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International Journal of Differential Equations
Volume 2013 (2013), Article ID 740980, 11 pages
http://dx.doi.org/10.1155/2013/740980
Research Article

Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

1Ehime University, Matsuyama 790-8577, Japan
2Department of Mathematics, University of Zagreb, FER, 10000 Zagreb, Croatia

Received 2 April 2013; Accepted 21 May 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Yūki Naito and Mervan Pašić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at .

1. Introduction

We are concerned with the two-dimensional nonautonomous linear differential systems: where , and , is a matrix-valued function such that , where denotes the space of all real matrices. A solution of linear system (1) is a function , such that .

In appendix of the paper, we show, there exists a unique solution of linear system (1). By the uniqueness of solution of (1), it is clear that if , then for all . It is called the zero solution of linear system (1). The zero solution of linear system (1) is said to be attractive as if for every the corresponding solution of (1) satisfies as , where denotes the standard Euclidean norm on . Since we are dealing with the attractive zero solution of (1), from we conclude that . For a continuous function , let denote a curve in corresponding to determined by where . It is often said that is parametrized by .

Let . The curve is said to be a Jordan curve if is injective (one-to-one) on . Hence, is nonself-intersecting; it has two ends and it is a compact connected set in . In appendix of the paper, see Theorem A.3; we give some simple sufficient conditions on the matrix-valued function such that the solution curve of every nontrivial solution of (1) is a Jordan curve in . The assumption that is a Jordan curve is important for the process of measuring the length of (the rectification of ) as follows. If is a Jordan curve, then the length of is denoted by length and it is defined as usual by (see [13]) where the supremum is taken over all finite dissections of the interval .

Definition 1. The zero solution of linear system (1) is said to be rectifiable (resp., nonrectifiable) attractive as ; if for every such that the of corresponding solution of (1) is a Jordan curve in , one has as and length (resp., length ).

Let , , be two complex conjugate eigenvalues of . The main purpose of the paper is to characterize attractivity, rectifiable and nonrectifiable attractivity, as of the zero solution of system (1) in the terms of regular and singular asymptotic behaviour of near ; see Theorems 2 and 3. It is illustrated on a simple two-dimensional linear differential systems with the polynomial coefficients; see Theorem 4. The key point of proofs of the main results is an asymptotic solution formula for system (1) obtained by its asymptotic integration near ; see Section 3. The proofs of the main results are given in Sections 4, 5, and 6. Such a kind of topics has been considered for the first time in [4] but only in the case of the so-called integrable systems (systems that allow all solutions in explicit forms), where the asymptotic integration near is not required. About the asymptotic integration near of differential equations and systems, we refer reader to [512]. On several asymptotic properties of two-dimensional differential systems near , let us see, for instance, [1318]. Recently interest for studying differential equations and systems on compact (finite time) intervals is rapidly increasing because of its application in atmosphere, fluid, and ocean dynamics and in biological science; see, for instance, [1921] and references therein.

2. Main Results and Consequences

Let for each the matrix elements of be denoted by . In the paper, we suppose the following kind of singular asymptotic behaviour of near : where is a constant matrix having two complex conjugate eigenvalues.

Some essential consequences of (4) are presented in Section 3. Since the eigenvalues of are continuous on , by (4) we may assume that has two complex conjugate eigenvalues for .

The first two main results of the paper are the following theorems.

Theorem 2. Let (4) be satisfied, and let , , be two complex conjugate eigenvalues of . Assume that near . Then the zero solution of linear system (1) is attractive as if and only if

Theorem 3. Let (4) be satisfied, and let , , be two complex conjugate eigenvalues of . Assume that near , and that (5) holds. Then the zero solution of linear system (1) is rectifiable attractive as if and only if

Previous theorem allows us to study the rectifiable and nonrectifiable attractivity for the following model system with polynomial singular coefficients near : where the parameters and coefficients satisfy, respectively, According to (8), (9), and (10), we will show that the model system (7) satisfies all assumptions of Theorem 3. Hence, we are able to derive and prove the following conditions for the rectifiable and nonrectifiable attractivity of the zero solution of model system (7).

