Multiple solutions for nonlinear boundary-value problems of ODE
We consider a Hamilton system related to the Trott curve in Harnack’s theorem. This theorem says that the maximal number of ovals for the fourth order curve is four. We treat the related Hamilton system which has more ovals that is prescribed by Harnack’s theorem. We give explanation and consider th...
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Інститут математики НАН України
2014
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irk-123456789-1747132021-01-28T01:26:36Z Multiple solutions for nonlinear boundary-value problems of ODE Kirichuka, A. We consider a Hamilton system related to the Trott curve in Harnack’s theorem. This theorem says that the maximal number of ovals for the fourth order curve is four. We treat the related Hamilton system which has more ovals that is prescribed by Harnack’s theorem. We give explanation and consider the Dirichlet boundary-value problem for the system. Precise estimation is given for the number of solutions to the Dirichlet problem. Розглянуто гамiльтонову систему, пов’язану з кривою Тротта в теоремi Харнака, яка стверджує, що максимальна кiлькiсть овалiв кривої четвертого порядку дорiвнює 4. Розглянуто гамiльтонову систему, що має бiльшу кiлькiсть овалiв, нiж стверджується в теоремi Харнака. Наведено пояснення цього факту та розглянуто граничну задачу Дiрiхле для вiдповiдної системи. Отримано точнi оцiнки кiлькостi розв’язкiв задачi Дiрiхле. 2014 Article Multiple solutions for nonlinear boundary-value problems of ODE / A. Kirichuka // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 50-57. — Бібліогр.: 4 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174713 517.9 en Нелінійні коливання Інститут математики НАН України |
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We consider a Hamilton system related to the Trott curve in Harnack’s theorem. This theorem says that the maximal number of ovals for the fourth order curve is four. We treat the related Hamilton system which has more ovals that is prescribed by Harnack’s theorem. We give explanation and consider the Dirichlet boundary-value problem for the system. Precise estimation is given for the number of solutions to the Dirichlet problem. |
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Article |
| author |
Kirichuka, A. |
| spellingShingle |
Kirichuka, A. Multiple solutions for nonlinear boundary-value problems of ODE Нелінійні коливання |
| author_facet |
Kirichuka, A. |
| author_sort |
Kirichuka, A. |
| title |
Multiple solutions for nonlinear boundary-value problems of ODE |
| title_short |
Multiple solutions for nonlinear boundary-value problems of ODE |
| title_full |
Multiple solutions for nonlinear boundary-value problems of ODE |
| title_fullStr |
Multiple solutions for nonlinear boundary-value problems of ODE |
| title_full_unstemmed |
Multiple solutions for nonlinear boundary-value problems of ODE |
| title_sort |
multiple solutions for nonlinear boundary-value problems of ode |
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Інститут математики НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/174713 |
| citation_txt |
Multiple solutions for nonlinear boundary-value problems of ODE / A. Kirichuka // Нелінійні коливання. — 2014. — Т. 17, № 1. — С. 50-57. — Бібліогр.: 4 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT kirichukaa multiplesolutionsfornonlinearboundaryvalueproblemsofode |
| first_indexed |
2025-07-15T11:45:55Z |
| last_indexed |
2025-07-15T11:45:55Z |
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1837713280164954112 |
| fulltext |
UDC 517.9
MULTIPLE SOLUTIONS FOR NONLINEAR
BOUNDARY-VALUE PROBLEMS OF ODE*
КРАТНI РОЗВ’ЯЗКИ НЕЛIНIЙНИХ ГРАНИЧНИХ ЗАДАЧ
ДЛЯ ЗВИЧАЙНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
A. Kirichuka
Daugavpils Univ.
Parades str., 1, Daugavpils, Latvia
e-mail: anita.kiricuka@du.lv
We consider a Hamilton system related to the Trott curve in Harnack’s theorem. This theorem says that the
maximal number of ovals for the fourth order curve is four. We treat the related Hamilton system which
has more ovals that is prescribed by Harnack’s theorem. We give explanation and consider the Dirichlet
boundary-value problem for the system. Precise estimation is given for the number of solutions to the Di-
richlet problem.
Розглянуто гамiльтонову систему, пов’язану з кривою Тротта в теоремi Харнака, яка ствер-
джує, що максимальна кiлькiсть овалiв кривої четвертого порядку дорiвнює 4. Розглянуто га-
мiльтонову систему, що має бiльшу кiлькiсть овалiв, нiж стверджується в теоремi Харнака.
Наведено пояснення цього факту та розглянуто граничну задачу Дiрiхле для вiдповiдної сис-
теми. Отримано точнi оцiнки кiлькостi розв’язкiв задачi Дiрiхле.
