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

Abstract

We study a new kind of asymptotic behaviour near 𝑡=0 for the nonautonomous system of two linear differential equations: 𝐱^′(𝑡)=𝐴(𝑡)𝐱(𝑡), 𝑡∈(0,𝑡₀], where the matrix-valued function 𝐴=𝐴(𝑡) has a kind of singularity at 𝑡=0. It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that ‖𝐱(𝑡)‖₂→0 as 𝑡→0 and the length of the solution curve of 𝐱 is finite (resp., infinite) for every 𝐱≠𝟎. It is characterized in terms of certain asymptotic behaviour of the eigenvalues of 𝐴(𝑡) near 𝑡=0. Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at 𝑡=0.

1. Introduction

We are concerned with the two-dimensional nonautonomous linear differential systems: 𝑑𝐱(𝑡)𝑑𝑡=𝐴𝐱for𝑡∈0,𝑡₀,𝐱𝑡₀=𝐱₀∈ℝ², (1) where 𝑡₀>0, and 𝐴=𝐴(𝑡), 𝐴∶(0,𝑡₀]→ℳ₂ is a matrix-valued function such that 𝐴∈𝐶¹((0,𝑡₀];ℳ₂), where ℳ₂ denotes the space of all 2×2 real matrices. A solution of linear system (1) is a function 𝐱=𝐱(𝑡), 𝐱∶(0,𝑡₀]→ℝ² such that 𝐱∈𝐶¹((0,𝑡₀],ℝ²).

In appendix of the paper, we show, there exists a unique solution 𝐱 of linear system (1). By the uniqueness of solution 𝐱 of (1), it is clear that if 𝐱₀=(0,0), then 𝐱(𝑡)=(0,0) for all 𝑡∈(0,𝑡₀]. It is called the zero solution of linear system (1). The zero solution of linear system (1) is said to be attractive as 𝑡→0 if for every 𝐱₀∈ℝ² the corresponding solution 𝐱 of (1) satisfies ‖𝐱(𝑡)‖₂→0 as 𝑡→0, where ‖⋅‖₂ denotes the standard Euclidean norm on ℝ². Since we are dealing with the attractive zero solution of (1), from 𝐱∈𝐶¹((0,𝑡₀],ℝ²) we conclude that 𝐱∈𝐶([0,𝑡₀],ℝ²)∩𝐶¹((0,𝑡₀],ℝ²). For a continuous function 𝐱∶[0,𝑡₀]→ℝ², let Γ_𝐱 denote a curve in ℝ² corresponding to 𝐱 determined by Γ_𝐱=𝐱0,𝑡₀=𝑥,𝑦∈ℝ²∶𝑥=𝑥(𝑡),𝑦=𝑦(𝑡), (2) where (𝑥(𝑡),𝑦(𝑡))≡𝐱(𝑡). It is often said that Γ_𝐱 is parametrized by 𝐱.

Let 𝐱∈𝐶([0,𝑡₀];ℝ²). The curve Γ_𝐱⊆ℝ² is said to be a Jordan curve if 𝐱 is injective (one-to-one) on [0,𝑡₀]. Hence, Γ_𝐱 is nonself-intersecting; it has two ends and it is a compact connected set in ℝ². In appendix of the paper, see Theorem A.3; we give some simple sufficient conditions on the matrix-valued function 𝐴=𝐴(𝑡) such that the solution curve Γ_𝐱 of every nontrivial solution of (1) is a Jordan curve in ℝ². The assumption that Γ_𝐱 is a Jordan curve is important for the process of measuring the length of Γ_𝐱 (the rectification of Γ_𝐱) as follows. If Γ_𝐱 is a Jordan curve, then the length of Γ_𝐱 is denoted by length (Γ_𝐱) and it is defined as usual by (see [1–3]) lengthΓ_𝐱=sup^𝑚_𝑘=1𝐱𝑡_𝑘−𝐱𝑡_𝑘−12∶𝑚∈ℕ,0=𝑡_𝑚<⋯<𝑡₁<𝑡₀, (3) where the supremum is taken over all finite dissections 0=𝑡_𝑚<⋯<𝑡₁<𝑡₀ of the interval [0,𝑡₀].

