Integrable Systems

Abstract

Consider a conservative Hamiltonian system. One may specify the state of such a system by giving all the position and momentum coordinates (q,p) of a point in the system phase space, and the time evolution of the system is described by a trajectory lying on a surface described by the conservation of energy in the phase space. A dynamical system is said to be ergodic, if left to itself for long enough, it will pass in an erratic manner close to nearly all the dynamical states compatible with conservation of energy.

Appendix: The Problem of Internal Resonances in Nonlinearly-Coupled Systems

The problem of internal resonances in nonlinearly-coupled oscillator systems is of much interest in connection with redistribution of energy among the various natural modes of such a system. This energy sharing is usually brought about by resonant interactions among the natural modes of the system. The nature of the couplings among the latter plays a crucial role in such interactions.

Systems such as the one considered by Fermi et al. (1955) (see Chap. 7), namely, a system of one-dimensional weakly-nonlinearly-coupled chain of oscillators with the Hamiltonian of the form

$$ H=\frac{1}{2}\sum^{64}_{k=1} \bigl(p^2_k+\omega^2_k q^2_k \bigr)+\varepsilon \sum ^{64}_{j,k,l=1}A_{jkl}q_j q_k q_l. $$

(4.165)

ε being chosen sufficiently small (so that the nonlinear terms can be treated as a small perturbation on a primarily linear problem), are non-ergodic for small values of ε and are amenable to analysis using a perturbation theory. One property of such systems is that there is, in general, no effective energy sharing without an internal resonance. One would therefore desire to analyze a nonlinear system of equations possessing internal resonances and show that the latter lead to an effective exchange of energy.

Jackson (1963a,b) and Ford and Waters (1963) made such attempts for a problem of two coupled oscillators. In a straightforward perturbation theory, as we saw in Sect. 4.5, the internal resonances lead to the problem of small divisors, which are associated with terms having $(\sum^{N}_{k=1}n_{k}\omega_{k} )^{-1}$ as factors. Jackson (1963a,b) tried to remove the small divisors in the solution by introducing shifts in the frequencies ω _k so that the corrected frequencies were no longer commensurable. However, the frequency shifts arise whether or not small divisors in the solution exist so that this approach cannot describe adequately what happens during an internal resonance—such as the manner in which energy is exchanged among the oscillators.

In order to describe the internal resonances adequately, it is obvious that such a perturbation theory should include modulations of the amplitudes of the oscillators along with any corrections to the frequencies ω _k —which were most clearly seen in the numerical calculations of Ford and Waters (1963). Some perturbation theories for treating the problem of internal resonances in a system of two nonlinearly-coupled oscillators adequately have been given by Kabakow (1968), van der Burgh (1976), Verhulst (1979), and Kevorkian (1980). These studies revealed that whereas the actions of the individual oscillators are nearly constant when the system does not show an internal resonance, the total action of the uncoupled system is found to be nearly constant when the system undergoes an internal resonance and allows for a significant redistribution of action between two oscillators. The problem of higher-order internal resonances was considered by Shivamoggi and Varma (1988). The approximate invariant mentioned above to exist for the lowest-order internal resonance was found to continue to hold for the higher-order internal resonances.

Let us consider a system of two nonlinearly-coupled oscillators given by the generalized Hénon-Heiles Hamiltonian,

$$ H=\frac{1}{2} \bigl(p^2_1+p^2_2 \bigr)+\frac{1}{2} \bigl(\omega^2_1 q^2_1 + \omega^2_2 q^2_2 \bigr)+ \varepsilon \biggl( q^2_1 q_2 -\frac{1}{3}q^3_2 \biggr) $$

(4.166)

where ε is a small parameter characterizing the weakness of the couplings between the two oscillators.

Hamilton’s equations for the two oscillators then follow from (4.166),

$$\begin{aligned} q_{1{tt}}+\omega^2_1 q_1&=-\varepsilon\cdot 2 q_1q_2 \end{aligned}$$

(4.167)

$$\begin{aligned} q_{2_{tt}}+\omega^2_2 q_2&=\varepsilon \bigl(q_2^2 - q_1^2 \bigr). \end{aligned}$$

(4.168)

Let us look for solutions of the form,

$$ \begin{aligned} q_1(t;\varepsilon)&= A_1(t_1,t_2,\ldots )\cos \phi_1(t, t_1, t_2,\ldots )+\varepsilon u_1(t,t_1,t_2,\ldots ) \\ &\quad {}+\varepsilon^2 u_2(t,t_1,t_2, \ldots ) \\ q_2(t;\varepsilon)&= A_2(t_1,t_2, \ldots )\cos \phi_2(t, t_1, t_2,\ldots )+ \varepsilon \nu_1(t,t_1,t_2,\ldots ) \\ &\quad {}+\varepsilon^2 \nu_2(t,t_1,t_2, \ldots ) \end{aligned} $$

(4.169)

where t _n =ε ⁿ t are the slow-time scales that characterize the slow variations introduced by the weak couplings among the oscillators, and

$$ \phi_s\equiv\omega_st-\theta_s(t_1,t_2);\quad s=1,2. $$

Note that (4.169) expresses the fact that the solutions of equations (4.167) and (4.168), for ε≪1, are very nearly equal to the set of harmonics represented by the first terms in the expansions of (4.169), which they would be, if ε=0. The perturbations induced by the terms of O(ε) in equations (4.167) and (4.168) may then be expected to show up as slow variations in the A’s and θ’s and as higher harmonics in the solution through the u _k ’s and ν _k ’s.

In order to fully specify the solution, one needs to specify some initial conditions. However, we will not do this keeping in mind that the periodic solutions derived in the following correspond to very special initial conditions.