1. Introduction. In this article we would like to consider two problems: the Harnack’s theory
about algebraic curves and the number of period annuli; two-dimensional Hamilton system and
the number of solutions of this system with the Dirichlet boundary-value problem on a finite
interval.
The paper has the following structure. In Section 2 we describe the Hamilton system, the
Harnack’s theorem and give a definition. In Section 3 we consider the Hamilton system of di-
fferential equations related to the Trott curve and analyze how many periodic solutions contain
this system. In Section 4 we formulate the theorem and lemmas about the number of solutions
for a system of the Trott curve with the Dirichlet boundary-value condition. In final Section 5
we summarize the results and make conclusions.
2. Preliminary results and definitions. Consider the two-dimensional nonlinear system
x′ = f(x, y),
(2.1)
y′ = g(x, y).
Critical points of such system are to be determined from the relationships
f(x, y) = g(x, y) = 0.
∗ This work has been supported by the European Social Fund within the Project “Support for the implementati-
on of doctoral studies at Daugavpils University” Agreement Nr 2009/0140/1DP/1.1.2.1.2/09/IPIA/V IAA/015.
c© A. Kirichuka, 2014
50 ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
MULTIPLE SOLUTIONS FOR NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 51
Suppose the critical points of system (2.1) are at (xi; yi).
By linearization of the system the types of critical points can be determined and description
can be given in the phase plane.
A critical point of system (2.1) is a center if it has a punctured neighborhood covered with
non trivial cycles.
Definition 2.1 [2]. A central region is the largest connected region covered with cycles around
a center type critical point.
Definition 2.2 [2]. A period annulus is every connected region covered with nontrivial concent-
ric cycles.
Definition 2.3 [3]. We will call a period annulus associated with a central region a trivial
period annulus. Periodic trajectories of a trivial period annulus encircle exactly one critical point
of the type center.
Definition 2.4 [3]. Respectively a period annulus enclosing several (more than one) critical
points will be called a nontrivial period annulus.
When studying periodic solutions of planar systems it is reasonable to consider the Hamil-
ton systems which can be integrated [1].
A Hamiltonian function H(x, y) is the first integral of the Hamilton’s system
x′ =
∂H(x, y)
∂y
,
y′ = −∂H(x, y)
∂x
because
dH
dt
=
∂H
∂x
dx
dt
+
∂H
dy
dy
dt
=
∂H
∂x
∂H
∂y
+
∂H
∂y
(
−∂H
∂x
)
= 0.
The Hamilton function can be interpreted as the total energy of the system being described.
For a closed system, it is the sum of the kinetic and potential energy in the system. There is a
set of differential equations known as the Hamilton’s equations which give the time evolution
of the system and H(x, y) = C. Function H(x, y) is often called Hamiltonian.
The Harnack’s theorem about algebraic curves can be used in studying period annuli. It
allows to construct fairly simple examples of existence of multiple period annuli for polynomials
of lowest degree. We construct two-dimensional planar systems called Hamilton systems which
have many period annuli as defined in the Harnack’s theorem.
Theorem 2.1 [4]. For any algebraic curve of degree n in the real projective plane, the number
of components c is bounded by
1− (−1)n
2
≤ c ≤ (n− 1)(n− 2)
2
+ 1.
Any number of components in this range of possible values can be attained.
Definition 2.5. A curve which attains the maximum number of real components is an M -
curve.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
52 A. KIRICHUKA
Fig. 3.1. The Trott curve.
There are the Hamilton systems with a number of period annuli greater than the number of
ovals prescribed by the Harnack’s theorem for related curves.
3. The system related to the Trott curve and periodic solutions. As an example, we consider
a curve of degree four, which according to the Harnack’s theorem has the maximal number of
components four. It is a fourth degree curve, namely the Trott curve, which has the maximal
number of components, exactly four.
Definition 3.1. The Trott curve is an algebraic curve which satisfies the equation
144(x4 + y4)− 225(x2 + y2) + 350x2y2 + 81 = 0.
The Trott curve (Fig. 3.1) has four separated ovals, the maximal number for a curve of
degree four, and hence it is an M -curve.
Let us consider a functionH1(x, y), a part of the polynomial Trott curve,H1(x, y) = 144(x4+
+y4)− 225(x2 + y2) + 350x2y2, and the Hamilton system
x′ =
∂H1
∂y
= 576y3 − 450y + 700x2y,
(3.1)
y′ = −∂H1
∂x
= −576x3 + 450x− 700xy2.
System (3.1) has 9 critical points, 5 of them are the points of type “center” and 4 are points
of type “saddle”:(
15√
638
,
15√
638
)
,
(
− 15√
638
,− 15√
638
)
,
(
− 15√
638
,
15√
638
)
,
(
15√
638
,− 15√
638
)
.