Definition 1. The zero solution of linear system (1) is said to be rectifiable (resp., nonrectifiable) attractive as 𝑡→0; if for every 𝐱₀∈ℝ²⧵{𝟎} such that the Γ_𝐱 of corresponding solution 𝐱 of (1) is a Jordan curve in ℝ², one has ‖𝐱(𝑡)‖₂→0 as 𝑡→0 and length (Γ_𝐱)<∞ (resp., length (Γ_𝐱)=∞).

Let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡), 𝑡∈(0,𝑡₀], be two complex conjugate eigenvalues of 𝐴(𝑡). The main purpose of the paper is to characterize attractivity, rectifiable and nonrectifiable attractivity, as 𝑡→0 of the zero solution of system (1) in the terms of regular and singular asymptotic behaviour of 𝐴(𝑡) near 𝑡=0; 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 𝑡=0; 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 𝑡=0 is not required. About the asymptotic integration near 𝑡=∞ of differential equations and systems, we refer reader to [5–12]. On several asymptotic properties of two-dimensional differential systems near 𝑡=∞, let us see, for instance, [13–18]. 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, [19–21] and references therein.

2. Main Results and Consequences

Let for each 𝑡∈(0,𝑡₀] the matrix elements of 𝐴(𝑡) be denoted by (𝑎_𝑖,𝑗(𝑡))_{𝑖,𝑗≤2}. In the paper, we suppose the following kind of singular asymptotic behaviour of 𝐴(𝑡) near 𝑡=0: lim_𝜀→0^𝑡0_𝜀𝑑𝑡𝑑𝑡²𝑎_𝑖,𝑗(𝑡)𝑑𝑡<∞,𝑖,𝑗∈{1,2},lim_𝑡→0𝑡²𝐴(𝑡)=𝐴₀, (4) where 𝐴₀ is a constant matrix having two complex conjugate eigenvalues.

Some essential consequences of (4) are presented in Section 3. Since the eigenvalues of 𝐴(𝑡) are continuous on 𝑡∈(0,𝑡₀], by (4) we may assume that 𝐴(𝑡) has two complex conjugate eigenvalues 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡) for 𝑡∈(0,𝑡₀].

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

Theorem 2. Let (4) be satisfied, and let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡), 𝑡∈(0,𝑡₀], be two complex conjugate eigenvalues of 𝐴(𝑡). Assume that Re𝜆(𝑡)≥0 near 𝑡=0. Then the zero solution of linear system (1) is attractive as 𝑡→0 if and only if lim_𝑡→0^𝑡0_𝑡Re𝜆(𝑠)𝑑𝑠=∞. (5)

Theorem 3. Let (4) be satisfied, and let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡), 𝑡∈(0,𝑡₀], be two complex conjugate eigenvalues of 𝐴(𝑡). Assume that Re𝜆(𝑡)≥0 near 𝑡=0, and that (5) holds. Then the zero solution of linear system (1) is rectifiable attractive as 𝑡→0 if and only if Im𝜆(𝑡)𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}∈𝐿¹0,𝑡₀. (6)

