DOI 10.1007/s10958-015-2282-z Journal of Mathematical Sciences, Vol. 205, No. 6, March, 2015

MULTIPLE SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS A. Kirichuka

UDC 517.9

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 consider the related Hamilton system which has more ovals than prescribed by Harnack’s theorem. We give an explanation and consider the Dirichlet boundary-value problem for the system. Precise estimation is given for the number of solutions to the Dirichlet problem.

1. Introduction In the present paper, we consider two problems: Harnack’s theory of 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 condition on a finite interval. The paper has the following structure. In Sec. 2, we describe the Hamilton system, Harnack’s theorem and present definitions. In Sec. 3, we consider the Hamilton system of differential equations related to the Trott curve and analyze how many periodic solutions has this system. In Sec. 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 the final Sec. 5, we summarize the results and make conclusions.

2. Preliminary Results and Definitions Consider a two-dimensional nonlinear system x 0 D f .x; y/; y 0 D g.x; y/:

(2.1)

The critical points of this system are determined from the relations f .x; y/ D g.x; y/ D 0: Suppose that the critical points of system (2.1) are located at .xi I yi /:

The types of the critical points can be determined as a result of linearization of the system. Moreover, their description can be given in the phase plane. Daugavpils University, Parades str., 1, Daugavpils, Latvia; e-mail: [email protected]

Published in Neliniini Kolyvannya, Vol. 17, No. 1, pp. 50–57, January–March, 2014. Original article submitted September 12, 2013; revision submitted December 18, 2013. 768

1072-3374/15/2056–0768

c 2015 �

Springer Science+Business Media New York

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A critical point of system (2.1) is a center if it has a punctured neighborhood covered with nontrivial 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 concentric cycles. Definition 2.3 [3]. A period annulus associated with the central region is called a trivial period annulus. Periodic trajectories of a trivial period annulus encircle exactly one critical point of the type “center.” Definition 2.4 [3]. A period annulus enclosing several (more than one) critical points is called a nontrivial period annulus. In studying periodic solutions of planar systems, it is reasonable to consider the Hamilton systems that can be integrated [1]. A Hamiltonian function H.x; y/ is the first integral of the Hamiltonian system x0 D

@H.x; y/ ; @y

y0 D �

@H.x; y/ @x

because dH @H dx @H dy @H @H @H D C D C dt @x dt dy dt @x @y @y

✓

@H � @x

◆

D 0:

The Hamilton function can be interpreted as the total energy of the described system. For a closed system, it is the sum of the kinetic and potential energies in the system. There is a set of differential equations known as the Hamiltonian equations, which describe the time evolution of the system with H.x; y/ D C: The function H.x; y/ is often called Hamiltonian. The Harnack theorem on algebraic curves can be used to study period annuli. It allows one to construct fairly simple examples of existence of multiple period annuli for polynomials of the lowest degree. We now construct two-dimensional planar systems called Hamilton systems which have many period annuli, as shown in 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 .n � 1/.n � 2/ 1 � .�1/n c C 1: 2 2 Any number of components can be attained in this range of possible values. Definition 2.5. A curve attaining the maximum number of real components is an M -curve. There are Hamilton systems for which the number of period annuli is greater than the number of ovals prescribed by the Harnack theorem for related curves.

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Fig. 1. The Trott curve.

3. System Related to the Trott Curve and Periodic Solutions As an example, we consider a curve of degree four, which (according to Harnack’s theorem) has the maximal number of components equal to four. This is a fourth-degree curve, namely, the Trott curve. It has the maximal number of components exactly equal to four. Definition 3.1. The Trott curve is an algebraic curve described by the equation 144.x 4 C y 4 / � 225.x 2 C y 2 / C 350x 2 y 2 C 81 D 0: The Trott curve (Fig. 1) has four separated ovals, i.e., the maximal possible number for a curve of degree four. Hence, it is an M -curve. We now consider a function H1 .x; y/; a part of the polynomial Trott curve described by the equation H1 .x; y/ D 144.x 4 C y 4 / � 225.x 2 C y 2 / C 350x 2 y 2 ; and the Hamilton system x0 D

@H1 D 576y 3 � 450y C 700x 2 y; @y (3.1)

y0 D �

@H1 D �576x 3 C 450x � 700xy 2 : @x

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Fig. 2. The phase portrait of system (3.1).