Theorem 4. Let (8), (9), and (10) be fulfilled.  If , then the zero solution of (7) is rectifiable attractive as . If , then the zero solution of (7) is rectifiable attractive as provided that one of the following conditions is satisfied:  If , then the zero solution of (7) is nonrectifiable attractive as provided that one of the following conditions is satisfied:

As a consequence of previous theorem, we realize that the rectifiable and nonrectifiable attractivity of the zero solution of model system (7) depends on the order of growth of singular behaviour of coefficients near determined by parameters and . Let us remark that if, besides (8), (9), and (10), it is supposed that , , and , then system (7) is integrable, that is, all solutions of system (7) are explicitly determined, see appendix of the paper. In such a particular case of (7), Theorem 4 can be explicitly verified.

Remark 5. We suggest reader to consider the rectifiable and nonrectifiable attractivity of the zero solution of system (7) in the case of . It is required to assume that if as well as if and if .

The most difficult point of this paper is to derive and prove the nonrectifiable attractivity of the zero solution of systems (1) and (7); see Sections 3 and 4. On the (non)rectifiability of the solution curve of scalar second-order differential equations, we refer reader to [2227].

3. Some Asymptotic Properties Near

As a consequence of the main assumption (4), we derive in this section some important asymptotic properties of as well as of its eigenvalues and solutions of system (1), which will be used in the next sections in the proof of the main results. Since the eigenvalues of are continuous on , by (4) we may assume that has two complex conjugate eigenvalues , .

Lemma 6. Let (4) be satisfied. Then the eigenvalues of satisfy where means that as .

Proof. Let , , be matrix elements of . Also, let and be matrix elements and eigenvalues of , respectively, where appears in (4). Rewriting (4) in terms of matrix elements and eigenvalues, we get Hence, we can conclude the following: It proves that as . Hence (18) and the second claim in (17) hold. In order to complete (17), it rests to show that as . In fact, from the assumption , we observe that for at least one matrix element we have and, therefore, . Moreover, since every matrix norm is a continuous function, from (4) we especially obtain . It completes the proof of (17).

As the main consequence of (4) we have the following asymptotic formula near for all solutions of linear system (1): where , , and there are such that It is the subject of the next lemma.

Lemma 7. Let be two eigenvalues of . If satisfies (4), then for every solution of linear system (1) with there exist two functions and two constants such that (21) and (22) are fulfilled.

Proof. Let be a matrix satisfying all assumptions of this lemma. Let be a new matrix depending on a new variable defined by Then . Let be a constant matrix determined in (4). If is a new matrix defined by , then from (4), (23), and , we conclude where is a constant matrix having two complex conjugate eigenvalues.
Also, the eigenvalues of and , of are obviously related by the equality and hence, Putting into system (1) and denoting by we obtain the equivalent system of (1) where matrix is defined in (23). For system (28), we will use the next generalized Matell’s theorem appearing as Theorem 11 in Coppel’s book [5, Chapter 4] and Theorem 6.5 in Kiguradze’s monograph [7] (see also [12, Theorem 7] and [8]).
Theorem (see [7]). Let and , be two eigenvalues of an arbitrary matrix , , with the matrix elements which are absolutely continuous functions on every compact set and Let be a constant such thatLet as , where is a constant matrix having two different eigenvalues. Then linear differential system (28) has a fundamental system of solutions and such that where are two eigenvectors of constant matrix .
Now, with the help of (24) and (26), it is easy to check that the matrix defined in (23) and its eigenvalues and defined in (25) satisfy all required assumptions of Theorem 8. Hence, by Theorem 8, we observe that linear system (1) has a fundamental system of solutions and satisfying Indeed, from (31) and using , , (25), (27), and , we obtain Analogously, the second equality in (32) can be proved. Now, from (32) we particularly obtain
Denoting from (34), we get and near , Since , , and , previous statements verify the desired asymptotic formula (21). The statement (22) holds because and are two eigenvectors and hence, .