Using (4.169), equations (4.167) and (4.168) give

$$\begin{aligned} &\bigl(2\varepsilon\omega_1A_1 \theta_{1_{t_1}}+2\varepsilon^2 \omega_1 A_1\theta_{1_{t_2}}+2\varepsilon^3 \omega_1A_1\theta_{1_{t_3}}-\varepsilon^2A_1 \theta^2_{1_{t_1}} \\ &\qquad{} -2\varepsilon^3A_1\theta_{1_{t_1}} \theta_{1_{t_2}}+\varepsilon^2A_{1_{{t_1}{t_1}}}+2 \varepsilon^3A_{1_{{t_1}{t_2}}} \bigr)\cos \phi_1 \\ &\qquad{}+ \bigl(-2\varepsilon\omega_1A_{1_{t_1}}-2 \varepsilon^2 \omega_1 A_{1_{t_2}}-2 \varepsilon^3 \omega_1 A_{1_{t_3}}+2 \varepsilon^2A_{1_{t_1}}\theta_{1_{t_1}} \\ &\qquad{}+ 2\varepsilon^3A_{1_{t_2}}\theta_{1_{t_1}}+2 \varepsilon^3A_{1_{t_1}}\theta_{1_{t_2}}+ \varepsilon^2A_1\theta_{1_{{t_1}{t_1}}}+2 \varepsilon^3A_1\theta_{1_{{t_1}{t_2}}} \bigr)\sin \phi_1 \\ &\qquad{}+\varepsilon \bigl(u_{1_{tt}}+\omega^2_1u_1 \bigr)+\varepsilon^2 \bigl(2u_{1_{tt_1}}+u_{2_{tt}}+ \omega^2_1u_2 \bigr) \\ &\qquad{}+\varepsilon^3 \bigl(2u_{1_{tt_2}}+2u_{2_{tt_1}}+u_{1_{{t_1}{t_1}}}+u_{3_{tt}}+ \omega^2_1u_3 \bigr)+\cdots \\ &\quad{}=-\varepsilon (2A_1A_2\cos \phi_1\cos \phi_2 )-\varepsilon^2 (2A_1 \cos \phi_1\cdot \nu_1+2A_2\cos\phi_2 \cdot u_1 ) \\ &\qquad{}-\varepsilon^3 (2A_1 \cos\phi_1\cdot \nu_2+2A_2\cos\phi_2\cdot u_2+2u_1 \nu_1 )+\cdots \end{aligned}$$

(4.170)

$$\begin{aligned} &\bigl(2\varepsilon\omega_2A_2 \theta_{2_{t_1}}+2\varepsilon^2\omega_2A_2 \theta_{2_{t_2}}+2\varepsilon^3\omega_2A_2 \theta_{2_{t_3}}-\varepsilon^2A_2 \theta^2_{2{t_1}} \\ &\qquad{}-2\varepsilon^3A_2\theta_{2_{t_1}} \theta_{2_{t_2}}+\varepsilon^2 A_{2_{{t_1}{t_1}}}+2 \varepsilon^3A_{2_{{t_1}{t_2}}} \bigr)\cos\phi_2 \\ &\qquad{}+ \bigl(-2\varepsilon\omega_2A_{2_{t_1}}-2 \varepsilon^2\omega_2A_{2_{t_2}}-2 \varepsilon^3\omega_2A_{2_{t_3}}+2 \varepsilon^2A_{2_{t_1}}\theta_{2{t_1}} \\ &\qquad{}+2\varepsilon^3A_{2_{t_2}}\theta_{2_{t_1}}+2 \varepsilon^3 A_{2_{t_1}}\theta_{2_{t_2}}+ \varepsilon^2A_2\theta_{2_{{t_1}{t_1}}}+2 \varepsilon^3A_2 \theta_{2_{t_2t_2}} \bigr)\sin \phi_2 \\ &\qquad{}+\varepsilon \bigl(\nu_{1_{tt}}+\omega^2_2 \nu_1 \bigr)+\varepsilon^2 \bigl(2\nu_{1_{tt_1}}+ \nu_{2_{tt}}+\omega^2_2\nu_2 \bigr) \\ &\qquad{}+\varepsilon^3 \bigl(2\nu_{1_{tt_2}}+2\nu_{2_{tt_1}}+ \nu_{1_{t_1t_1}}+\nu_{3_{tt}}+\omega^2_2 \nu_3 \bigr)+\cdots \\ &\quad{}=\varepsilon \bigl(A^2_2\cos^2 \phi_2-A^2_1\cos^2 \phi_1 \bigr)+\varepsilon^2 (2A_2 \cos \phi_2\cdot\nu_1-2A_1\cos\phi_1 \cdot u_1 )\qquad\quad \\ &\qquad{}+\varepsilon^3 \bigl(\nu^2_1 +2A_2\cos \phi_2\cdot \nu_2-u^2_1-2A_1 \cos\phi_1\cdot u_2 \bigr)+\cdots . \end{aligned}$$

(4.171)

By equating the coefficients of sinϕ ₁, cosϕ ₁, sinϕ ₂, cosϕ ₂ and the rest to zero separately, one obtains from equations (4.170) and (4.171) to O(ε):

$$\begin{aligned} &\theta_{1_{t_1}}=0 \end{aligned}$$

(4.172)

$$\begin{aligned} &A_{1_{t_1}}=0 \end{aligned}$$

(4.173)

$$\begin{aligned} &u_{1_{tt}}+\omega^2_1 u_1 =-2A_1A_2\cos\phi_1\cdot\cos \phi_2 \end{aligned}$$

(4.174)

$$\begin{aligned} &\theta_{2_{t_1}}=0 \end{aligned}$$

(4.175)

$$\begin{aligned} &A_{2_{t_1}}=0 \end{aligned}$$

(4.176)

$$\begin{aligned} &\nu_{1_{tt}}+\omega^2_2 \nu_1 =A^2_2 \cos^2 \phi_2-A^2_1\cos^2 \phi_1 \end{aligned}$$

(4.177)

On solving equation (4.172)–(4.177), one obtains

$$ \begin{aligned} u_1&=\frac{A_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1}\cos ( \phi_1+\phi_2 )+\frac{A_1A_2}{ (\omega_1 -\omega_2 )^2-\omega^2_1}\cos ( \phi_1-\phi_2 ) \\ \nu_1&=\frac{A^2_2-A^2_1}{2\omega^2_2}-\frac{A^2_2}{6 \omega^2_2}\cos 2 \phi_2+\frac{A^2_1 /2}{ (2 \omega_1 )^2 -\omega^2_2}\cos 2 \phi_1 \end{aligned} $$

(4.178)

where the A’s and θ’s are constants to O(ε). Thus, in general, the oscillators move as if they were uncoupled effectively, and there is no appreciable energy sharing among them. Note, however, that this solution breaks down when ω ₂=2ω ₁, which leads to small divisors in (4.178). This condition corresponds to a first-order internal resonance in the system which, as we will see below, leads to considerable energy sharing in the system.

