Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument
Gespeichert in:
| Datum: | 1999 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
1999
|
| Schriftenreihe: | Нелинейные граничные задачи |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/169289 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument / I.I. Klevchuk // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 168-173. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-169289 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1692892020-06-10T01:26:19Z Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument Klevchuk, I.I. 1999 Article Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument / I.I. Klevchuk // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 168-173. — Бібліогр.: 6 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169289 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| format |
Article |
| author |
Klevchuk, I.I. |
| spellingShingle |
Klevchuk, I.I. Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument Нелинейные граничные задачи |
| author_facet |
Klevchuk, I.I. |
| author_sort |
Klevchuk, I.I. |
| title |
Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument |
| title_short |
Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument |
| title_full |
Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument |
| title_fullStr |
Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument |
| title_full_unstemmed |
Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument |
| title_sort |
bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169289 |
| citation_txt |
Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument / I.I. Klevchuk // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 168-173. — Бібліогр.: 6 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT klevchukii bifurcationofanequilibriumpointinasystemofnonlinearparabolicequationswithtransformedargument |
| first_indexed |
2025-07-15T04:02:44Z |
| last_indexed |
2025-07-15T04:02:44Z |
| _version_ |
1837684140123619328 |
| fulltext |
BIFURCATION OF AN EQUILIBRIUM POINT
IN A SYSTEM OF NONLINEAR PARABOLIC
EQUATIONS WITH TRANSFORMED ARGUMENT
c© I.I.Klevchuk
1.Introduction. We consider the system of nonlinear parabolic equations with
transformed argument
∂u
∂t
= D(t, ε)
∂2u
∂x2
+ A(t, ε)u + B(t, ε)u∆ + f(t, u, u∆, ε) (1)
with periodic condition
u(t, x + 2π) = u(t, x). (2)
Here u∆ = u(t, x − ∆),∆ is a transformation of the argument, the matrices D(t, ε),
A(t, ε), B(t, ε) and function f : R2n+p+1 → Rn are fourfold continuously differentiable
with respect to all arguments and 2π - periodic with respect to t, f(t, u, v, ε) = O(|u|2 +
|v|2) as |u|+ |v| → 0. Therefore the function f(t, u, v, ε) satisfies to the conditions
f(t, 0, 0, ε) = 0, |f(t, u, v, ε)− f(t, u′, v′, ε)| ≤ ν(|u− u′|2 + |v − v′|2)1/2,
|u| ≤ ρ, |u′| ≤ ρ, |v| ≤ ρ, |v′| ≤ ρ, (3)
where |u|2 = u2
1 + . . . + u2
n, Lipschitz constant ν may be make sufficiently small under
decreasing ρ. Function f(t, u, v, ε) can be determined outside the region |u| ≤ ρ, |v| ≤ ρ,
so that the conditions (3) valid overall the space. Let the matrix D(t, ε) is positive
definite.
System (1) is used for modelling of nonlinear effects in optics [1]. The autonomous
parabolic equation with transformed argument was considered in paper [2].
We consider the linear system
∂u
∂t
= D(t, ε)
∂2u
∂x2
+ A(t, ε)u + B(t, ε)u∆. (4)
We will search the solution of the problem (4),(2) in the form of complex Fourier series
u(t, x) =
∞∑
k=−∞
yk(t)exp(−ikx), y−k(t) = ȳk(t). (5)
Substituting (5) into (4) and comparising the coefficients under exp(−ikx), we obtain
the countable system of differential equations in Fourier coefficients
dyk(t)
dt
= [−k2D(t, ε) + A(t, ε) + B(t, ε)exp(ik∆)]yk(t), k = 0,±1, . . . (6)
System (6) is a one of linear differential equations with periodic coefficients. According
to the Floquet theorem, a matrix Hk(t, ε), detHk(t, ε) 6= 0, Hk(t + 2π, ε) = Hk(t, ε)
exists, such that the substitution yk = Hk(t, ε)zk transforms system (6) to the form
dzk
dt
= Ck(ε)zk, C−k(ε) = C̄k(ε), k = 0,±1, . . . (7)
Suppose that the characteristic equation det(Ck(ε) − λE) = 0, k ∈ Z, has the simple
roots αm(ε) ± iβm(ε), αm(0) = 0, βm(0) = λm > 0,m = 1, ..., p, and the remaining of
roots satisfies to the condition |Re λ| > γ + δ, γ > δ > 0. Suppose that ε is the p -
dimensional parameter.
