm{sn}}}(u,k) \, {{\mathrm{cn}}}(u,k)\nonumber \\&-\, \frac{k}{k'^2} \, {{\mathrm{sn}}}^{2}(u,k) \, {{\mathrm{dn}}}(u,k)\nonumber \\&+\,\frac{k}{k'^2} \mathbf{E}(u,k)\,{{\mathrm{sn}}}(u,k) \, {{\mathrm{cn}}}(u,k),\nonumber \\ \end{aligned}$$

(11.5)

where $k'^2=1-k^2$.

Finding the value of $\Delta$ for rotations in Sect. 2 requires use of

$$\begin{aligned} \int \limits _0^{x_1} {{\mathrm{dn}}}^2(x,k) \mathrm{d}x \!=\! \int \limits _0^{{{\mathrm{sn}}}(x_1,k)} \frac{\sqrt{1 \!-\! k^2\hat{x}^2}}{\sqrt{1 \!-\! \hat{x}^2}} \mathrm{d}\hat{x} \!=\! \mathbf{E}(x_1,k).\nonumber \\ \end{aligned}$$

(11.6)

In the case of librations, we also require the relation $k^2 {{\mathrm{cn}}}^2(\cdot ) + (1 - k^2) = {{\mathrm{dn}}}^2(\cdot )$.

The integral for in Eq. (2.20) is found by

(11.7)

where .

The Jacobi elliptic functions can be expanded in a Fourier series as

$$\begin{aligned}&{{\mathrm{sn}}}(u,k) \!=\! \frac{2\pi }{k\mathbf{K}(k)}\sum _{n=1}^\infty \frac{{\mathfrak {q}}^{n - 1/2}}{1 \!-\! {\mathfrak {q}}^{2n - 1}}\sin \left( \frac{(2n \!-\! 1)\pi u}{2\mathbf{K}(k)}\right) ,\nonumber \\&{{\mathrm{cn}}}(u,k)\!=\! \frac{2\pi }{k\mathbf{K}(k)}\sum _{n=1}^\infty \frac{{\mathfrak {q}}^{n - 1/2}}{1 \!+\! {\mathfrak {q}}^{2n - 1}}\cos \left( \frac{(2n \!-\! 1)\pi u}{2\mathbf{K}(k)}\right) ,\nonumber \\&{{\mathrm{dn}}}(u,k) \!=\! \frac{\pi }{2\mathbf{K}(k)}\!+\!\frac{2\pi }{\mathbf{K}(k)}\sum _{n=1}^\infty \frac{{\mathfrak {q}}^{n}}{1 \!-\! {\mathfrak {q}}^{2n}}\cos \left( \frac{2n\pi u}{2\mathbf{K}(k)}\right) \!, \end{aligned}$$

(11.8)

where ${\mathfrak {q}}$ is the nome, defined as

$$\begin{aligned} {\mathfrak {q}} = \exp \left( - \frac{ \pi \mathbf{K}(k')}{\mathbf{K}(k)}\right) , \end{aligned}$$

with $k'=\sqrt{1-k^2}$.

In the calculation of $\langle R^{(n)} \rangle$ for $n \ge 2$, when the pendulum is in libration, we require the evaluation of the integrals

$$\begin{aligned} \frac{1}{T}\int \limits _0^T \frac{2\mathbf{K}(k_1)}{\sqrt{\alpha }\pi } \frac{\partial }{\partial \tau }\Bigl ({{\mathrm{cn}}}^2(\sqrt{\alpha }\tau )\Bigr ) \mathrm{d}\tau = 0, \end{aligned}$$

(11.9)

and

$$\begin{aligned}&\frac{1}{T}\int \limits _0^T \frac{2\mathbf{K}(k_1)}{\sqrt{\alpha }\pi }\frac{\partial }{\partial \tau } \Bigl ({{\mathrm{sn}}}(\sqrt{\alpha }\tau ){{\mathrm{cn}}}(\sqrt{\alpha }\tau ){{\mathrm{dn}}}(\sqrt{\alpha }\tau ) \Bigr )\cos (\tau - \tau _0) \mathrm{d}\tau \nonumber \\&\quad = \frac{1}{T}\int \limits _0^T \frac{2\mathbf{K}(k_1)}{\sqrt{\alpha }\pi }{{\mathrm{sn}}}(\sqrt{\alpha }\tau ) {{\mathrm{cn}}}(\sqrt{\alpha }\tau ){{\mathrm{dn}}}(\sqrt{\alpha }\tau )\sin (\tau - \tau _0) \mathrm{d}\tau \nonumber \\&\quad = \frac{\cos (\tau _0)}{T}\int \limits _0^T \frac{2\mathbf{K}(k_1)}{\sqrt{\alpha }\pi } {{\mathrm{sn}}}(\sqrt{\alpha }\tau ){{\mathrm{cn}}}(\sqrt{\alpha }\tau ){{\mathrm{dn}}}(\sqrt{\alpha }\tau )\sin (\tau ) \mathrm{d}\tau \nonumber \\&\qquad - \frac{\sin (\tau _0)}{T}\int \limits _0^T \frac{2\mathbf{K}(k_1)}{\sqrt{\alpha }\pi } {{\mathrm{sn}}}(\sqrt{\alpha }\tau ){{\mathrm{cn}}}(\sqrt{\alpha }\tau ){{\mathrm{dn}}}(\sqrt{\alpha }\tau )\cos (\tau ) \mathrm{d}\tau , \nonumber \\ \end{aligned}$$

(11.10)

where . The integral multiplying $\sin (\tau _0)$ vanishes due to parity, and hence,

(11.11)