On a problem for a system of two second-order differential equations via the theory of vector ﬁelds

. We consider Dirichlet boundary value problem for systems of two second-order differential equations with nonlinear continuous and bounded functions in right-hand sides. We prove the existence of a nontrivial solution to the problem comparing behaviors of solutions of auxiliary Cauchy problems at zero solution and at inﬁnity.


Introduction
The system x = −k 2 x, y = − 2 y (1) describes two independent harmonic oscillators.Each of equations in (1) is elementary solvable and the general solution of (1) is known.A couple of functions (x(t), y(t)), which satisfy (1) when visualized on the xy-plane can form somewhat complicated figures.These figures are known as Lissajeux ones and they can be shown using oscilloscopes.
A system x + k 2 x = ϕ(x, y), y is much more complicate due to possibly nonlinear terms in the right-hand sides.This system can be interpreted as description of motion of a particle of unit mass in a force field defined by the vector −k 2 x + ϕ(x, y), − 2 y + ψ(x, y) .
If we are interested in motions which start and end at the origin, we should consider system (2) together with the boundary conditions c Vilnius University, 2015 In this paper, we consider the boundary value problem (2), (3).A vector function (x(t), y(t)) with C 2 [0, 1] components is a solution if it turns (2) to identity and satisfies conditions (3).

Assumption (A1) implies that the homogeneous problem
x + k 2 x = 0, y + 2 y = 0, x(0) = y(0) = x(1) = y(1) = 0 has only the trivial solution.Then in view of the boundedness of the functions ϕ, ψ (A2), problem (2), ( 3) is known to be solvable [5,Chap. 2,§2].Under the assumption (A3), it has the trivial solution.Therefore, conditions for the existence of a nontrivial solution of (2), (3) are desirable.In this paper, we provide the conditions for nontrivial solvability of (2), (3) exploiting properties of two-dimensional vector fields.The paper is organized as follows.Section 2 contains basic theorems of the theory of vector fields and some propositions concerning the vector field induced by solutions of the linear systems.In Section 3, the homotopy results for vector field of a given nonlinear system and vector fields of the respective limiting systems at zero and at infinity are formulated and proved.In the next section, different cases with respect to the coefficients of the limiting linear systems are considered and a rotation of a vector field (winding number) is calculated.The main result about nontrivial solvability of boundary value problem under consideration is formulated in the third subsection of Section 4. In Section 5, an application of the main result is shown, the corresponding examples are analyzed.

Tools
We use the theory of vector fields based on the popular sources [2,3,4].
It is convenient for our purposes to consider the initial conditions (4) in the form The initial values are located on circles C R of radius R, where R varies from zero to infinity: In what follows, we make use of Brower degree deg(Φ, C R , (0, 0)), which is identical with γ(Φ; C R ) -the rotation of a vector field Φ on a circle C R .γ(Φ; C R ) always is an integer, it is the number of revolutions made by a vector Φ(α, β) during point (α, β) runs through a circle C R .
Since our considerations are based on the theory developed in the book [3], we prefer formulate the results, both auxiliary and main ones, in terms of rotations of the vector fields.In order to count a rotation of the vector field Φ, let us transfer the vectors Φ(α, β) such that their beginning will be at the origin O = (0, 0).Then the ends of these vectors will depict a closed curve So, a rotation of the vectors Φ(α, β) is the same as a rotation of the radius-vectors of the points on the curve Γ R , therefore, it is equal to a winding number of a closed curve ) Under a continuous deformation of a vector field and a closed curve, the winding number does not change as long as curve does not pass through a critical point of a field.
So, in accordance with the general theory of planar vector fields, any change in γ(Φ; C R ) = n(Γ R , O), when R varies from zero to infinity, means that the vector field Φ passes through a critical point which is not the origin and this, in turn, means that a nontrivial solution to problem (2), (3) emerges.
with a center at the origin.
Remark.Using substitutions ξ = A 2 x − A 1 y and η = B 2 x − B 1 y, Eq. ( 9) can be reduced to the form Corollary 2. Mapping Φ lin : R 2 → R 2 defined by (5) for the linear system (7) (whatever the coefficients a, b, c, d be) maps every circle C R centered at the origin into either an ellipse or a segment of a straight line centered at the origin.
If a closed Jordan curve Γ in vector field Φ is given by then a winding number of this curve around the origin may be calculated by Poincaré formula [3], [4] Proof.The proof follows from (8), (10) by direct computation.

Comparison of vector fields
We produce our results by comparison of the vector field Φ(α, β) on circles C R of small and large radiuses R.