Previous theorem allows us to study the rectifiable and nonrectifiable attractivity for the following model system with polynomial singular coefficients near 𝑡=0: 𝑥^′(𝑡)=𝑒(𝑡)𝑡^−𝑝𝑥(𝑡)+𝑓(𝑡)𝑡⁻²𝑦(𝑡),𝑦^′(𝑡)=−𝑔(𝑡)𝑡⁻²𝑥(𝑡)+ℎ(𝑡)𝑡^−𝑞𝑦(𝑡),𝑡∈0,𝑡₀, (7) where the parameters 𝑝,𝑞∈ℝ and coefficients 𝑒,𝑓,𝑔,ℎ∈𝐶([0,𝑡₀])∩𝐶¹((0,𝑡₀]) satisfy, respectively, 𝑓(0)𝑔(0)(𝑡)(𝑡)1≤max𝑝,𝑞<2,>0,𝑒>0,ℎ>0,𝑡∈0,𝑡₀,∫^𝑡0₀𝑓^′(𝑡)𝑑𝑡<∞,∫^𝑡0₀𝑔^′(𝑡∫)𝑑𝑡<∞,_𝑡00𝑑𝑡𝑑𝑡^2−𝑝𝑒(𝑡)∫𝑑𝑡<∞,_𝑡00𝑑𝑡𝑑𝑡^2−𝑞ℎ(𝑡)𝑑𝑡<∞. (8)(9)(10) 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. (i) If max{𝑝,𝑞}>1, then the zero solution of (7) is rectifiable attractive as 𝑡→0.(ii) If max{𝑝,𝑞}=1, then the zero solution of (7) is rectifiable attractive as 𝑡→0 provided that one of the following conditions is satisfied: (𝑡)(𝑡)(𝑡)(𝑡)𝑝=1,𝑞<1,𝑒>2;𝑝<1,𝑞=1,ℎ>2;𝑝=1,𝑞=1,𝑒+ℎ>2. (11)(12)(13)(iii) If max{𝑝,𝑞}=1, then the zero solution of (7) is nonrectifiable attractive as 𝑡→0 provided that one of the following conditions is satisfied: (𝑡)(𝑡)(𝑡)(𝑡)𝑝=1,𝑞<1,𝑒≤2;𝑝<1,𝑞=1,ℎ≤2;𝑝=1,𝑞=1,𝑒+ℎ≤2. (14)(15)(16)

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 𝑡=0 determined by parameters 𝑝 and 𝑞. Let us remark that if, besides (8), (9), and (10), it is supposed that 𝑒(𝑡)≡ℎ(𝑡), 𝑓(𝑡)≡𝑔(𝑡), and 𝑝=𝑞, 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 max{𝑝,𝑞}=2. It is required to assume that 4𝑓(0)𝑔(0)−𝑒²(0)>0 if 𝑝=2>𝑞 as well as 4𝑓(0)𝑔(0)−ℎ²(0)>0 if 𝑞=2>𝑝 and 4𝑓(0)𝑔(0)−(𝑒(0)−ℎ(0))²>0 if 𝑝=𝑞=2.

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 [22–27].

3. Some Asymptotic Properties Near 𝑡=0

As a consequence of the main assumption (4), we derive in this section some important asymptotic properties of 𝐴(𝑡) 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 𝐴(𝑡) are continuous on (0,𝑡₀], by (4) we may assume that 𝐴(𝑡) has two complex conjugate eigenvalues 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡), 𝑡∈(0,𝑡₀].

Lemma 6. Let (4) be satisfied. Then the eigenvalues of 𝐴(𝑡) satisfy ‖𝐴(𝑡)‖∼|Im𝜆(𝑡)|∼𝑡⁻²(𝑡)𝑎𝑠𝑡⟶0,Im𝜆≠0𝑛𝑒𝑎𝑟𝑡=0, (17)(18) where 𝑓(𝑡)∼𝑔(𝑡) means that 𝑓(𝑡)/𝑔(𝑡)→1 as 𝑡→0.