Now, according to the asymptotic formula (21), we are able to prove Theorem 2.

Proof of Theorem 2. Because of (4) and Lemmas 6 and 7, we can use (17) and (18) and the asymptotic solution’s formula (21)-(22).
At first, we suppose that (5) holds. Taking the norm on both sides in (21) and using (22), we obtain the upper estimate with some constant . Now putting the limit as on both sides in (37), and using (5), we get as . Thus, it is shown that the zero solution of system (1) is attractive as provided that (5) holds.
The proof of the reverse claim is slightly complicated. Suppose that Next, from (17), and (18), we especially obtain Remarking that in (21) and (22), we use the following notations: By (22), at least one of and is nonzero. So, for instance, , which together by (22) gives Now, from (21) and (40), we derive the following inequality: Next, because of (18) and (39), there is a sequence , as , such that Now, putting into (42) and using (41), (43), and , we get the inequality Taking the limit on both sides in previous inequality, with the help of (38), we conclude that that is, Since near , the function is continuous (since matrix elements of are continuous functions on ) and monotone. Hence, statement (46) proves (5). Thus, the statement (5) holds provided that the zero solution of system (1) is attractive.

4. Rectifiable Attractivity of the Zero Solution

Unlike the nonrectifiable attractivity which will be studied in the Section 5, the rectifiable attractivity of the zero solution is proved without any essential difficulties. It is because we can use here the following known result.

Lemma 9. Let be a Jordan curve in . Then one has if and only if .

For the proof of previous lemma, we suggest reader to consult the appropriate scalar case shown in [28].

The following theorem derives some properties of and its eigenvalues which ensure that the zero solution of system (1) is rectifiable attractive.

Theorem 10. Let be two complex conjugate eigenvalues of for . Assume that (5) holds. Let every solution of system (1) satisfies (37). If then the zero solution of (1) is rectifiable attractive as .

Proof. It is clear that the attractivity of the zero solution of system (1) follows from (5), (21), and (37). Next, by taking the matrix norm in (1) and using a priori estimate (37), we obtain, Now, from the assumption (47) and the inequality (48), we observe that Hence, Lemma 9 proves this theorem.

If admits the asymptotic behaviour given in (4), then a priori estimate (37) is fulfilled because of the asymptotic formula (21) and thus, the rectifiable attractivity of the zero solution of system (1), in such a case, holds without supposing (37), as follows.

Theorem 11. Let (4) be satisfied, and let be two complex conjugate eigenvalues of for . Assume that (5) holds. If the condition (47) is fulfilled, then the zero solution of linear system (1) is rectifiable attractive as .

Proof. It is clear that all assumptions of Theorem 11 ensure that Lemma 7 can be applied. Hence, by Lemma 7, that is, by asymptotic formula (21), one can easily show that a priori estimate (37) holds for all solutions of system (1). Thus, all assumptions of Theorem 10 are fulfilled. Now, Theorem 10 proves Theorem 11.

5. Nonrectifiable Attractivity of the Zero Solution

In this section we study the nonrectifiable attractivity for linear system (1). Unlike the rectifiable attractivity, the nonrectifiable attractivity is a more difficult case. It is because the required property cannot be derived from system (1) and therefore, we are not in an opportunity to use Lemma 9. Hence, instead of Lemma 9, we state and prove the next new lemma, which plays an essential role in the proof of the main results of this section.

Lemma 12. Let , , and let there be a sequence , a number , and two functions , , , such that sequence is decreasing, as and for all ; function , , is strictly monotone near , that is, near and there is a constant such that  function , , is nondecreasing and there is a constant such that .
Then the graph of function is a nonrectifiable curve in .