Let us now try to determine whether there are any higher-order resonances in this system. Towards this end, let us first assume that the lowest-order resonance is inoperative, i.e., ω ₂≠2ω ₁. Then, to O(ε ²), equations (4.170) and (4.171) give

$$\begin{aligned} &2\omega_1A_1\theta_{1_{t_2}}\cos \phi_1-2\omega_1A_{1_{t_2}}\sin\phi_1+2u_{1_{tt_1}}+u_{2_{tt}}+ \omega^2_1u_2 \\ &\quad = -2A_1 \cos \phi_1 \cdot \nu_1 -2A_2 \cos \phi_2\cdot u_1 \end{aligned}$$

(4.179)

$$\begin{aligned} &2\omega_2A_2\theta_{2_{t_2}}\cos \phi_2-2\omega_2A_{2_{t_2}}\sin\phi_2+2 \nu_{1_{tt_1}}+\nu_{2_{tt}}+\omega^2_2 \nu_2 \\ &\quad = 2A_2 \cos \phi_2 \cdot \nu_1 -2A_1 \cos \phi_1\cdot u_1 \end{aligned}$$

(4.180)

Using (4.172), (4.173), (4.175), (4.176) and (4.178), equations (4.179) and (4.180) become

$$\begin{aligned} &2\omega_1A_1\theta_{1_{t_2}}\cos \phi_1-2\omega_1A_{1_{t_2}}\sin\phi_1+u_{2_{tt}}+ \omega^2_1u_2 \\ &\quad =- \biggl[A_1\frac{A^2_2-A^2_1}{\omega^2_2}+ \frac{A^3_1/2}{ (2\omega_1 )^2-\omega^2_2} \\ &\qquad{} +\frac{A_1A^2_2}{ (\omega_1+ \omega_2 )^2-\omega^2_1}+\frac{A_1A^2_2}{ (\omega_1- \omega_2 )^2-\omega^2_1} \biggr]\cos \phi_1 \\ &\qquad{} - \biggl[\frac{A_1A^2_2}{6\omega^2_2}-\frac{A_1A^2_2}{ (\omega_1+ \omega_2 )^2-\omega^2_1} \biggr]\cos ( \phi_1 + 2\phi_2 ) \\ &\qquad{}- \biggl[\frac{A_1A^2_2}{6\omega^2_2}-\frac{A_1A^2_2}{ (\omega_1- \omega_2 )^2-\omega^2_2} \biggr]\cos ( \phi_1 -2\phi_2 )-\frac{A^3_1/2}{ (2\omega_1 )^2-\omega^2_2}\cos3 \phi_1\qquad \end{aligned}$$

(4.181)

$$\begin{aligned} &2 \omega_2 A_2 \theta_{2_{t_2}} \cos \phi_2 -2 \omega_2 A_{2_{t_2}} \sin \phi_2 + \nu_{2_{tt}} + \omega^2_2 \nu_2 \\ &\quad = \biggl[ A_2 \frac{A^2_2 -A^2_1}{2\omega^2_2}- \frac{A^3_2}{6\omega^2_2}-\frac{A^2_1 A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_1 A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr] \cos \phi_2 - \frac{A^3_2}{6\omega^2_2}\cos 3\phi_2 \\ &\qquad{}- \biggl[\frac{A^2_1 A_2 /2}{ (2\omega_1 )^2-\omega^2_2}- \frac{A^2_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr]\cos (2 \phi_1 + \phi_2 ) \\ &\qquad{}- \biggl[\frac{A^2_1 A_2 /2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A_2}{ (\omega_1-\omega_2 )^2- \omega^2_1} \biggr]\cos (2 \phi_1 - \phi_2 ) \end{aligned}$$

(4.182)

from which, one obtains

$$\begin{aligned} &\theta_{1_{t_2}}= - \frac{1}{2\omega_1} \biggl[ \frac{A^2_2 -A^2_1}{2\omega^2_2}\,{+}\,\frac{A^2_1/2}{ (2\omega_1 )^2- \omega^2_2}\,{+}\,\frac{A^2_2 }{ (\omega_1+\omega_2 )^2-\omega^2_1}\,{+}\, \frac{A^2_2 }{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr] \end{aligned}$$

(4.183)

$$\begin{aligned} &A_{1_{t_2}}=0 \end{aligned}$$

(4.184)

$$\begin{aligned} &u_2 = \biggl[\frac{A_1A^2_2}{6\omega^2_2}- \frac{A_1A^2_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (\phi_1+ 2\phi_2 )}{ (\omega_1+2\omega_2 )^2-\omega^2_1} \\ &\qquad {}+ \biggl[\frac{A_1A^2_2}{6\omega^2_2}-\frac{A_1A^2_2}{ (\omega_1-\omega_2 )^2- \omega^2_1} \biggr] \frac{\cos (\phi_1-2\phi_2 )}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \\ &\qquad {}+\frac{A^3_1 /2}{ (2\omega_1 )^2-\omega^2_2}\frac{\cos 3\phi_1}{8\omega^2_1} \end{aligned}$$

(4.185)

$$\begin{aligned} &\theta_{2_{t_2}}= \frac{1}{2\omega_2} \biggl[ \frac{A^2-A^2_1}{\omega^2_2}- \frac{A^2_2}{6\omega^2_2}-\frac{A^2_1 }{ (\omega_1+\omega_2 )^2-\omega^2_1}-\frac{A^2_1 }{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr] \end{aligned}$$

(4.186)

$$\begin{aligned} &A_{2_{t_2}}=0 \end{aligned}$$

(4.187)

$$\begin{aligned} &\nu_2=\frac{A^3_2}{6\omega^2_2}\frac{\cos3\phi_2}{8\omega^2_2}+ \biggl[\frac{A^2_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A_2}{ (\omega_1+ \omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (2\phi_1+\phi_2 )}{ (2\omega_1+\omega_2 )^2- \omega^2_2} \\ &\qquad {}+ \biggl[\frac{A^2_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A_2}{ (\omega_1- \omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (2\phi_1-\phi_2 )}{ (2\omega_1-\omega_2 )^2- \omega^2_2} \end{aligned}$$

(4.188)

Inspection of (4.185) and (4.188) shows that in the O(ε)² problem, one has a second-order internal resonance at ω ₂=ω ₁. This corresponds to the case considered by Hénon and Heiles (1964).