Proof. Let 𝑎_𝑖,𝑗=𝑎_𝑖,𝑗(𝑡), 𝑎_𝑖,𝑗∈𝐶¹((0,𝑡₀]), be matrix elements of 𝐴(𝑡). Also, let 𝑎⁰_𝑖,𝑗 and 𝜆₀=Re𝜆₀±𝑖Im𝜆₀ be matrix elements and eigenvalues of 𝐴₀, respectively, where 𝐴₀ appears in (4). Rewriting (4) in terms of matrix elements and eigenvalues, we get lim_𝑡→0𝑡²𝑎_𝑖,𝑗(𝑡)=𝑎⁰_𝑖,𝑗,Im𝜆₀≠0. (19) Hence, we can conclude the following: lim_𝑡→0𝑡²|(𝑡=1Im𝜆)|2lim_𝑡→0𝑡²−𝑎_1,1(𝑡)−𝑎_2,2(𝑡)²−4𝑎_1,2(𝑡)𝑎_2,1(𝑡=1)2lim_𝑡→0−𝑡²𝑎_1,1(𝑡)−𝑡²𝑎_2,2(𝑡)²−4𝑡²𝑎_1,2(𝑡)𝑡²𝑎_2,1(𝑡)=12−𝑎⁰_1,1−𝑎⁰_2,2²−4𝑎⁰_1,2𝑎⁰_2,1=Im𝜆₀≠0. (20) It proves that |Im𝜆(𝑡)|∼𝑡⁻² as 𝑡→0. Hence (18) and the second claim in (17) hold. In order to complete (17), it rests to show that ‖𝐴(𝑡)‖∼𝑡⁻² as 𝑡→0. In fact, from the assumption Im𝜆₀≠0, we observe that for at least one matrix element 𝑎⁰_𝑖,𝑗 we have 𝑎⁰_𝑖,𝑗≠0 and, therefore, ‖𝐴₀‖≠0. Moreover, since every matrix norm is a continuous function, from (4) we especially obtain lim_𝑡→0[𝑡²‖𝐴(𝑡)‖]=‖𝐴₀‖≠0. It completes the proof of (17).

As the main consequence of (4) we have the following asymptotic formula near 𝑡=0 for all solutions 𝐱 of linear system (1): 𝐱(𝑡)=𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}𝐡₁(𝑡)cos^𝑡0_𝑡Im𝜆(𝑠)𝑑𝑠+𝐡₂(𝑡)sin^𝑡0_𝑡Im𝜆(𝑠)𝑑𝑠, (21) where 𝐡₁=𝐡₁(𝑡), 𝐡₂=𝐡₂(𝑡), 𝐡_𝑖∶(0,𝑡₀]→ℝ² and there are 𝐜₁,𝐜₂∈ℝ² such that lim_𝑡→0𝐡_𝑖(𝑡)=𝐜_𝑖≠𝟎,𝑖∈{1,2}. (22) It is the subject of the next lemma.

Lemma 7. Let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡) be two eigenvalues of 𝐴(𝑡). If 𝐴(𝑡) satisfies (4), then for every solution 𝐱 of linear system (1) with 𝐱₀≠{𝟎} there exist two functions 𝐡_𝑖(𝑡) and two constants 𝐜₁,𝐜₂∈ℝ² such that (21) and (22) are fulfilled.