Proof. Because of , there is a sequence such that for all and hence we have Let be a sequence defined by Obviously for all , the set makes a finite partition of interval and since is a supremum of all sums taken over all finite partitions of ; we conclude that Next, let, for instance, be strictly decreasing near (the case when is strictly increasing could be analogously considered). Hence, from and , it follows that Now from (55) and previous inequality, for all we obtain Letting in previous inequality, the hypothesis proves this lemma.

Now, we are able to prove the following result.

Theorem 13. Let be two complex conjugate eigenvalues of for . Assume that and near and that (5) holds. Let every solution of system (1) satisfy (37) as well as the following assumption: let be defined by or and let there be two constants , a number , and a decreasing sequence , as , which all depend on function , such that for all , one has If then the zero solution of (1) is nonrectifiable attractive as .

Proof. At first, the attractivity of the zero solution of linear system (1) follows from the assumptions (5) and (37).
Next, let and denote two new functions defined by Let or , where is a solution of system (1). With the help of notation (60) and assumption (58), it is easy to check that the function satisfies all hypotheses , , and from Lemma 12. Indeed, the existence of a sequence satisfying follows from the first inequality in (58). Also, from (18), (58), and (60) we obtain which show that and from Lemma 12 are fulfilled too. Moreover, the equality together with assumption (59) ensure that the hypothesis from Lemma 12 is also fulfilled. Now, by Lemma 12 we obtain that the graph of or is a nonrectifiable curve in . In particular, it follows that the solution curve is also a nonrectifiable curve in , where .

In the next theorem, the nonrectifiable attractivity is obtained without supposing the a priori estimates (37) and (58), since they immediately follow from the asymptotic behaviour of near given in (4).

Theorem 14. Let (4) be satisfied, and let be two complex conjugate eigenvalues of the matrix for . Assume that near and that (5) holds. If the condition (59) holds, then the zero solution of (1) is nonrectifiable attractive as .

Proof. This proof is a consequence of Theorem 13 and the asymptotic formula (21). At first, according to Lemma 7, every solution of system (1) satisfies (21). If we denote by or , then (21) ensures that could be written in the form where with , , , and are defined in (60).
Since is a continuous complex-valued function on , from (18), we conclude that Therefore, there are two real constants and such that , and
Because of (59), we have as and therefore there is a decreasing sequence and a number such that Putting (66) into (65) and retaking as sufficiently large, we get and because of (64) and (66), Now, statements (67) and (68) show that the function and sequence defined, respectively, in (65) and (66) satisfy the required condition (58). Therefore, the main conclusion of this theorem immediately follows from Theorem 13.

6. Proof of Theorem 4

In this section, we prove the rectifiable and nonrectifiable attractivity for the zero solution of model system (7) as a direct consequence of Theorem 3.

Let be the matrix of the model system (7) given by where the real numbers , , and functions , , , and satisfy (8), (9), and (10), respectively. We will show that such a matrix together with its eigenvalues satisfy the required conditions near , (5), and (4).

At first, because of (9), we have .

Next, since , let, for instance, , and ; from (9) we obtain a constant such that Now, it is clear that (70) proves the desired statement (5).

Finally, since and , where is a constant matrix. Since , matrix has two different eigenvalues. So, the condition (4) is fulfilled too.

Thus, matrix together with its eigenvalues satisfy all required conditions so that Theorem 3 can be applied to the model system (7) in all cases of and given in (11)–(16). Since by Theorem 3 the rectifiable and nonrectifiable attractivity of the zero solution of (1) depends on the integrability of function we can conclude that the zero solutions of (7) is rectifiable attractive if and nonrectifiable attractive if . Let us remark that in Lemma 6, it has been proved that

(i) Assume that . For instance, we assume . Since on , we have as . In particular, there exist and such that Since we have where the constant . Now from (72), (73), and (76), we observe that Thus, it is shown that .

