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On the proper form of the amplitude modulation equations for resonant systems. (English) Zbl 1047.70042

The complex amplitude modulation equations (AME) of a discrete dynamical system are derived under general conditions of simultaneous internal and external resonances. The AME read: \[ A_m'={\mathcal L}_m\Biggl(A_m(A_n\overline A_n)^k,\dots, \prod_{n\in\mathbb{N}^\pm} A_n^{l_{smn}} e^{\mp i\sigma_st},\dots\Biggr),\quad m= 1,2,\dots, N, \] in which the exponents \(l_{smn}\), are (generally not unique) solutions of the equations \[ l_{smn}- l_{sm,-n}= k_{smn},\;\sum_{n\in\mathbb{N}^\pm} l_{smn}= K,\;l_{smn}\in\mathbb{N},\;K\in [K_s, K_{\max}]. \] The prime denotes differentiation with respect to the reconstituted true time \(t\), and \({\mathcal L}_m\) is a complex linear operator with constant coefficients. The so-called Cartesian rotating form is introduced which makes it possible to evaluate periodic solutions and analyze their stability, even if they are incomplete. A mixed polar-Cartesian form is presented; it is proved that the mixed form leads to standard form equations with the same dimension as the polar form. Finally, some illustrative examples are presented.

MSC:

70K30 Nonlinear resonances for nonlinear problems in mechanics
70K20 Stability for nonlinear problems in mechanics
70K60 General perturbation schemes for nonlinear problems in mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34E13 Multiple scale methods for ordinary differential equations
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