Proof. Let 𝐴(𝑡) be a matrix satisfying all assumptions of this lemma. Let 𝐵(𝜏)=(𝑏_𝑖,𝑗(𝜏)_{𝑖,𝑗≤2}) be a new matrix depending on a new variable 𝜏∶=1/𝑡 defined by 𝐵(𝜏)=−𝜏⁻²𝐴𝜏⁻¹∀𝜏≥𝜏₀1∶=𝑡₀. (23) Then 𝑏_𝑖,𝑗∈𝐶¹([𝜏₀,∞)). Let 𝐴₀ be a constant matrix determined in (4). If 𝐵₀ is a new matrix defined by 𝐵₀=−𝐴₀, then from (4), (23), and 𝑡=1/𝜏, we conclude lim_𝑀→∞^𝑀_𝜏0𝑑𝑏𝑑𝜏^𝑖,𝑗(𝜏)𝑑𝜏<∞,𝑖,𝑗∈{1,2},lim_𝜏→∞𝐵(𝜏)=𝐵₀, (24) where 𝐵₀ is a constant matrix having two complex conjugate eigenvalues.
Also, the eigenvalues 𝜇₁(𝜏),𝜇₂(𝜏) of 𝐵(𝜏) and 𝜆₁(𝑡)=Re𝜆(𝑡)+𝑖Im𝜆(𝑡), 𝜆₂(𝑡)=Re𝜆(𝑡)−𝑖Im𝜆(𝑡) of 𝐴(𝑡) are obviously related by the equality 𝜇_𝑖(𝜏)=−𝜏⁻²𝜆_𝑖𝜏⁻¹=−𝜏⁻²Re𝜆𝜏⁻¹±𝑖Im𝜆𝜏⁻¹,𝑖=1,2, (25) and hence, Re𝜇₁(𝜏)−𝜇₂(𝜏)=0∀𝜏≥𝜏₀. (26) Putting 𝑡=1/𝜏 into system (1) and denoting by 𝐰(𝜏)1∶=𝐱𝜏for𝜏≥𝜏₀, (27) we obtain the equivalent system of (1) 𝐰^′(𝜏)=𝐵(𝜏)𝐰(𝜏)for𝜏≥𝜏₀, (28) where matrix 𝐵(𝜏) 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 8 (see [7]). Let 𝜇₁=𝜇₁(𝜏) and 𝜇₂=𝜇₂(𝜏), 𝜇₁,𝜇₂∶[𝜏₀,∞)→ℂ be two eigenvalues of an arbitrary 2×2 matrix 𝐵(𝜏), 𝜏∈[𝜏₀,∞), with the matrix elements 𝑏_𝑖,𝑗(𝜏) which are absolutely continuous functions on every compact set 𝐼⊂[𝜏₀,∞) and ^∞_𝜏0𝑑𝑏𝑑𝜏^𝑖,𝑗(𝜏)𝑑𝜏<∞,𝑖,𝑗∈{1,2}. (29)Let 𝑐>0 be a constant such thatRe𝜇^𝜏∗_𝜎∗1(𝜏)−𝜇₂(𝜏)𝑑𝜏≤𝑐,𝜏₀≤𝜎_∗<𝜏_∗,𝑜𝑟_∞𝜏0Re𝜇₁(𝜏)−𝜇₂(𝜏)d𝜏=∞,_{𝜏∗𝜎∗}Re𝜇₁(𝜏)−𝜇₂(𝜏)𝑑𝜏≥−𝑐,𝜏₀≤𝜎_∗<𝜏_∗. (30)Let 𝐵(𝜏)→𝐵₀ as 𝜏→∞, where 𝐵₀ is a constant matrix having two different eigenvalues. Then linear differential system (28) has a fundamental system of solutions 𝐰₁(𝜏)∈ℂ² and 𝐰₂(𝜏)∈ℂ² such that lim_𝜏→∞𝐰₁(𝜏)𝑒^{−∫𝜏𝜏0𝜇1(𝜎)𝑑𝜎}=𝐛₁,lim_𝜏→∞𝐰₂(𝜏)𝑒^{−∫𝜏𝜏0𝜇2(𝜎)𝑑𝜎}=𝐛₂, (31) where 𝐛₁,𝐛₂∈ℂ² are two eigenvectors of constant matrix 𝐵₀.
Now, with the help of (24) and (26), it is easy to check that the matrix 𝐵(𝜏) defined in (23) and its eigenvalues 𝜇₁(𝜏) and 𝜇₂(𝜏) 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 𝐱₁(𝑡) and 𝐱₂(𝑡) satisfying lim_𝑡→0𝐱₁(𝑡)𝑒^{∫𝑡0𝑡𝜆1(𝑠)𝑑𝑠}=𝐛₁,lim_𝑡→0𝐱₂(𝑡)𝑒^{∫𝑡0𝑡𝜆2(𝑠)𝑑𝑠}=𝐛₂. (32) Indeed, from (31) and using 𝜏=1/𝑡, 𝜏₀=1/𝑡₀, (25), (27), and 𝜎=1/𝑠, we obtain 𝐛₁=lim_𝜏→∞𝐰₁(𝜏)𝑒^{−∫𝜏𝜏0𝜇1(𝜎)𝑑𝜎}=lim_𝑡→0𝐰₁1𝑡𝑒^{−∫01/𝑡1/𝑡𝜇1(𝜎)𝑑𝜎}=lim_𝑡→0𝐰₁1𝑡𝑒^{∫01/𝑡1/𝑡𝜎−2𝜆1(𝜎−1)𝑑𝜎}=lim_𝑡→0𝐱₁(𝑡)𝑒^{∫𝑡0𝑡𝜆1(𝑠)𝑑𝑠}. (33) Analogously, the second equality in (32) can be proved. Now, from (32) we particularly obtain lim_𝑡→0𝐱₁(𝑡)𝑒^{∫𝑡0𝑡𝜆1(𝑠)𝑑𝑠}−𝐛₁=𝟎,lim_𝑡→0𝐱₂(𝑡)𝑒^{∫𝑡0𝑡𝜆2(𝑠)𝑑𝑠}−𝐛₂=𝟎. (34)
Denoting 𝐠₁(𝑡)=𝐱₁(𝑡)𝑒^{∫𝑡0𝑡𝜆1(𝑠)𝑑𝑠}−𝐛₁,𝐠₂(𝑡)=𝐱₂(𝑡)𝑒^{∫𝑡0𝑡𝜆2(𝑠)𝑑𝑠}−𝐛₂, (35) from (34), we get lim_𝑡→0𝐠_𝑖(𝑡)=𝟎,𝑖∈{1,2} and near 𝑡=0, 𝐱₁(𝑡)=𝐛₁+𝐠₁(𝑡)𝑒^{−∫𝑡0𝑡𝜆1(𝑠)𝑑𝑠},𝐱₂(𝑡)=𝐛₂+𝐠₂(𝑡)𝑒^{−∫𝑡0𝑡𝜆2(𝑠)𝑑𝑠}. (36) Since 𝜆₁(𝑡)=Re𝜆(𝑡)+𝑖Im𝜆(𝑡), 𝜆₂(𝑡)=Re𝜆(𝑡)−𝑖Im𝜆(𝑡), 𝐛_𝑖∈ℂ² and 𝐠_𝑖∶(0,𝑡₀]→ℂ², previous statements verify the desired asymptotic formula (21). The statement (22) holds because 𝐛₁ and 𝐛₂ are two eigenvectors and hence, 𝐛_𝑖≠𝟎.