Next, let us determine the internal resonances of the O(ε)³ problem. First, let us assume that there are no internal resonances in the O(ε) and O(ε)² problems, i.e., ω ₂≠2ω ₁ and ω ₂≠ω ₁. Then, to O(ε ³), equations (4.170) and (4.171) give

$$\begin{aligned} &2\omega_1 A_1 \theta_{1_{t_3}} \cos \phi_1-2\omega_1 A_{1_{t_3}}\sin \phi_1+2u_{1_{tt_2}}+u_{1_{t_1t_1}}+u_{3_{tt}}+ \omega^2_1u_3 \\ &\quad =-2A_1\cos\phi_1\cdot\nu_2-2A_2 \cos\phi_2\cdot u_2-2u_1\nu_1 \end{aligned}$$

(4.189)

$$\begin{aligned} &2\omega_2 A_2 \theta_{2_{t_3}} \cos \phi_2-2\omega_2 A_{2_{t_3}}\sin \phi_2+2\nu_{1_{tt_2}}+\nu_{1_{t_1t_1}}+\nu_{3_{tt}}+ \omega^2_2\nu_3 \\ &\quad =\nu^2_1+2A_2\cos\phi_2\cdot \nu_2 -u^2_1 -2 A_1\cos \phi_1\cdot u_1 \end{aligned}$$

(4.190)

Using (4.172), (4.173), (4.175), (4.176), (4.178), (4.183)–(4.188), and proceeding as before, one obtains from equations (4.189) and (4.190),

$$\begin{aligned} &\theta_{1_{t_3}}=0 \end{aligned}$$

(4.191)

$$\begin{aligned} &A_{1_{t_3}}=0 \end{aligned}$$

(4.192)

$$\begin{aligned} &u_3 = \biggl[\frac{2A_1A_2 (\omega_1+\omega_2 )}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggl\{ \frac{1}{2\omega_1} \biggl[\frac{A^2_2-A^2_1}{\omega^2_2}+ \frac{A^2_1/2}{ (2\omega_1 )^2-\omega^2_2}+\frac{A^2_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}+\frac{A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]-\frac{1}{2\omega_2} \biggl[ \frac{A^2_2-A^2_1}{\omega^2_2}-\frac{A^2_2}{6\omega^2_2}-\frac{A^2_1}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_1}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr] \biggr\} - \biggl\{ \frac{A^3_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^3_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr\} \\ &\qquad{}\times\frac{1}{ (2\omega_1+\omega_2 )^2-\omega^2_2} \\ &\qquad{}- \biggl\{ \frac{A_1A^3_2}{6\omega^2_2}-\frac{A_1A^3_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1+2\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_2-A^2_1}{\omega^2_2}\frac{A_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1}+\frac{A^2_2}{6\omega^2_2}\frac{A_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_1 /2}{ (2\omega_1 )^2-\omega^2_2}\frac{A_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (\phi_1+\phi_2 )}{\omega^2_1- (\omega_1+\omega_2 )^2} \\ &\qquad{}+ \biggl[\frac{2A_1A_2 (\omega_1-\omega_2 )}{ (\omega_1-\omega_2 )^2-\omega^2_1}\biggl\{ \frac{1}{2\omega_1} \biggl[\frac{A^2_2-A^2_1}{\omega^2_2}+\frac{A^2_1/2}{ (2\omega_1 )^2-\omega^2_2}+\frac{A^2_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}+\frac{A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]+\frac{1}{2\omega_2}\biggl[ \frac{A^2_2-A^2_1}{\omega^2_2}-\frac{A^2_2}{6\omega^2_2}-\frac{A^2_1}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_1}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr] \biggr\} - \biggl\{ \frac{A^3_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^3_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \\ &\qquad{}\times\frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\qquad{}- \biggl\{ \frac{A_1A^3_2}{6\omega^2_2}-\frac{A_1A^3_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \\ &\qquad{}+\frac{A^2_2}{6\omega^2_2}\frac{A_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1}-\frac{A^2_1 /2}{ (2\omega_1 )^2-\omega^2_2}\frac{A_1A_2}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_2-A^2_1}{\omega^2_2}\frac{A_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (\phi_1-\phi_2 )}{\omega^2_1- (\omega_1-\omega_2 )^2} \\ &\qquad{}+ \biggl[-\frac{A_1A^3_2}{48\omega^4_2}- \biggl\{ \frac{A_1A^3_2}{6\omega^2_2}-\frac{A_1A^3_2}{ (\omega_1 +\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1 +2\omega_2 )^2-\omega^2_1} \\ &\qquad{}+\frac{A^2_2}{6\omega^2_2}\frac{A_1A_2}{ (\omega_1 +\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (\phi_1+3\phi_2 )}{\omega^2_1- (\omega_1+3\omega_2 )^2} \\ &\qquad{}+ \biggl[-\frac{A_1A^3_2}{48\omega^4_2}- \biggl\{ \frac{A_1A^3_2}{6\omega^2_2}-\frac{A_1A^3_2}{ (\omega_1 -\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1 -2\omega_2 )^2-\omega^2_1} \\ &\qquad{}+\frac{A^2_2}{6\omega^2_2}\frac{A_1A_2}{ (\omega_1 -\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (\phi_1-3\phi_2 )}{\omega^2_1- (\omega_1-3\omega_2 )^2} \\ &\qquad{}+ \biggl[- \biggl\{ \frac{A^3_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^3_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1+\omega_2 )^2-\omega^2_2} \\ &\qquad{}-\frac{1}{16\omega^2_1} \frac{A_2 A^3_1}{ (2\omega_1 )^2-\omega^2_2} -\frac{A^2_1/2}{ (2\omega_1 )^2-\omega^2_2} \frac{A_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr] \frac{\cos (3\phi_1+\phi_2 )}{\omega^2_1- (3\omega_1+\omega_2 )^2} \\ &\qquad{}+ \biggl[- \biggl\{ \frac{A^3_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^3_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\qquad{}-\frac{1}{16\omega^2_1}\frac{A_2A^3_1}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1 /2}{ (2\omega_1 )^2-\omega^2_2} \frac{A_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos (3\phi_1-\phi_2 )}{\omega^2_1- (3\omega_1-\omega_2 )^2} \\ \end{aligned}$$

