tokens: must come from combinations of the harmonics contained in . Recall that we defined containing both synodic terms with harmonics λ 1 − λ 2 and λ 2 − λ 3 , which we wrote in equation ( 16 ) in terms of the new variables . Therefore, the harmonics that are included in are combinations of these synodic harmonics; more specifically, they come from the products of their cosines. 6 Using the standard trigonometric identity , the resulting harmonics are so the harmonic with the lowest fraction of is . Specializing now to the case of a chain with the same resonance index k (1) = k (2) = k , this simply gives With the aid of an algebraic manipulator, one can compute the full expression of and select the desired harmonic term (we used the software package wolfram mathematica ), thus obtaining its coefficient (actually, one can see that this term emerges solely from the term ). We avoid writing here the full expression, which is rather cumbersome; moreover, as in ( 35 ), we evaluate it at the reference values of the actions so the term multiplying the cosine becomes a numerical coefficient, and we write this term as Since we want to compare the frequency of to that of , we need to consider the resonant Hamiltonian in the variables, expand the ‘barred’ variables around the equilibrium point characterized by the equilibrium actions and the equilibrium angles (equation 20 ) as in Section 4.3 , and then introduce the transformation to the action-angle variables , which transforms into the sum of decoupled harmonic oscillators plus higher order terms (equation 22 ). It is also useful to translate the value of around its initial reference value introducing . Therefore, we write in terms of the variables . The Hamiltonian will now contain harmonic terms of type where is an integer combination with coefficients of the angles φ 1 , ..., φ 4 , which we can calculate explicitly. Therefore, whenever , a secondary resonance is crossed. We can rewrite this expression as . Since the Hamiltonian has d’Alembert characteristics in each pair ( I l , φ l ), and the values of the actions are initially (that is, before their excitation) small, the strongest secondary resonances will come from lowest integer combinations , that is, where most h l are zero. We also note that since and the frequencies ω l are all negative, a secondary resonance term can only appear when , which, together with the requirement that be small, is tantamount to requiring that all non-zero integers h l are positive. Since we calculated ω ( m pl ) in Section 4.3 , we can calculate for each the relative frequency as a function of m pl , and check if any of these vanish for some value of m pl , which corresponds to crossing a secondary resonance. We carried out the calculation with the aid of the mathematica software in the reference case k = 3 and a 1 ≃ 0.1, which corresponds to the evolution shown in Fig. 1(b) (and also Figs 2 and 3 ). We found that contains, among many others, the following
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