Resonance terms

Those terms $ f _ {PQ} X ^ {P} \mathop{\rm exp} \{ i \langle Q, Y \rangle \} $ in the Taylor–Fourier series

$$ \tag{1 } f( X, Y) = \sum f _ {PQ} X ^ {P} \mathop{\rm exp} \{ i \langle Q, Y \rangle \} , $$

$$ P \in \mathbf Z ^ {m} ,\ P \geq 0 ,\ Q \in \mathbf Z ^ {n} , $$

$$ X ^ {P} = x _ {1} ^ {p _ {1} } \dots x _ {m} ^ {p _ {m} } , $$

whose indicators $ P $ and $ Q $ satisfy a linear relation as follows:

$$ \tag{2 } \langle P, \Lambda \rangle + i \langle Q, \Omega \rangle = c. $$

Here $ f _ {PQ} $ are constant coefficients, $ \langle Q, Y\rangle $ is the scalar product of $ Q $ and $ Y $; the constants $ ( \lambda _ {1} \dots \lambda _ {m} ) = \Lambda $ and $ ( \omega _ {1} \dots \omega _ {n} ) = \Omega $ are usually the eigenvalues and the basis of frequencies of a specific system of ordinary differential equations; the constant $ c $ is independent of $ P $ and $ Q $ and it is defined by the role of the series (1) in the problem under analysis.

If in a linear system

$$ \tag{3 } \dot{x} _ {j} = \lambda _ {j} x _ {j} ,\ \ j = 1 \dots m,\ \ \dot{y} _ {k} = \omega _ {k} ,\ \ k = 1 \dots n, $$

all $ \lambda _ {j} $ are purely imaginary and in (2) $ c= 0 $, then the total resonance term of the series (1) coincides with the average of this series along the solutions of the system (3). A system of ordinary differential equations in a neighbourhood of invariant manifolds can be reduced to a normal form in which the series contains only resonance terms (see [1]). Thus, for a Hamiltonian system in a neighbourhood of a fixed point, the Hamiltonian function is reducible to the form (1) where $ n= 0 $ and (2) is fulfilled with $ c= 0 $, whence $ \Lambda = ( \lambda _ {1} \dots \lambda _ {l} , - \lambda _ {1} \dots - \lambda _ {l} ) $ is the vector of eigenvalues of the linearized system (see ). In this case, the terms $ p _ {j} = p _ {j+} l $, $ j = 1 \dots l $, are sometimes called secular (for them (2) is fulfilled trivially), and the remaining terms of the series (1) for which (2) is fulfilled are called the resonance terms.

The separation of resonance terms, derived in problems with a small parameter, can often be based on a normal form (see [1]). For a point transformation with multipliers $ ( \mu _ {1} \dots \mu _ {m} ) = M $ the indices of the resonance terms of the series (1) with $ n= 0 $ satisfy the relation $ M ^ {P} = 1 $; if one assumes that $ \Lambda = \mathop{\rm ln} M $ and $ \omega _ {1} = 1 $, then (2) is obtained with $ c= 0 $.

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