(ii) Assume that (11) is fulfilled. Since on , there exists such that on . Since and on , from (75) we have where the constant . Now from (72), (73), and (78), we obtain . When (12) is fulfilled, we obtain by the similar argument.

Assume that (13) is fulfilled. Since on , there exists such that on . From (75) we have where the constant . Thus we obtain by the argument above.

(iii) Let (14) be fulfilled. Since , there is a constant such that on , and since there is a constant such that

Hence, Now, by combining (72), (73), and (81), we observe Thus, . When (15) is fulfilled, we similarly obtain .

Let (16) be fulfilled. Then, from (75) we have By combining (72), (73), and (83), we obtain. Thus, the proof is completed.

Appendix

At first, we state and prove the existence and uniqueness of solutions of linear system (1).

Theorem A.1. There exists a unique solution of linear system (1).

Proof. Putting into system (1) and denoting , we obtain the equivalent system where and the matrix is defined by (23). By the standard theory for system (A.1), we obtain the existence and uniqueness of the solution to (1).

At the second, we give some simple sufficient conditions on the matrix-valued function such that the solution curve of every nontrivial solution of linear system (1) (as well as of (7)) is a Jordan curve.

Definition A.2. A solution of linear system (1) is said to be trivial solution of (1) near if there is such that for all . Otherwise, is said to be nontrivial solution of (1) near .

Theorem A.3. Let the matrix elements of satisfy If is a nontrivial solution of linear system (1) near , then is strictly positive and decreasing on . In particular, the solution curve is a Jordan curve.

Remark A.4. It is clear that if , and on , then condition (A.2) is fulfilled.

Proof of Theorem A.3. Let be a nontrivial solution of system (1) near . The assumption (A.2) can be rewritten as
Let be a function defined by Taking the derivative in (A.4), it follows that Multiplying the first and second equation in system (1), respectively, by and , we obtain Because of inequality , we have which together with (A.6) derives that where denotes the function Putting (A.8) into (A.5), we obtain Because of (A.3), we have on . Hence, from (A.10), we observe on . Integrating this inequality over the interval for all , we get . If we suppose for a moment that , then from previous inequality we get for all , which is not possible because of Definition A.2. Therefore, for all , which implies for all because of (A.4). It proves the first statement of this theorem.
Next, with the help of (A.3) and for all , we deduce that on , which together with (A.10) gives on . Integrating this inequality over the interval for all , we get If, for instance, there exist such that and , then involving this equality into (A.4), we obtain , which is not possible because of (A.11). Therefore, . Thus, it is shown that is a Jordan curve near . In the same way, for all such that .

According to Remark A.4, previous theorem can be applied to model system (7).

Corollary A.5. Let , , , and near . If is a nontrivial solution of linear system (7) near , then is strictly positive and decreasing on . In particular, the solution curve is a Jordan curve.

Finally, we point out that in some essential cases, the model system (7) allows the explicit form of all its solutions. Such systems are called the integrable systems; for details, see [4]. Hence, the rectifiable and nonrectifiable attractivity of the zero solution of (7) in such cases can be reproven in an explicit way. That is, the statements of Theorem 4 can be confirmed once more, but explicitly.

Proposition A.6. Let and on . The fundamental system of all solutions of linear differential system is explicitly given by the formula

Obviously for , and on , the system (A.12) is a particular case of our model system (7). Hence, according to Theorem 4, we can state the rectifiable and nonrectifiable attractivity near for the system (A.12) as follows.

Corollary A.7. Let and on .(i)If , then the zero solution of (A.12) is rectifiable attractive as .(ii)If , then the zero solution of (A.12) is rectifiable attractive as provided that.(iii)If , then the zero solution of (A.12) is nonrectifiable attractive as provided that.

However, since by Proposition A.6 the system (A.12) allows the explicit form of all its solutions, this means that Corollary A.7 can be also shown by using Lemma 9, Lemma 12, and Proposition A.6.

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