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(Çevrimiçi): http://www.jmulti.de Erisim Tarihi:
25.06.2011
ĐSPAT: LOTKA-VOLTERRA DENKLEM
SĐSTEMĐNĐN LYAPUNOV ANLAMINDA
KARARLILIĞI
Teorem:
* X , bir diferansiyel denklem sisteminin sabit
noktası ve L(X) :U →ℜ, U açık kümesi üzerinde
tanımlanan ve W komsuluğunda, , ⊂ ⊂ ℜn
U W ,
türevlenebilir bir fonksiyon olsun. Eğer,
i) L(X* ) = 0 ve ≠ * X X için L(X) > 0 ,
ii) −{ *} U X açık kümesinde,
dL(X) / dt ≤ 0 ise,
* X kararlıdır. L(X) ise Lyapunov fonksiyonudur.
(Lorenz, 1993:34-35).
Çözüm:
x& = kx − axy (1)
y& = −rx + bxy (2)
Lotka Volterra diferansiyel denklem sistemini
olusturmaktadır. Sistemin sabit noktaları x& = 0 için
y* = k / a ve y& = 0 için x* = r / b dir. Denklem (1),
denklem (2)’ye bölünürse,dx k ay x
dy r bx y
−
= −
−
(3)
elde edilir. Denklem (3) asağıdaki gibi düzenlenebilir:
(r−bx)ydx=−(k−ay)xdy⇒(r−bx)ydx+(k−ay)xdy=0
(4)
Denklem (4), xy ’ye bölündüğünde değiskenlerine
ayrılabilir hale gelir.
(rx−1 − b)dx + (ky−1 − a)dy = 0 (5)
Denklem (5)’ in integrali alındığında,
∫ (rx−1 − b)dx + ∫ (ky −1 − a)dy = r ln x − bx + k ln y − ay = F ( x, y )
(6)
elde edilir. Sabit noktalar denklem (6)’ nın türevinde
yerine konduğunda,
*
*
0
x x * *
y y
dF r k
b a
dt = x y
=
= − + − =
olduğu kanıtlanır.
F ( x, y) = r ln x − bx + k ln y − ay , Lotka-
Volterra diferansiyel denklem sistemi için Lyapunov
fonksiyonudur. Goodwin modeli için sabit noktalar
Lyapunov anlamında kararlıdır.
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