The Hamilton system (3.1) contains five trivial period annuli around points of type “center”:
(0, 0),
(
−5
4
√
2
, 0
)
,
(
5
4
√
2
, 0
)
,
(
0,
5
4
√
2
)
,
(
0,
−5
4
√
2
)
and one nontrivial periodic annulus-around all critical points, totally six period annuli (Fig. 3.2).
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
MULTIPLE SOLUTIONS FOR NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 53
Fig. 3.2. The phase portrait of system (3.1).
The Harnack’s theorem says about 4 ovals for the Trott curve. For the respective system
(3.1) we have more, six period annuli. This is because we consider curves given by the relation
H1(x, y) = C where C is arbitrary, and a specific Trott curve we obtain for C = −81.
4. A system related to the Trott curve and the Dirichlet boundary-value problem. We are
interested in the question of how many solutions of the system satisfy the given boundary condi-
tions.
Consider system (3.1) with the Dirichlet boundary conditions
x(0) = 0, x(T ) = 0. (4.1)
Denote parts of the phase portrait of system (3.1): Region 1 in Fig. 4.1 (a); Region 2 in
Fig. 4.1 (b); Region 3 in Fig. 4.1 (c); Region 4 in Fig. 4.1 (d):
Make the linearization of system (3.1).
f ′x = 1400xy,
f ′y = 1728y2 − 450 + 700x2,
g′x = −1728x2 + 450− 700y2,
g′y = −1400xy.
The linearized system for system (3.1) at a critical point (x∗, y∗) is
u′ = (1400x∗y∗)u+ (1728y∗2 − 450 + 700x∗2)v,
v′ = (−1728x∗2 + 450− 700y∗2)u+ (−1400x∗y∗)v.
Consider Region 1 and closed trajectories which are in a close proximity of the boundary of
Region 1.
This boundary consists of the critical points
(
− 15√
638
,
15√
638
)
,
(
15√
638
,
15√
638
)
which are
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
54 A. KIRICHUKA
(a) (b)
(c) (d)
Fig. 4.1. The phase portrait of system (3.1).
of the “saddle” type, and of two heteroclinic solutions connecting them. Motion along heterocli-
nic solutions is very “slow” in the meaning that the time needed to pass from one critical point
to another is infinity.
Denote a point of intersection of the “upper” heteroclinic solution with y axis (0, u∗). The
calculation gives u∗ ≈ 1.01.
For periodic solutions of system (3.1), which satisfy the initial conditions (0, u0),where u0 ∼
∼ u∗, u0 < u∗, we have that the time τ(u0) needed to reach the next intersection point with
the y axis is arbitrarily large in the meaning that τ(u0) → +∞ as u0 → u∗.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
MULTIPLE SOLUTIONS FOR NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 55
Fig. 4.2. A solution of system (3.1) in Region 1,
x(t) ∼ u(t), u0 = 0.9, u0 = 0.95, u0 = 1.0.
On the other hand, trajectories in the interior of Region 1 surrounded the critical point(
0,
5
4
√
2
)
are relatively “fast” (Fig. 4.2). We can compute the time τ(u0), needed to pass the
from y axis to another point on the y axis, in fact the half-period for u0 →
5
4
√
2
≈ 0.884.
For this, consider the linearized system of system (3.1) at the critical point
(
0,
5
4
√
2
)
u′ = 900v,
(4.2)
v′ = −96.875u
(the value −96.875 is precise).
The eigenvalues of the linearized system (4.2) are λ1,2 = ±295.27529i and the point
(
0,
5
4
√
2
)
is a point of type “center”.
System (4.2) can be rewritten in the form
u′′ = 900v′ = −96.875 · 900u = −87187.5u (4.3)
and the solution of equation (4.3),
u(t) = sin
√
87187.5t, (4.4)
generates an approximation (u(t), u′(t)) to a solution of Cauchy problem (3.1), x(0) = 0,
y(0) =
5
4
√
2
± ε where ε is a small value.
Therefore, in the Dirichlet problem (4.1), (4.4), T =
τ
2
=
π√
87187.5
, where τ is a period of
solution.
Lemma 4.1. Let n be the largest integer such that
nπ√
87187.5
≤ T. The Dirichlet problem
(3.1), (4.1) in Region 1 has at least 2n nontrivial solutions.
Proof. By considering the initial value problems (3.1),
(x(0), y(0)) = (0, u0),
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
56 A. KIRICHUKA
where u0 ∈
(
5
4
√
2
, u∗
)
, we get n solutions and, for u0 <
5
4
√
2
, the symmetric solutions which
gives n more solutions.