(4.193)

$$\begin{aligned} &\theta_{2_{t_3}}=0 \end{aligned}$$

(4.194)

$$\begin{aligned} &A_{2_{t_3}}=0 \end{aligned}$$

(4.195)

$$\begin{aligned} &\nu_3=- \biggl[\frac{4A^2_2}{6\omega^2_2} \biggl\{ \frac {A^2_2-A^2_1}{\omega^2_2}-\frac{A^2_2}{6\omega^2_2}-\frac{A^2_1}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_1}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} +\frac{A^4_2}{48\omega^4_2}-\frac{A^2_2 (A^2_2 -A^2_1 )}{6\omega^4_2} \\ &\qquad{}- \biggl\{ \frac{A^2_1A^2_2}{6\omega^2_2}-\frac{A^2_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1+2\omega_2 )^2-\omega^2_1} \\ &\qquad{}- \biggl\{ \frac{A^2_1A^2_2}{6\omega^2_2}-\frac{A^2_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \\ &\qquad{}-\frac{A^2_1A^2_2}{ \{ (\omega_1+\omega_2 )^2-\omega^2_1 \} \{ (\omega_1-\omega_2 )^2-\omega^2_1 \}} \biggr]\frac{\cos2\phi_2}{3\omega^2_2} \\ &\qquad{}+ \biggl[\frac{2A^2_1}{ (2\omega_1 )^2-\omega^2_2} \biggl\{ \frac{A^2_2-A^2_1}{\omega^2_2}+ \frac{A^2_1/2}{ (2\omega_1 )^2-\omega^2_2}+\frac{A^2_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \\ &\qquad{}+\frac{A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} + \biggl\{ \frac{A^2_1 A^2_2 /2}{ (2\omega_1 )^2-\omega^2_2}- \frac{A^2_1 A^2_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr\} \\ &\qquad{}\times \frac{1}{ (2\omega_1+\omega_2 )^2-\omega^2_2}+ \biggl\{ \frac{A^2_1 A^2_2 /2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1 A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \\ &\qquad{}\times\frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2}-\frac{1}{2\omega^2_2}\frac{A^2_1 (A^2_2-A^2_1 )}{ (2\omega_1 )^2-\omega^2_2}+ \frac{1}{16\omega^2_1}\frac{A^4_1}{ (2\omega_1 )^2-\omega^2_2} \\ &\qquad{}-\frac{A^2_1A^2_2}{ \{ (\omega_1+\omega_2 )^2-\omega^2_1 \} \{ (\omega_1-\omega_2 )^2-\omega^2_1 \}} \biggr]\frac{\cos2\phi_1}{\omega^2_2- (2\omega_1 )^2} \\ &\qquad{}- \biggl[\frac{A^4_2}{48\omega^4_2}+\frac{A^4_2}{72\omega^4_2} \biggr]\frac{\cos4\phi_2}{15\omega^2_2} \\ &\qquad{}+ \biggl[ \biggl\{ \frac{A^2_1A^2_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1 A^2_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1+\omega_2 )^2-\omega^2_2} \\ &\qquad{}-\frac{1}{12\omega^2_2}\frac{A^2_1A^2_2}{ (2\omega_1 )^2\,{-}\,\omega^2_2}\,{-}\,\frac{A^2_1A^2_2/2}{ \{ (\omega_1\,{+}\,\omega_2 )^2\,{-}\,\omega^2_1 \}^2}\,{-}\,\biggl\{ \frac{A^2_1A^2_2}{6\omega^2_2}\,{-}\,\frac{A^2_1A^2_2}{ (\omega_1\,{+}\,\omega_2 )^2\,{-}\,\omega^2_1} \biggr\} \\ &\qquad{}\times\frac{1}{ (\omega_1 + 2\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos 2 (\phi_1+\phi_2 )}{\omega^2_2-4 (\omega_1+\omega_2 )^2} \\ &\qquad{}+ \biggl[ \biggl\{ \frac{A^2_1A^2_2/2}{ (2\omega_1 )^2-\omega_2^2}-\frac{A^2_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\qquad{}-\frac{1}{12\omega^2_2}\frac{A^2_1A^2_2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A^2_2/2}{ \{ (\omega_1-\omega_2 )^2-\omega^2_1 \}^2} \\ &\qquad{}- \biggl\{ \frac{A^2_1A^2_2}{6\omega^2_2}-\frac{A^2_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \biggr]\frac{\cos2 (\phi_1-\phi_2 )}{\omega^2_2-4 (\omega_1-\omega_2 )^2} \\ &\qquad{}+\frac{1}{\omega^2_2} \biggl[\frac{A^2_2-A^2_1}{4\omega^4_2}+\frac{A^4_2}{72\omega^4_2}+ \frac{A^4_1/8}{ \{ (2\omega_1 )^2-\omega^2_2 \}^2} \\ &\qquad{}-\frac{A^2_1 A^2_2 /2}{ \{ (\omega_1+\omega_2 )^2-\omega^2_1 \}^2}-\frac{A^2_1 A^2_2 /2}{ \{ (\omega_1-\omega_2 )^2-\omega^2_1 \}^2} \biggr] \\ &\qquad{}+ \biggl[\frac{A^4_1 /8}{ \{ (2\omega_1 )^2-\omega^2_2 \}}-\frac{1}{16\omega^2_1}\frac{A^4_1}{ (2\omega_1 )^2-\omega^2_2} \biggr] \frac{\cos4\phi_1}{\omega^2_2- (4\omega_1 )^2}. \end{aligned}$$

(4.196)

Inspection of (4.193) and (4.196) shows that in the O(ε ³) problem, one has third-order internal resonances at $\omega_{2}=\frac{2}{3}\omega_{1}$ and ω ₂=4ω ₁.

These higher-order resonances, as is confirmed in the following, become weaker successively, so they do not lead to as much exchange of energy among the oscillators as the first-order resonance does. Let us first consider the latter case.