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 ‖𝐱(𝑡)‖₂≤𝑐𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}near𝑡=0 (37) with some constant 𝑐>0. Now putting the limit as 𝑡→0 on both sides in (37), and using (5), we get ‖𝑥(𝑡)‖₂→0 as 𝑡→0. Thus, it is shown that the zero solution of system (1) is attractive as 𝑡→0 provided that (5) holds.
The proof of the reverse claim is slightly complicated. Suppose that ‖𝐱(𝑡)‖₂⟶0as𝑡⟶0. (38) Next, from (17), and (18), we especially obtain lim_𝑡→0^𝑡0_𝑡Im𝜆(𝑠)𝑑𝑠=∞. (39) Remarking that in (21) and (22), we use the following notations: 𝐱(𝑡)=𝑥(𝑡),𝑦(𝑡),𝐡_𝑖(𝑡)=ℎ_𝑖,1(𝑡),ℎ_𝑖,2(𝑡),𝐜_𝑖=𝑐_𝑖,1,𝑐_𝑖,2,𝑖∈{1,2}. (40) By (22), at least one of 𝑐_𝑖,1 and 𝑐_𝑖,2 is nonzero. So, for instance, 𝑐_1,1≠0, which together by (22) gives lim_𝑡→0ℎ_1,1(𝑡)=𝑐_1,1>0. (41) Now, from (21) and (40), we derive the following inequality: ‖𝐱(𝑡)‖≥‖𝐱(𝑡)‖=𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}ℎ_1,1(𝑡)cos^𝑡0_𝑡Im𝜆(𝑠)𝑑𝑠+ℎ_1,2(𝑡)sin^𝑡0_𝑡Im𝜆(𝑠)𝑑𝑠. (42) Next, because of (18) and (39), there is a sequence 𝑡_𝑛∈(0,𝑡₀], 𝑡_𝑛→0 as 𝑛→∞, such that Im𝜆(^𝑡0_𝑡𝑛𝑠)𝑑𝑠=𝑛𝜋,𝑛∈ℕ. (43) Now, putting 𝑡=𝑡_𝑛 into (42) and using (41), (43), and sin(±𝑛𝜋)=0, we get the inequality 𝐱𝑡_𝑛2≥𝑐_1,12𝑒^{−∫𝑡0𝑡𝑛Re𝜆(𝑠)𝑑𝑠}∀large𝑛∈ℕ. (44) Taking the limit on both sides in previous inequality, with the help of (38), we conclude that lim_𝑛→∞𝑒^{−∫𝑡0𝑡𝑛Re𝜆(𝑠)𝑑𝑠}=0, (45) that is, lim_𝑛→∞^𝑡0_𝑡𝑛Re𝜆(𝑠)𝑑𝑠=∞. (46) Since Re𝜆(𝑡)≥0 near 𝑡=0, the function ∫^𝑡0_𝑡Re𝜆(𝑠)𝑑𝑠 is continuous (since matrix elements of 𝐴(𝑡) are continuous functions on (0,𝑡₀]) 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 Γ_𝐱 be a Jordan curve in ℝ². Then one has 𝑙𝑒𝑛𝑔𝑡ℎ(Γ_𝐱)<∞ if and only if 𝐱^′∈𝐿¹((0,𝑡₀),ℝ²).