We will search the solution of the problem (1),(2) in the form of series (5). Substi-
tuting (5) into (1) and comparising the coefficients under exp(−ikx), k ∈ Z, we obtain
the countable system of differential equations in Fourier coefficients
dy
dt
= M(t, ε)y + F (t, y, ε), (8)
where y = (y0, y1, y−1, ...)T , M(t, ε) is infinite blocks- diagonal matrix with the blocks
Mk(t, ε) = −k2D(t, ε) + A(t, ε) + B(t, ε) exp(ik∆), k = 0,±1, ...; F (t, y, ε) = (f0, f1,
f−1, ...)T is nonlinear function, where fk are the Fourier coefficients of the function
f(t, u, u∆, ε) under exp(−ikx).
We will show that the function F (t, y, ε) satisfies to the Lipschitz condition. Let us
introduse in the space of sequences the following norm |y| = (
∑∞
k=−∞ |yk|2)1/2. We
consider another vector z = (z0, z1, z−1, ...)T of Fourier coefficiets for solution v(t, x) of
equation (1) and the corresponding vector F (t, z, ε) = (g0, g1, g−1, ...)T . Using Parseval
equation, we obtain
|F (t, y, ε)− F (t, z, ε)| = (
∞∑
k=−∞
|fk − gk|2)1/2 = (
1
2π
2π∫
0
|f(t, u, u∆, ε)−
−f(t, v, v∆, ε)|2dx)1/2 ≤ ν(
1
2π
2π∫
0
(|u− v|2 + |u∆ − v∆|2)dx)1/2 =
= ν(2
∞∑
k=−∞
|yk − zk|2)1/2 =
√
2ν|y − z|.
Therefore the function F satisfies the Lipschitz condition with the constant
√
2ν.
In the system (8), we make the substitution yk = Hk(t, ε)zk, k = 0,±1, ..., then we
obtain the system
dz
dt
= C(ε)z + G(t, z, ε), (9)
where z = (z0, z1, z−1, ...)T , C(ε) = diag(C0(ε), C1(ε), C−1(ε), ...), G(t, z, ε) = H−1
(t, ε)F (t,H(t, ε)z, ε), H(t, ε) = diag (H0,H1,H−1, ...). We reduce the matrices Ck(ε)
with eigenvalues αm(ε)± iβm(ε) and eigenvalues with positive real parts to the Jordan
canonical form. Under this transformation, we obtain the system
dw1
dt
= A1(ε)w1 + G1(t, w, ε),
(10)
dw2
dt
= A2(ε)w2 + G2(t, w, ε),
where w = (w1, w2)T , A1(ε) = diag(A3(ε), A4(ε)), w1 ∈ Rl+2p, w2 belong to the some
Banach space M , the eigenvalues of the matrix A3(ε) lie on the half-plane Re λ >
γ + δ, A4(ε) is the diagonal matrix with number αm(ε) ± iβm(ε) on diagonal, and
the eigenvalues of the infinite blocks-diagonal matrix A2(ε) lieing on half-plane Re λ <
−γ − δ. Since the vector-function G satisfies the Lipschitz condition and G(t, 0, ε) = 0,
we obtain
G1(t, 0, ε) = G2(t, 0, ε) = 0, (|G1(t, v, ε)−G1(t, w, ε)|2+
(11)
+|G2(t, v, ε)−G2(t, w, ε)|2)1/2 ≤ ν1|v − w|.