(i) First-Order Internal Resonance (ω ₂=2ω ₁)

Let us write the O(ε) terms on the right in equations (4.170) and (4.171) as follows:

$$\begin{aligned} &-2A_1A_2\cos\phi_1\cdot\cos \phi_2=-2A_1A_2\cos\phi_1\cdot \cos \bigl[ (\phi_2-2\phi_1 )+2\phi_1 \bigr] \\ &\hphantom{-2A_1A_2\cos\phi_1\cdot\cos \phi_2\ \,}=-A_1A_2\cos (\phi_2-2\phi_1 ) \cdot\cos\phi_1 \\ &\hphantom{-2A_1A_2\cos\phi_1\cdot\cos \phi_2=\ \; }{}-A_1A_2\cos (\phi_2-2\phi_1 ) \cdot\cos3\phi_1 \\ &\hphantom{-2A_1A_2\cos\phi_1\cdot\cos \phi_2=\ \; }{}+A_1A_2\sin (\phi_2-2\phi_1 ) \cdot\sin\phi_1 \\ &\hphantom{-2A_1A_2\cos\phi_1\cdot\cos \phi_2=\ \; }{}+A_1A_2\sin (\phi_2-2\phi_1 ) \cdot\sin 3\phi_1 \\ &-A^2_1\cos^2\phi_1=- \frac{A^2_1}{2}-\frac{A^2_1}{2}\cos (\phi_2-2 \phi_1 )\cdot\cos\phi_2 \\ &\hphantom{-A^2_1\cos^2\phi_1=\ \,}{}-\frac{A^2_1}{2}\sin (\phi_2-2\phi_1 )\cdot\sin \phi_2. \end{aligned}$$

Using these expressions, equations (4.170) and (4.171) give

$$\begin{aligned} &2\omega_1A_1\theta_{1_{t_1}}=-A_1A_2 \cos (\phi_2-2\phi_1 ) \end{aligned}$$

(4.197)

$$\begin{aligned} &-2\omega_1 A_{1_{t_1}}=A_1A_2 \sin (\phi_2-2\phi_1 ) \end{aligned}$$

(4.198)

$$\begin{aligned} &u_{1_{tt}}+\omega^2_1u_1=-A_1A_2 \cos (\phi_2-2\phi_1 )\cdot\cos 3\phi_1 \\ &\hphantom{u_{1_{tt}}+\omega^2_1u_1=}{}+A_1A_2\sin (\phi_2-2\phi_1 ) \cdot\sin 3\phi_1 \end{aligned}$$

(4.199)

$$\begin{aligned} &2\omega_2A_2\theta_{2_{t_1}}=- \frac{A^2_1}{2}\cos (\phi_2-2\phi_1 ) \end{aligned}$$

(4.200)

$$\begin{aligned} &-2\omega_2 A_{2_{t_1}}=-\frac{A^2_1}{2} \sin (\phi_2-2\phi_1 ) \end{aligned}$$

(4.201)

$$\begin{aligned} &\nu_{1_{tt}}+\omega^2_2 \nu_1=\frac{1}{2} \bigl(A^2_2-A^2_1 \bigr)+\frac{1}{2}A^2_2\cos 2\phi_2. \end{aligned}$$

(4.202)

Solving equations (4.199) and (4.202), one obtains

$$ \begin{aligned} u_1&=\frac{A_1A_2}{8\omega^2_1} \bigl[\cos (\phi_2-2\phi_1 )\cdot\cos 3\phi_1-\sin ( \phi_2-2\phi_1 )\cdot\sin3\phi_1 \bigr] \\ \nu_1&=\frac{A^2_2- A^2_1}{2\omega^2_2}-\frac{A^2_2}{6\omega^2_2}\cos2 \phi_2. \end{aligned} $$

(4.203)

Observe that (4.203) no longer exhibits any small divisors.

Also, one obtains from equations (4.198) and (4.201),

$$ \omega_1 A_1 A_{1_{t_1}}+2 \omega_2 A_2 A_{2_{t_1}}=0 $$

(4.204)

or

$$ \frac{1}{2}\omega_1A^2_1+ \omega_2A^2_2=\mbox{const} $$

(4.205)

which dictates the manner in which action is exchanged among the oscillators under an internal resonance. Note that (4.205) is only an approximate constant of motion, because it has been derived under conditions of weak nonlinearities in the coupled system, and this constant of motion will cease to exist when the nonlinearities become strong. If one introduces the action J _k by

$$ J_k\equiv A^2_k \omega_k $$

(4.206)

then, (4.205) becomes

$$ J_1\omega_1+J_2 \omega_2=\mbox{const}=E. $$

(4.207)

This result was first given by Kabakow (1968) and van der Burgh (1976), and is similar to the “adelphic” integral discussed by Whittaker (1944). That the action exchange at the first-order resonance (ω ₂=2ω ₁) follows according to E ₁+E ₂=const=E, where E _k ≡J _k ω _k , was demonstrated by the computer calculations of Ford and Waters (1963), as shown in Fig. 4.6.

Fig. 4.6

Variation of the individual actions with time during the first-order internal resonance (due to Ford and Waters (1963)) (By courtesy of the American Institute of Physics)

(ii) Second-Order Internal Resonance (ω ₂=ω ₁)

Let us write

$$\begin{aligned} &\cos (\phi_1-2\phi_2 )=\cos 2(\phi_1- \phi_2)\cos \phi_1+\sin 2(\phi_1- \phi_2)\sin \phi_1 \\ &\cos (2\phi_1-\phi_2 )=\cos 2(\phi_1- \phi_2)\cos \phi_2-\sin 2(\phi_1- \phi_2)\sin \phi_2 \end{aligned}$$

on the right hand sides in equations (4.181) and (4.182); one then obtains,

$$\begin{aligned} &2\omega_1A_1\theta_{1_{t_2}}=- \biggl[A_1 \frac{A^2_2-A^2_1}{\omega^2_2}+ \frac{A^3_1/2}{ (2\omega_1 )^2-\omega^2_2}+\frac{A_1A^2_2}{ (\omega_1+ \omega_2 )^2-\omega^2_1} \\ &\hphantom{2\omega_1A_1\theta_{1_{t_2}}=}{} +\frac{A_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]- \biggl[\frac{A_1A^2_2}{6\omega_2^2}- \frac{A_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr]\cos 2(\phi_1-\phi_2) \end{aligned}$$

(4.208)

$$\begin{aligned} &-2\omega_1A_{1_{t_2}}=- \biggl[ \frac{A_1A^2_2}{6\omega^2_2}-\frac{A_1A^2_2}{ (\omega_1-\omega_2 )^2- \omega^2_1} \biggr]\sin 2(\phi_1- \phi_2) \end{aligned}$$