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 𝐴(𝑡) and its eigenvalues 𝜆(𝑡) which ensure that the zero solution of system (1) is rectifiable attractive.

Theorem 10. Let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡) be two complex conjugate eigenvalues of 𝐴(𝑡) for 𝑡∈[0,𝑡₀]. Assume that (5) holds. Let every solution 𝐱 of system (1) satisfies (37). If ‖𝐴(𝑡)‖𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}∈𝐿¹0,𝑡₀, (47) then the zero solution of (1) is rectifiable attractive as 𝑡→0.

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, 𝐱^′(𝑡)₂=‖𝐴(𝑡)𝐱(𝑡)‖₂≤‖𝐴(𝑡)‖‖𝐱(𝑡)‖₂≤𝑐‖𝐴(𝑡)‖𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}. (48) Now, from the assumption (47) and the inequality (48), we observe that lim_𝜀→0^𝑡0_𝜀𝐱^′(𝑡)₂𝑑𝑡≤lim_𝜀→0𝑐^𝑡0_𝜀‖𝐴(𝑡)‖𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}𝑑𝑡<∞. (49) Hence, Lemma 9 proves this theorem.

If 𝐴(𝑡) 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 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡) be two complex conjugate eigenvalues of 𝐴(𝑡) for 𝑡∈[0,𝑡₀]. Assume that (5) holds. If the condition (47) is fulfilled, then the zero solution of linear system (1) is rectifiable attractive as 𝑡→0.

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 𝐱 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 𝐱^′∉𝐿¹(0,𝑡₀) 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 𝑢=𝑢(𝑡), 𝑢∈𝐶([0,𝑡₀]), and let there be a sequence 𝑠_𝑛∈(0,𝑡₀], a number 𝑛₀∈ℕ, and two functions 𝜑=𝜑(𝑡), 𝑎=𝑎(𝑡), 𝑡∈(0,𝑡₀], such that(𝐻₁) sequence 𝑠_𝑛 is decreasing, 𝑠_𝑛→0 as 𝑛→∞ and 𝑢(𝑠_𝑛)𝑢(𝑠_𝑛+1)<0 for all 𝑛≥𝑛₀;(𝐻₂) function 𝜑(𝑡), 𝜑∈𝐶¹((0,𝑡₀]), is strictly monotone near 𝑡=0, that is, 𝜑^′(𝑡)≠0 near 𝑡=0 and there is a constant 𝑐₂>0 such that 𝜑𝑠_𝑛+1−𝜑𝑠_𝑛≤𝑐₂∀𝑛≥𝑛₀; (50)(𝐻₃) function 𝑎(𝑡), 𝑎∈𝐶([0,𝑡₀]), is nondecreasing and there is a constant 𝑐₁>0 such that 𝑢𝑠_𝑛≥𝑐₁𝑎𝑠_𝑛∀𝑛≥𝑛₀; (51)(𝐻₄) lim_𝜀→0∫^𝑡0_𝜀𝑎(𝑡)|𝜑^′(𝑡)|𝑑𝑡=∞.
Then the graph Γ_𝑢=𝑥,𝑦∈ℝ²∶𝑥=𝑡,𝑦=𝑢(𝑡)for𝑡∈0,𝑡₀ (52) of function 𝑢(𝑡) is a nonrectifiable curve in ℝ².