The vector field at zero
We can get information about solutions around the trivial one using the system of equations of variations Consider the equations of variations (11) together with the initial conditions on the unit circle Since we operate with the initial data given on a circle, we will refer to this problem as a generalized Cauchy problem for Eqs.(11).We can prove the following auxiliary result about solutions of the Cauchy problems (2), (6).Lemma 3. Let (x(t), y(t)) be a solution of the nonlinear Cauchy problem (2), (6) (θ is fixed).If R → 0, then (x(t), y(t)) tends to (Ru(t), Rv(t)), where (u(t), v(t)) is a solution of the linear problem (11), (12).
Proof.A solution (x(t), y(t)) of system (2) subject to the initial conditions (6) can be treated as a solution of the system where ε 1 and ε 2 tend to zero as R tends to zero uniformly in (t, x, y), t ∈ [0, 1], Functions ũ(t) = x(t) − Ru(t) and ṽ(t) = y(t) − Rv(t) satisfy Since ε 1 and ε 2 tend to zero as R → 0, the solution (ũ(t), ṽ(t)) of ( 13), ( 14) uniformly tends to the trivial one as a solution of homogeneous linear Cauchy problem.Therefore, Consider a vector field Φ defined by solutions of ( 2), ( 6) on a circle C R of fixed radius R. Denote by Φ 0 a vector field defined by solutions of the system of variations (11) and consider this field on the unit circle C 1 (denote by Φ 0 | C1 ) and on a circle of fixed radius C R (denote by Φ 0 | C R ).Compare the fields Φ and Φ 0 when radius R tends to zero.Theorem 5. A vector field Φ 0 is a principal part of Φ for sufficiently small R and Proof.Let (x(t; θ), y(t; θ)) = (x(t; R cos θ, R sin θ), y(t; R cos θ, R sin θ)) be a solution of problem (2), ( 6), then on a circle of fixed radius R, Φ = x(1; θ), y(1; θ) .
Proof.The functions z(t) = x(t)/R − z(t) and w(t) = y(t)/R − w(t) satisfy Under assumption (A2), the right-hand sides of the equations in (17) tend to zero as R → ∞, then solution (z(t), w(t)) of ( 17), (18) uniformly tends to the trivial one as a solution of homogeneous linear Cauchy problem.Therefore, Denote by Φ ∞ a vector field defined by solutions of system (15) and consider this field on the unit circle C 1 (denote by Φ ∞ | C1 ) and on a circle of fixed radius C R (denote by Φ ∞ | C R ).Compare the fields Φ and Φ ∞ when radius R tends to infinity.Theorem 6.A vector field Φ ∞ is a principal part of Φ for large enough R and Proof.Let (x(t; θ), y(t; θ)) = (x(t; R cos θ, R sin θ), y(t; R cos θ, R sin θ)) be a solution of problem ( 2), ( 6), then on a circle of fixed radius R, Φ = x(1; θ), y(1; θ) .
If (z(t; θ), w(t; θ)) = (z(t; cos θ, sin θ), w(t; cos θ, sin θ)) is a solution of system (15), which satisfies normalized initial conditions (16), then on the unit circle C 1 , Since system (15) is linear and Lemma 1 is valid, For any point M on a circle C R , a vector field Φ can be represented as In accordance with Lemma 4, a solution (x(t), y(t)) tends to (Rz(t), Rw(t)) as R tends to infinity, thus,

Investigation of limiting linear systems
We have two limiting linear systems, namely, system (11), which describes the behavior of solutions of nonlinear system (2) near the trivial solution and uncoupled linear system (15), which describes the behavior of solutions of (2) at infinity.

Investigation of uncoupled linear system at infinity
The vector field Φ ∞ on the unit circle C 1 can be computed explicitly for solutions of system (15).

Investigation of limiting linear system at zero
The limiting system at zero is given by (11) and after regrouping the terms it looks where The characteristic equation for (20) is A form of solution of the system (20) and therefore a vector field Φ 0 depends on values of a, b, c, d, but in all cases the following results are valid.
Proposition 4. Let u(t; θ), v(t; θ)) be a solution of system (20), which satisfy (12), then , then for 0 θ < 2π, Eqs.(23) define an ellipse Γ 0 = Φ 0 (C 1 ) centered at the origin and Calculations of a winding number of an ellipse Γ 0 around the origin (or calculations of a rotation of a vector field Φ 0 on the unit circle C 1 ) in particular cases was conducted in the work [6].Therefore, we omit the proofs for the cases below and provide only formulations.
hold, then the characteristic equation ( 22) has two pairs of conjugate purely imaginary roots Proposition 5.If conditions (C1) are fulfilled, then hold, then the characteristic equation ( 22) has a pair of opposite real roots and a pair of conjugate purely imaginary roots Proposition 6.If conditions (C2) are fulfilled, then hold, then the characteristic equation ( 22) has a pair of conjugate purely imaginary roots and a double zero Proposition 7. If conditions (C3) are fulfilled, then hold, then the characteristic equation ( 22) has two pairs of opposite real roots hold, then the characteristic equation ( 22) has multiple real roots

Main result
It follows from the above considerations that the following result is true.
A behavior of solutions of given system at infinity may be described by solutions of the linear problem z + 50z = 0, w + 22w = 0.

Conclusions
Solutions of systems of the type (2) define mappings Φ : R 2 → R 2 .These mappings map the circles C R of initial values into closed curves in x(1)y(1)-plane.The image of circle C R is simply ellipse Γ R in case system (2) is linear.A winding number of an ellipse Γ R around the origin O is equal to 1 or −1.
http://www.mii.lt/NAA behavior of solutions of nonlinear system (2) near the trivial solution and at infinity can be described by two limiting linear systems (11) and (15), respectively.Under some additional conditions with respect to the coefficients of limiting linear systems, the winding numbers of ellipses Γ 0 and Γ ∞ are different.This means that the boundary value problem ( 2), (3) has at least one nontrivial solution.
In a case the winding numbers of Γ 0 and Γ ∞ around the origin are equal, no conclusion can be made about nontrivial solvability of problem ( 2), (3), further investigation is needed.