218 - Österreichische Mathematische Gesellschaft

with multiple indexes, i.e. m ≡ (m 1 ,..., m N ), a j ≡ a j (m 1 ,..., m N ), ψ m ≡
ψ m 1
···ψ m N
and a minimal term in this sum is of the order ≥ 2, i.e. m 1 +···+m N =
|m| ≥ 2. The change of variables
ψ j = ϕ j + ∑ b j (m)ϕ m j (6)
|m|≥2
with some (not known yet) coefficients b j , yields a new system
˙ϕ j = λ j ϕ j + ∑ c j (m)ϕ m j . (7)
|m|≥2
Let us try to choose the coefficients b j in such a way that c j (m) = 0 ∀ j, m. If
this is possible, the change of variables (7) will linearize the system of nonlinear
ODEs (5). Direct substitution of (7) into (5) allows to determine b i only if the corresponding
coefficient −λ i + ∑ N j=1 (m jλ j ) ≠ 0. Otherwise, (7) remains nonlinear.
If there exist N positive integers m 1 ,m 2 ,...,m N , such that
N
∑ m j ≥ 2
j=1
and
N
∑ m j λ j − λ i = 0, (8)
j=1
then (8) is called in mathematics resonance conditions for the system (5). The
number N is the order of resonance while the change of variables (6) is called
Poincaré transformation.
Combination of the Theorem on vector field linearization with the Poincaré transformation
yields the fact of the utmost importance which is indeed the sound
ground for the entire nonlinear resonance analysis:
A system of nonlinear ODEs, if not linearizable, can be transformed
into normal form, with the resonance term having the smallest order.
To see the significance of this issue, let us suppose that ϕ in (3) is small, 0 < ϕ ∼
ε ≪ 1, and the term with ϕ 2 ∼ ε 2 is resonant. Then all other terms, being of order
ε 3 , ε 4 ,..., can be neglected and a solution of
˙ϕ = λϕ + 1 − λa
1 + 2aϕ ϕ2 (9)
will give an approximate solution of (7), with terms of the next order of smallness
omitted.
The importance of studying resonances theoretically has been pointed out by
Poincaré who regarded this as “the fundamental problem of dynamics” [18].
18