Text 4. Linearized and Spectral Stability

There are two other ways of treat­ing stability. First of all, one can linearize equation (1.7.1); if δu denotes a variation in u and X' (u_e)denotes the linearization of X at u_e (the matrix of partial derivatives in the case of finitely many degrees of freedom), the linearized equations describe the time evolution of "infinitesimal" distur­bances of u_e:

 

(1.7.2)

 

Equation (1.7.1), on the other hand, describes the nonlinear evolution of finite disturbances Δ u = u – u_e. We say that u_e is linearly stable if (1.7.2) is stable at δu = 0, in the sense defined above. Intuitively, this means that there are no infinitesimal disturbances that are growing in time. If (δu)₀is an eigenfunction of X' (u_e), that is, if

 

(1.7.3)

for a complex number λ, then the corresponding solution of (1.7.2) with initial condition (δu)₀ is

(1.7.4)

The right side of this equation is growing when λ has positive real part. This leads us to the third notion of stability: We say that (1.7.1) or (1.7.2) is spectrally stable if the eigenvalues (more precisely, points in the spec­trum) all have nonpositive real parts. In finite dimensions and, under ap­propriate technical conditions in infinite dimensions, one has the following implications

(stability) => (spectral stability)

and

(linear stability) => (spectral stability).

If the eigenvalues all lie strictly in the left half-plane, then a classical re­sult of Liapunov guarantees stability. (See, for instance, Hirsch and Smale [1974] for the finite-dimensional case and Marsden and McCracken [1976] or Abraham, Marsden, and Ratiu [1988] for the infinite-dimensional case.) However, in many systems of interest, the dissipation is very small and are modeled as being conservative. For such systems the eigenvalues must be symmetrically distributed under reflection in the real and imaginary axes (We prove this later in the text). This implies that the only possibility for spectral stability occurs when the eigenvalues lie exactly on the imagi­nary axis. Thus, this version of the Liapunov theorem is of no help in the Hamiltonian case.

 

Слова к тексту

linearized	[,lınıə´raızd]	линеаризованный
partial	[´p a:∫l]	частный
derivative	[dı´rıvətıv]	производная
finitely	[´faınaıtlı]	конечно
degree	[dı´gri:]	степень
evolution	[,i:və´lu:∫ən]	развитие
infinitesimal	[,ınfını´tesım ə l]	бесконечно малый
disturbance	[dıs´tз:bəns]	возмущение
linearly	[´lınıəlı]	линейно
eigenfunction	[´aıgən´fΛŋk∫ ə n]	собственная функция
solution	[sə´lu:∫n]	решение
initial	[ı´nı∫ ə l]	начальный, исходный
real	[rıəl]	действительный
spectrally	[´spektrəlı]	спектрально
eigenvalue	[´aıgən´vælju:]	собственное значение
spectrum (pl spectra)	[´spektrəm], [´spektrə]	спектр
appropriate	[ə´prəυprıət]	соответствующий
implication	[ımplı´keı∫n]	импликация
guarantee	[,gær ə n´ti:]	гарантировать
finite-dimensional	[´faınaıtd(а)ı´men∫ ə nl]	конечномерный
infinite-dimensional	[´ınfınıtd(а)ı´men∫ ə nl]	бесконечномерный
dissipation	[dısı´peı∫ ə n]	рассеяние
axis (pl axes)	[´æksıs], [´æksi:z]	ось
imply	[ım´plaı]	значить, означать

 

Letters and Sounds

 

Task 1. Answer the following questions

 

1. Вспомните как можно больше слов, где буква о читается [u:]. В каком типе слога (открытый/закрытый) это может иметь место?

 

2. Какой дифтонг произносится в словах either, neither, height, eigenvalue, eingenfunction?

 

3. Как читаются буквосочетания alf, alve, alm, alk? Приведите примеры.

 

4. Как читается буква u в открытом слоге? Как она читается в открытом слоге после [l], [r], [dζ]? Приведите примеры.

 

Task 2. Say and write what sound is pronounced in the following words:

 

1) m o ve, 2) c al f, 3) ei ther, 4) h al ves, 5) m u te, 6) ch al k, 7) m o ve, 8) p al m, 9) l o se, 10) evol u tion, 11) ei genvalue, 12) excl u sion, 13) und o, 14) r u diment, 15) h ei ght, 16) w ei ght, 17) j u nior, 18) concl u de, 19) n ei ther, 20) c al m.

 

Task 3. Read aloud the following words from Text 4

 

Guarantee, above, version, eigenvalues, other, notion, growing, half, axes, dimension, stable, solution, partial, matrix, prove, evolution, condition, following.

 

Grammar Activity