The following estimations are valid
| exp[A3(ε)t] ≤ N exp[(γ + δ)t], t ≤ 0, | exp[A2(ε)t]| ≤
(12)
≤ N exp[−(γ + δ)t], t ≥ 0, | exp[A4(ε)t]| ≤ N exp[(γ − δ)|t|], t ∈ R.
2.Existence and properties of integral manifolds.
Theorem 1. Let the estimates (11),(12) holds. Thus, if
ν1 <
δ
N(1 + 2N)
, (13)
then there exists a function h : Rl+3p+1 → M ,
h(t, 0, ε), |h(t, w1, ε)− h(t, w′1, ε)| ≤
1
2
|w1 − w′1|, (14)
such that the set S− = {(t, w1, w2)|t ∈ R, w1 ∈ Rl+2p, w2 = h(t, w1, ε), w2 ∈ M} is the
integral manifold of the system (10). For any solution w(t) = (w1(t), h(t, w1(t), ε)) of
the system (10) on manifold S−, the following estimate is valid
|w(t)| ≤ 2N |w1(σ)| exp[γ(σ − t)], t ≤ σ.
Theorem 2. Let the conditions (11)-(13) are satisfied. Then there exists a function
g : Rp+1 × M → Rl+2p, g(t, 0, ε) = 0, |g(t, w, ε) − g(t, w′, ε)| ≤ 1
2 |w − w′|, such that
the set S+ = {(t, w1, w2)|t ∈ R,w2 ∈ M, w1 = g(t, w2, ε), w1 ∈ Rl+2p} is the integral
manifold of the system (10). For any solution w(t) = (g(t, w2(t), ε), w2(t)) of the system
(10) on manifold S+, the folloving estimate is valid |w(t)| ≤ 2N |w2(σ)|exp[γ(σ − t)],
t ≥ σ.
Let t = σ is some number (initial value). We show that the integral manifold S− is
exponential stable.
Note that the equation on manifold S− is of the following form
dv
dt
= A1(ε)v + G1(t, v, h(t, v, ε), ε). (15)
Theorem 3. Let w(t) = (w1(t), w2(t)) be arbitrary solution of the system (10) with
initial value w(σ) under t = σ. If the condition (13) is satisfied, then there exists a
solution ξ(t) = (v(t), h(t, v(t), ε) on manifold S−, such that the following estimate is
valid
|w(t)− ξ(t)| ≤ 2N |w2(σ)− h(σ, v(σ), ε)|exp[γ(σ − t)], t ≥ σ.
The equation (15) can be represented in the form
dw3
dt
= A3(ε)w3 + G3(t, w3, w4, h(t, w3, w4, ε), ε),
(16)
dw4
dt
= A4(ε)w4 + G4(t, w3, w4, h(t, w3, w4, ε), ε).
where v = (w3, w4), G1 = (G3, G4). If the condition (13) is satisfies, then the integral
manifold S+
1 = {(t, w3, w4)|t ∈ R,w4 ∈ R2p, w3 = r(t, w4, ε), w3 ∈ Rl} of the system
(16) exists [3,4]. The function r(t, w, ε) satisfies the following estimate
r(t, 0, ε) = 0, |r(t, w, ε)− r(t, v, ε)| ≤ 1
2
|w − v|, w ∈ R2p, v ∈ R2p.
We denote r1(t, w, ε) = h(t, r(t, w, ε), w, ε).
Theorem 4. Let the conditions (11)-(13) be satisfied. Then there exists the cen-
tral manifold S = {(t, w3, w4, w2)|t ∈ R, w4 ∈ R2p, w3 = r(t, w4, ε), w3 ∈ Rl, w2 =
r1(t, w4, ε), w2 ∈ M} of the system (10).