Trajectories in Region 2 (Fig. 4.1 (b)), which are in a close proximity of the external boundary
of Region 2 are very “slow”. This is because the boundary contains four critical points of the type
“saddle”:(
15√
638
,
15√
638
)
,
(
− 15√
638
,− 15√
638
)
,
(
− 15√
638
,
15√
638
)
,
(
15√
638
,− 15√
638
)
.
The linearized system of system (3.1) at the origin is
u′ = −450v,
(4.5)
v′ = 450u.
The eigenvalues of linearized system (4.5) are λ1,2 = ±450i and the origin is a point of type
“center”.
System (4.5) can be rewritten in the form u′′ = −450v′ = −4502u and u(t) = sin 450t
generates an approximation (u(t), u′(t)) to a solution of Cauchy problem (3.1), x(0) = 0,
y(0) = ±ε where ε is a small value.
Therefore, in the Dirichlet condition (4.1), T =
τ
2
=
π
450
, where τ is a period of soluti-
on (4.5).
Lemma 4.2. Let m be the largest integer such that
mπ
450
≤ T. The Dirichlet problem (3.1),
(4.1) in Region 2 has at least 2m nontrivial solutions.
Proof of Lemma 4.2 is similar to the proof of Lemma 4.1.
Consideration of Region 3 (Fig. 4.1 (c)) is similar to that for Region 1. The results can be
formulated in the following lemma.
Lemma 4.3. Let n be the largest integer such that
nπ√
87187.5
≤ T. The Dirichlet problem
(3.1), (4.1) in Region 3 has at least 2n nontrivial solutions.
Proof of Lemma 4.3 is similar to the proof of Lemma 4.1.
The curves which are in Region 4 (Fig. 4.1 (d)) are closed. We want to evaluate the speed of
rotation along these trajectories. For this, we consider the principal (cubic) part of system (3.1)
x′ = 576y3 + 700x2y,
(4.6)
y′ = −576x3 − 700xy2.
In the polar coordinates
x(t) = r(t) sin Θ(t),
y(t) = r(t) cos Θ(t),
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
MULTIPLE SOLUTIONS FOR NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 57
system (4.6) takes the form
r′(t) = −31r3(t) sin 4Θ(t),
(4.7)
Θ′(t) = r2(t)(576 + 62 sin2 2Θ(t)).
Lemma 4.4. In Region 4 trajectories with large u0 = r0 move arbitrarily “fast” .
Proof. Consider Region 4 where u > u0, u0 > 0 may be arbitrarily large. It is true that
Θ′(t) ≥ 576 ∀t ∈ [0, T ].
The solution of system (3.1) with the initial condition (x(0), y(0)) = (0, u0) has a period
τ(u0) and τ(u0) → 0 as u0 → 0. Since the expression in the parentheses in the right-hand side
in the second equation (4.7) is never zero, we get that Θ′(t) is arbitrarily large as r2 is large.
Therefore in Region 4 trajectories with large u0 move arbitrarily “fast” .
Lemma 4.5. Dirichlet problem (3.1), (4.1) for any T in Region 4 has infinitely many nontrivial
solutions (a countable set).
Therefore, we can formulate the following theorem which follows from Lemma 4.1 to Lem-
ma 4.4.
Theorem 4.1. The Dirichlet boundary-value problem (3.1), (4.1) in Regions 1, 2, 3 has at least
2(2n+m) nontrivial solutions. In external Region 4 there are infinitely many nontrivial solutions
(a countable set).
5. Conclusions. When analyzing the results it can be concluded that the number of peri-
od annuli of the Hamilton systems is significantly greater than the number of components of
the respective curve in the Harnack’s theorem. However this is not in contradiction with the
Harnack’s theorem, because in the latter equation the total number of component exceeds the
number four which is the prescribed number of ovals in the Harnack’s theory for a single curve
given by H1 = C.
For the system of differential equations (3.1) with the Dirichlet boundary conditions (4.1)
there is a nontrivial solution and the number of solutions is infinite (a countable set).
1. Lynch St. Dynamical systems with applications using mathematica. — Boston: Birkhäuser, 2007. — P. 111 –
123.
2. Atslega S. Bifurcations in nonlinear boundary value problems and multiplicity of solutions: PhD Thesis. —
Daugavpils, 2010.
3. Kozmina Y., Sadyrbaev F. On a maximal number of period annuli // Abstract and Appl. Anal. — 2011. —
2011. — Article ID 393875. — 8 p.
4. Harnack C. G. A. Über Vieltheiligkeit der ebenen algebraischen Curven // Math. Ann. — 1876. — 10. —
S. 189 – 199.
Received 12.09.13,
after revision — 18.12.13
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 1
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