System (3.1) has nine critical points, five of them are points of the type “center” and four are points of the type“saddle”: ✓

◆ ✓ ◆ ✓ ◆ ✓ ◆ 15 15 15 15 15 15 15 15 ;p ; �p ; �p ; �p ;p ; p ; �p : p 638 638 638 638 638 638 638 638

The Hamilton system (3.1) contains five trivial period annuli around the center-type points: .0; 0/;

✓

◆ ✓ ◆ ✓ ◆ ✓ ◆ 5 5 �5 �5 p ;0 ; p ; 0 ; 0; p ; 0; p 4 2 4 2 4 2 4 2

and one nontrivial periodic annulus (around all critical points). Thus, in total, we have six period annuli (Fig. 2). Harnack’s theorem speaks about four ovals for the Trott curve. For the corresponding system (3.1), we have more annuli, namely, six period annuli. This is explained by the fact that we consider curves given by the formula H1 .x; y/ D C; where C is arbitrary and a specific Trott curve is obtained for C D �81: 4. A System Related to the Trott Curve and the Dirichlet Boundary-Value Problem We are interested in the number of solutions of the analyzed system satisfying given boundary conditions. Consider system (3.1) with the Dirichlet boundary conditions x.0/ D 0;

x.T / D 0:

(4.1)

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(a)

(b)

(c)

(d )

Fig. 3. The phase portrait of system (3.1). The parts of the phase portrait of system (3.1) are denoted as follows: Region 1 in Fig. 3a; Region 2 in Fig. 3b; Region 3 in Fig. 3c, and Region 4 in Fig. 3d: We perform the linearization of system (3.1). fx0 D 1400xy; fy0 D 1728y 2 � 450 C 700x 2 ; gx0 D �1728x 2 C 450 � 700y 2 ; gy0 D �1400xy:

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The linearized system for system (3.1) at a critical point .x ⇤ ; y ⇤ / has the form u0 D .1400x ⇤ y ⇤ /u C .1728y ⇤2 � 450 C 700x ⇤2 /v; v 0 D .�1728x ⇤2 C 450 � 700y ⇤2 /u C .�1400x ⇤ y ⇤ /v: Consider Region 1 and closed trajectories, which are in a close proximity of the boundary of Region 1. ✓ ◆ ✓ ◆ 15 15 15 15 This boundary consists of the critical points � p ; of the saddle type and ;p ;p p 638 638 638 638 two heteroclinic solutions connecting these points. The motion along heteroclinic solutions is very “slow” in a sense that the time period required to pass from one critical point to another is infinitely large. Denote the point of intersection of the “upper” heteroclinic solution with the y -axis by .0; u⇤ /: As a result of calculations, we get u⇤ ⇡ 1:01:

For periodic solutions of system (3.1) satisfying the initial conditions .0; u0 /; where u0 ⇠ u⇤ ; u0 < u⇤ ; we conclude that the time ⌧ .u0 / required to reach the next point of intersection with the y -axis is arbitrarily large in a sense that ⌧.u0 / ! C1 as u0 ! u⇤ : ✓ ◆ 5 On the other hand, the trajectories in the interior of Region 1 surrounding the critical point 0; p are 4 2 relatively“fast” (Fig. 4). We can compute the time ⌧ .u0 / required to pass from the y -axis to another point on the 5 y -axis, i.e., in fact, the half period for u0 ! p ⇡ 0:884: 4 2 ✓ ◆ 5 To do this, we consider the linearized system for system (3.1) at the critical point 0; p 4 2 u0 D 900v;

(4.2)

v 0 D �96:875u (the value �96:875 is exact).

The eigenvalues of the linearized system (4.2) are �1;2 D ˙295:27529i

and the point

✓

5 0; p 4 2

◆

is a point of the type “center.”

System (4.2) can be rewritten in the form u00 D 900v 0 D �96:875 � 900u D �87187:5u

(4.3)

and the solution of equation (4.3), u.t / D sin

p 87187:5t;

(4.4)

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Fig. 4. Solution of system (3.1) in Region 1; x.t / ⇠ u.t /; u0 D 0:9; u0 D 0:95; and u0 D 1:0 . generates an approximation .u.t /; u0 .t // to the solution of the Cauchy problem for (3.1) with x.0/ D 0 and 5 y.0/ D p ˙ ✏; where ✏ is a small quantity. 4 2 Therefore, in the Dirichlet problem (4.1), (4.4), T D

⇡ ⌧ Dp ; 2 87187:5

where ⌧ is a period of the solution. n⇡  T: Then the Dirichlet problem (3.1), (4.1) Lemma 4.1. Let n be the largest integer such that p 87187:5 in Region 1 has at least 2n nontrivial solutions. Proof. By analyzing the initial-value problems (3.1), .x.0/; y.0// D .0; u0 /; where u0 2 solutions.