3.Bifurcation of equilibrium point. The equation on manifold S is of the follow-
ing form
dw4
dt
= A4(ε)w4 + G4(t, r(t, w4, ε), w4, r1(t, w4, ε), ε). (17)
The equation (17) can be represented in the form
dvk
dt
= [αk(ε) + iβk(ε)]vk + Vk(t, v, v̄, ε),
(18)
dv̄k
dt
= [αk(ε)− iβk(ε)]v̄k + V̄k(t, v, v̄, ε),
where vk is the complex variable, v = (v1, ..., vp)T , Vk(t + 2π, v, v̄, ε) = Vk(t, v, v̄, ε),
Vk(t, v, v̄, ε) = O(|v|2) as |v| → 0, k = 1, ..., p.
Let the following condition be satisfied
1) n1λ1 + ...+npλp 6= m as 0 < |n1|+ ...+ |np| < 6, where m,n1, ..., np are the integer
numbers.
Substituting the variable
v = x +
4∑
k=2
Wk(t, x, x̄, ε),
where W2,W3,W4 are the forms of the 2,3 and 4 order with periodic coefficients, we
transform the system (18) to the following form [5,6]
dxk
dt
= [αk(ε) + iβk(ε)]xk + xk
p∑
j=1
akj(ε)xj x̄j + Xk(t, x, x̄, ε),
dx̄k
dt
= [αk(ε)− iβk(ε)]x̄k + x̄k
p∑
j=1
ākj(ε)xj x̄j + X̄k(t, x, x̄, ε),
where Xk(t + 2π, x, x̄, ε) = Xk(t, x, x̄, ε), Xk(t, x, x̄, ε) = O(|x|5) as |x| → 0. Passing to
the polar coordinates xk = rk exp(iϕk), x̄k = rk exp(−iϕk), we obtain the real system
drk
dt
= αk(ε)rk + rk
p∑
j=1
bkj(ε)r2
j + Rk(t, r, ϕ, ε),
dϕk
dt
= βk(ε) +
p∑
j=1
ckj(ε)r2
j + Φk(t, r, ϕ, ε),
where bkj(ε) = Reakj(ε), ckj(ε) = Imakj(ε), Rk(t, r, ϕ, ε) = O(|r|5), Φk(t, r, ϕ, ε) =
O(|r|4) as |r| → 0.
We consider the bifurcation equation B(ε)r2 + a(ε) = 0, where B(ε) is the matrix
with elements bkj(ε), a(ε) and r2 are the vectors with elements αk(ε) and r2
j .
Theorem 5. Let detB(0) 6= 0, detda
dε (0) 6= 0, the all elements of vector B−1(ε)a(ε)
are negative and condition 1 is satisfied. Then, there exists an invariant torus of the
system (1).
The solutions on the torus are quasi-periodic if |(n, λ) + q| > γ|n|−p−1, λ = (λ1,
..., λp) = (β1(0), ..., βp(0)), where γ is some positive number, n = (n1, ..., np), q, n1, ..., np
are integer numbers.
References
1. Akhmanov S.A., Vorontsov M.A., Instabilities and structures in coherent nonlinear optical systems,
Nonlinear waves. Dynamics and evolution. Moskow:Nauka (1989), 228-237.
2. Kashchenko S.A., Asymptotic space-ihomogeneous structures in coherent nonlinear-optical systems,
J. Vychisl. Math. i Math. Phys. 31 (1991), no. 3, 467-473.
3. Plis V.A., Integral sets of periodic systems of differential equations, Moskow: Nauka (1977), 304.
4. Fodchuk V.I., Klevchuk I.I., Integral sets and reduction principle for differential-functional equa-
tions, Ukr. Math. J. 34 (1982), no. 3, 334-340.
5. Samoilenko A.M., Polesya I.V., The birth of the invariant sets in a neigborhood of equilibrium point,
Differentsial’nye Uravneniya 11 (1975), no. 8, 1409-1415.
6. Bibikov Yu.N., Hopf bifurcation for quasi-periodic motions, Differentsial’nye Uravneniya 16 (1980),
no. 9, 1539-1544.
Department of Mathematics,
Chernivtsi State University,
Kotsubinsky str.,2, Chernivtsi
E-mail: klevchuk@chsu.cv.ua
|