(4.209)

$$\begin{aligned} &u_{2_{tt}}\,{+}\,\omega^2_1u_2=- \frac{A^3_1/2}{ (2\omega_1 )^2\,{-}\,\omega^2_2}\cos 3\phi_1\,{-}\,\biggl[\frac{A_1A^2_2}{6\omega^2_2}\,{-}\,\frac{A_1A^2_2}{ (\omega_1\,{+}\,\omega_2 )^2\,{-}\,\omega^2_1} \biggr] \cos (\phi_1\,{+}\,2\phi_2 ) \end{aligned}$$

(4.210)

$$\begin{aligned} &2\omega_2A_2\theta_{2_{t_2}}=- \biggl[\frac{A^2_1A_2}{\omega^2_2}-\frac{5}{6}\frac{A^3_2}{\omega^2_2}+ \frac{A^2_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1}+\frac{A^2_1A_2}{ (\omega_1- \omega_2 )^2-\omega^2_1} \biggr] \\ &\hphantom{2\omega_2A_2\theta_{2_{t_2}}=}{}- \biggl[\frac{A^2_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A_2}{ (\omega_1- \omega_2 )^2-\omega^2_1} \biggr]\cos 2( \phi_1-\phi_2) \end{aligned}$$

(4.211)

$$\begin{aligned} &-2\omega_2A_{2_{t_2}}= \biggl[ \frac{A^2_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A_2}{ (\omega_1-\omega_2 )^2- \omega^2_1} \biggr]\sin 2 (\phi_1- \phi_2) \end{aligned}$$

(4.212)

$$\begin{aligned} &\nu_{2_{tt}}\,{+}\,\omega^2_2 \nu_2=-\frac{A^3_2}{6\omega^2_2}\cos 3\phi_2\,{-}\,\biggl[ \frac{A^2_1A_2/2}{ (2\omega_1 )^2\,{-}\,\omega^2_2}\,{-}\,\frac{A^2_1A_2}{ (\omega_1\,{+}\,\omega_2 )^2\,{-}\,\omega^2_1} \biggr] \cos (2\phi_1\,{+}\, \phi_2 ). \end{aligned}$$

(4.213)

Solving equations (4.210) and (4.211), one obtains

$$\begin{aligned} &u_2=\frac{A^3_1/16\omega^2_1}{ (2\omega_1 )^2- \omega^2_2}\cos 3\phi_1- \biggl[\frac{A_1A^2_2}{6\omega^2_2}-\frac{A_1A^2_2}{ (\omega_1+\omega_2 )^2- \omega^2_1} \biggr]\frac{\cos (\phi_1+2\phi_2 )}{\omega^2_1- (\omega_1+ 2\omega_2 )^2}\qquad \end{aligned}$$

(4.214)

$$\begin{aligned} &\nu_2=\frac{A^3_2}{48\omega^4_2}\cos 3\phi_2- \biggl[ \frac{A^2_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}- \frac{A^2_1A_2}{ (\omega_1+\omega_2 )^2-\omega^2_1} \biggr] \frac{\cos (2\phi_1+\phi_2 )}{\omega^2_2- (2\omega_1+\omega_2 )^2}. \end{aligned}$$

(4.215)

Next, one obtains from equations (4.209) and (4.212),

$$\begin{aligned} &2\omega_1A_1A_{1_{t_2}}+2 \omega_2A_2A_{2_{t_2}}=0 \end{aligned}$$

(4.216)

$$\begin{aligned} &\omega_1A^2_1+ \omega_2 A^2_2=\mbox{const} \end{aligned}$$

(4.217)

or

$$ J_1\omega_2+J_2 \omega_2=\mbox{const} $$

(4.218)

which is the same as the one, namely (4.207), deduced for the first-order resonance. This integral was also found by Verhulst (1979) and Kevorkian (1980) and was confirmed by the numerical calculations of Ford and Waters (1963)—see Fig. 4.7.

Fig. 4.7

Variation of the individual actions with time during the second-order internal resonance (due to Ford and Waters (1963)) (By courtesy of the American Institute of Physics)

Comparison of (4.209) and (4.212) with (4.198) and (4.201) shows that the action exchange at the first-order resonance is much greater than that at the second-order resonance. The numerical calculations of Ford and Waters (1963), also showed that (see Fig. 4.7) the action exchange at the second-order resonance was weaker than that at the first-order resonance.

(iii) Third-Order Resonance ($\omega_{2}=\frac{2}{3}\omega_{1}$ and ω ₂=4ω ₁)

For the resonance $w_{2} = \frac{2}{3} \omega_{1}$, let us write

$$\begin{aligned} &\cos (\phi_1-3\phi_2 )=\cos (3\phi_2-2 \phi_1 )\cos \phi_1-\sin (3\phi_2-2 \phi_1 )\sin \phi_1 \\ &\cos (2\phi_1-2\phi_2 )=\cos (3\phi_2-2 \phi_1 )\cos \phi_2-\sin (3\phi_2-2 \phi_1 )\sin \phi_2 \end{aligned}$$

on the right hand sides of equations (4.189) and (4.190); one obtains therefrom,

$$\begin{aligned} &2\omega_1A_1\theta_{1_{t_2}}= \biggl[-\frac{A_1A^3_2}{48\omega^4_2}- \biggl\{ \frac{A_1A^3_2}{6\omega^2_2}- \frac{A_1A^3_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \\ &\hphantom{2\omega_1A_1\theta_{1_{t_2}}=}{}+\frac{A^2_2}{6\omega^2_2}\frac{A_1A_2}{ (\omega_1-\omega_2 )^2- \omega^2_1} \biggr]\cos (3 \phi_2-2\phi_1 ) \end{aligned}$$

(4.219)

$$\begin{aligned} &-2\omega_1A_{1_{t_3}}=- \biggl[- \frac{A_1A^3_2}{48\omega^4_2}- \biggl\{ \frac{A_1A^3_2}{6\omega^2_2}- \frac{A_1A^3_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \\ &\hphantom{-2\omega_1A_{1_{t_3}}=\ \,}{}+\frac{A^2_2}{6\omega^2_2}\frac{A_1A_2}{ (\omega_1-\omega_2 )^2- \omega^2_1} \biggr]\sin (3 \phi_2-2\phi_1 ) \end{aligned}$$