✓

◆ 5 5 ⇤ p ; u ; we get n solutions and, for u0 < p ; symmetric solutions, which gives n more 4 2 4 2

The trajectories in Region 2 (Fig. 3b), 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 15 15 15 15 15 15 15 ; �p ; �p ; p : ;p ; �p ;p ; �p p 638 638 638 638 638 638 638 638

The linearized system for system (3.1) at the origin takes the form

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u0 D �450v;

775

(4.5)

v 0 D 450u: The eigenvalues of the linearized system (4.5) are �1;2 D ˙450i and the origin is a point of the type “center.” System (4.5) can be rewritten in the form u00 D �450v 0 D �4502 u; and the equality u.t / D sin 450t generates an approximation .u.t /; u0 .t // to the solution of the Cauchy problem (3.1), x.0/ D 0 , y.0/ D ˙✏; where ✏ is a small quantity. Therefore, in the Dirichlet condition (4.1), T D

⇡ ⌧ D ; 2 450

where ⌧ is a period of solution (4.5). Lemma 4.2. Let m be the largest integer for which m⇡  T: 450 The Dirichlet problem (3.1), (4.1) in Region 2 has at least 2m nontrivial solutions. Proof. The proof of Lemma 4.2 is similar to the proof of Lemma 4.1. The analysis of Region 3 (Fig. 3c) is similar to the analysis performed for Region 1. The results are formulated in the form of the following lemma: Lemma 4.3. Let n be the largest integer such that p

n⇡ 87187:5

 T:

The Dirichlet problem (3.1), (4.1) in Region 3 has at least 2n nontrivial solutions. Proof. The proof of Lemma 4.3 is similar to the proof of Lemma 4.1. The curves located in Region 4 (Fig. 3d) are closed. Our aim is to find the speed of rotation along these trajectories. To this end, we consider the principal (cubic) part of system (3.1) x 0 D 576y 3 C 700x 2 y; y 0 D �576x 3 � 700xy 2 : In the polar coordinates x.t / D r.t / sin ‚.t /;

(4.6)

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y.t / D r.t / cos ‚.t /; system (4.6) takes the form r 0 .t / D �31r 3 .t / sin 4‚.t /;

(4.7)

‚0 .t / D r 2 .t /.576 C 62 sin2 2‚.t //: Lemma 4.4. In Region 4, the trajectories with large u0 D r0 move arbitrarily “fast.” Proof. Consider Region 4, where u > u0 ; u0 > 0; may be arbitrarily large. It is true that ‚0 .t / � 576 8t 2 Œ0; T ç: The solution of system (3.1) with the initial condition .x.0/; y.0// D .0; u0 / has a period ⌧ .u0 / and, moreover, ⌧ .u0 / ! 0 as u0 ! 0: Since the expression in the parentheses on the right-hand side of the second equation in (4.7) is never equal to zero, we conclude that ‚0 .t / is arbitrarily large if r 2 is large. Therefore, in Region 4, the trajectories with large u0 move arbitrarily “fast.” Lemma 4.5. The 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 obtained as a consequence of Lemmas 4.1–4.4: Theorem 4.1. The Dirichlet boundary-value problem (3.1), (4.1) in Regions 1, 2, and 3 has at least 2.2nCm/ nontrivial solutions. In the external Region 4, problem (3.1), (4.1) possesses infinitely many nontrivial solutions (a countable set). 5. Conclusions In analyzing the accumulated results, we conclude that the number of period annuli of the Hamilton systems is much larger than the number of components of the corresponding curve in Harnack’s theorem. However, this does not contradict Harnack’s theorem because, in the last equation, the total number of components exceeds four, i.e., the prescribed number of ovals in Harnack’s theory for a single curve given by H1 D C: For the system of differential equations (3.1) with the Dirichlet boundary conditions (4.1), there exists a nontrivial solution and the number of solutions is infinite (a countable set). The present work was supported by the European Social Foundation within the Project “Support for the Implementation of Doctoral Studies at the Daugavpils University” Agreement No. 2009/0140/1DP/1.1.2.1.2/09/IPIA/ VIAA/015. REFERENCES 1. 2. 3. 4.

St. Lynch, in: Dynamical Systems with Applications Using Mathematica, Birkh¨auser, Boston (2007), pp. 111–123. S. Atslega, Bifurcations in Nonlinear Boundary Value Problems and Multiplicity of Solutions, PhD Thesis, Daugavpils (2010). Y. Kozmina and F. Sadyrbaev, “On a maximal number of period annuli,” in: Abstract Appl. Anal., 2011, Article ID 393,875 (2011). ¨ C. G. A. Harnack, “Uber Vieltheiligkeit der ebenen algebraischen Curven,” Math. Ann., 10, 189–199 (1876).