(4.220)

$$\begin{aligned} &2\omega_2A_2\theta_{2_{t_3}}= \biggl[ \biggl\{ \frac{A^2_1A^2_2/2}{ (2\omega_1 )^2-\omega^2_2}- \frac{A^2_1A^2_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\hphantom{2\omega_2A_2\theta_{2_{t_3}}=}{}-\frac{1}{12\omega^2_2}\frac{A^2_1A^2_2}{ (2\omega_1 )^2-\omega^2_2}- \frac{A^2_1A^2_2/2}{ \{ (\omega_1-\omega_2 )^2-\omega^2_1 \}^2}+ \\ &\hphantom{2\omega_2A_2\theta_{2_{t_3}}=}{}- \biggl\{ \frac{A^2_1 A^2_2}{6\omega^2_2}-\frac{A^2_1 A^2_2}{ (\omega_1-\omega_2 )^2 -\omega^2_1} \biggr\} \frac{1}{ (\omega_1-2\omega_2 )^2-\omega^2_1} \biggr] \\ &\hphantom{2\omega_2A_2\theta_{2_{t_3}}=}{}\times\cos (3\phi_2-2\phi_1 ) \end{aligned}$$

(4.221)

$$\begin{aligned} &-2\omega_2 A_{2_{t_3}}= \biggl[ \biggl\{ \frac{A^2_1A^2_2/2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1A^2_2}{ (\omega_1- \omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\hphantom{-2\omega_2 A_{2_{t_3}}=\ \,}{}-\frac{1}{12\omega^2_2}\frac{A^2_1 A^2_2}{ (2\omega_1 )^2-\omega^2_2}-\frac{A^2_1 A^2_2}{ \{ (\omega_1-\omega_2 )^2-\omega^2_1 \}^2}+ \\ &\hphantom{-2\omega_2 A_{2_{t_3}}=\ \,}{}- \biggl\{ \frac{A^2_1 A^2_2}{6 \omega^2_2}\,{-}\,\frac{A^2_1 A^2_2}{ (\omega_1\,{-}\,\omega_2 )^2\,{-}\,\omega^2_1} \biggr\} \frac{1}{ (\omega_1\,{-}\,2\omega_2 )^2\,{-}\,2\omega^2_1} \biggr]\sin (3\phi_2\,{-}\,2\phi_1). \end{aligned}$$

(4.222)

One obtains from equations (4.220) and (4.222),

$$ \omega_1A_1A_{1_{t_3}}+ \frac{2}{3}\omega_2A_2A_{2_{t_3}}=0 $$

(4.223)

or

$$ \omega_1A^2_1+ \frac{2}{3}\omega_2 A^2_2=\mbox{const} $$

(4.224)

or

$$ J_1\omega_1+J_2 \omega_2=\mbox{const} $$

(4.225)

as before, in (4.207) and (4.218).

Next, for the resonance ω ₂=4ω ₁, let us write

$$\begin{aligned} &\cos (3\phi_1-\phi_2 )=\cos (\phi_2-4 \phi_1 )\cos \phi_1-\sin (\phi_2-4 \phi_1 )\sin \phi_1 \\ &\cos 4\phi_1=\cos (\phi_2-4\phi_1 )\cos \phi_2+\sin (\phi_2-4\phi_1 )\sin \phi_2 \end{aligned}$$

on the right hand sides of equations (4.189) and (4.190); one obtains therefrom,

$$\begin{aligned} &2\omega_1A_1\theta_{1_{t_3}}=- \biggl[ \biggl\{ \frac{A^3_1A_2/2}{ (2\omega_1 )^2-\omega^2_2}- \frac{A^3_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\hphantom{2\omega_1A_1\theta_{1_{t_3}}= \,}{}+\frac{1}{16\omega^2_1}\frac{A_2A^3_1}{ (2\omega_1 )^2- \omega^2_2}+\frac{A^2_1}{ (2\omega_1 )^2-\omega^2_2} \frac{A_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr] \\ &\hphantom{2\omega_1A_1\theta_{1_{t_3}}= \,}{}\times\cos (\phi_2-4\phi_1 ) \end{aligned}$$

(4.226)

$$\begin{aligned} &-2\omega_1 A_{1_{t_3}} = \biggl[ \biggl\{ \frac{A^3_1 A_2/2}{ (2 \omega_1 )^2-\omega^2_2}-\frac{A^3_1A_2}{ (\omega_1- \omega_2 )^2-\omega^2_1} \biggr\} \frac{1}{ (2\omega_1-\omega_2 )^2-\omega^2_2} \\ &\hphantom{-2\omega_1 A_{1_{t_3}} =\ \,}{}+\frac{1}{16\omega^2_1}\frac{A_2 A^3_1}{ (2\omega_1 )^2-\omega^2_2}+ \frac{A^2_1/2}{ (2\omega_1 )^2-\omega^2_2} \frac{A_1A_2}{ (\omega_1-\omega_2 )^2-\omega^2_1} \biggr] \\ &\hphantom{-2\omega_1 A_{1_{t_3}} =\ \,}{}\times\sin (\phi_2-4\phi_1 ) \end{aligned}$$

(4.227)

$$\begin{aligned} &2\omega_2A_2\theta_{2_{t_3}}= \biggl[\frac{A^4_1/8}{ \{ (2\omega_1 )^2-\omega^2_2 \}^2}- \frac{1}{16\omega^2_1}\frac{A^4_1}{ (2\omega_1 )^2-\omega^2_2} \biggr]\cos ( \phi_2-4\phi_1) \end{aligned}$$

(4.228)

$$\begin{aligned} &-2\omega_2A_{2_{t_3}}= \biggl[ \frac{A^4_1/8}{ \{ (2\omega_1 )^2-\omega^2_2 \}^2}- \frac{1}{16\omega^2_1}\frac{A^4_1}{ (2\omega_1 )^2-\omega^2_2} \biggr]\sin( \phi_2-4\phi_1). \end{aligned}$$

(4.229)

One obtains from equations (4.227) and (4.229),

$$ \omega_1A_1A_{1_{t_3}}+4 \omega_2A_2A_{2_{t_3}}=0 $$

(4.230)

or

$$ \omega_1A^2_1+4 \omega_2 A^2_2=\mbox{const} $$

(4.231)

or

$$ J_1\omega_1+J_2 \omega_2=\mbox{const} $$

(4.232)

as before, in (4.207), (4.218) and (4.225). It thus appears that this result remains valid for resonances of all order in the system (4.166).