Proof. Because of (𝐻₁), there is a sequence 𝑎_𝑛∈(𝑠_𝑛+1,𝑠_𝑛) such that 𝑢(𝑎_𝑛)=0 for all 𝑛≥𝑛₀ and hence we have 0<⋯<𝑠_𝑛+1<𝑎_𝑛<𝑠_𝑛<𝑎_𝑛−1<⋯<𝑠_𝑛0+1<𝑎_𝑛0<𝑠_𝑛0<𝑡₀. (53) Let 𝑡_𝑛∈(0,𝑡₀] be a sequence defined by 𝑡_2𝑛−1∶=𝑠_𝑛,𝑡_2𝑛∶=𝑎_𝑛for𝑛≥1. (54) Obviously for all 𝑚≥𝑛₀, the set {0,𝑡_2𝑚−1,…,𝑡_2𝑛0−1,𝑡₀} makes a finite partition of interval [0,𝑡₀] and since length(Γ_𝑢) is a supremum of all sums ∑|𝑢(𝜏_𝑛)−𝑢(𝜏_𝑛−1)| taken over all finite partitions {𝜏_𝑛} of [0,𝑡₀]; we conclude that lengthΓ_𝑢≥𝑢(0)−𝑢𝑡_2𝑚−1+^2𝑚−1_𝑛=2𝑛0𝑢𝑡_𝑛−𝑢𝑡_𝑛−1+𝑢𝑡_2𝑛0−1−𝑢𝑡₀≥^2𝑚−1_𝑛=2𝑛0𝑢𝑡_𝑛−𝑢𝑡_𝑛−1=^𝑚_𝑛=𝑛0𝑢𝑠_𝑛. (55) Next, let, for instance, 𝜑(𝑡) be strictly decreasing near 𝑡=0 (the case when 𝜑(𝑡) is strictly increasing could be analogously considered). Hence, from (𝐻₂) and (𝐻₃), it follows that ^𝑚_𝑛=𝑛0𝑢𝑠_𝑛≥𝑐₁^𝑚_𝑛=𝑛0𝑎𝑠_𝑛≥𝑐₁𝑐₂^𝑚_𝑛=𝑛0𝑎𝑠_𝑛𝜑𝑠_𝑛+1−𝜑𝑠_𝑛=𝑐₁𝑐₂^𝑚_𝑛=𝑛0^{𝜑(𝑠𝑛+1)}_𝜑𝑠𝑛𝑎𝑠_𝑛=𝑐𝑑𝜏₁𝑐₂^𝑚_𝑛=𝑛0^{𝜑(𝑠𝑛+1)}_{𝜑(𝑠𝑛)}max_{𝑠𝑡∈𝑛+1,𝑠𝑛}𝑎(𝑡≥𝑐)𝑑𝜏₁𝑐₂^𝑚_𝑛=𝑛0^{𝜑(𝑠𝑛+1)}_{𝜑(𝑠𝑛)}𝑎𝜑⁻¹(𝜏)=𝑐𝑑𝜏₁𝑐₂^𝑚_𝑛=𝑛0^𝑠𝑛_𝑠𝑛+1𝑎(𝑡)𝜑^′(𝑡)𝑑𝑡. (56) Now from (55) and previous inequality, for all 𝑚≥𝑛₀ we obtain lengthΓ_𝑢≥𝑐₁𝑐₂^𝑚_𝑛=𝑛0^𝑠𝑛_𝑠𝑛+1𝑎(𝑡)𝜑^′(𝑡=𝑐)𝑑𝑡₁𝑐₂^𝑠𝑛0_𝑠𝑚+1𝑎(𝑡)𝜑^′(𝑡)𝑑𝑡. (57) Letting 𝑚→∞ in previous inequality, the hypothesis (𝐻₄) proves this lemma.

Now, we are able to prove the following result.

Theorem 13. Let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡) be two complex conjugate eigenvalues of 𝐴(𝑡) for 𝑡∈(0,𝑡₀]. Assume that Re𝜆(𝑡)≥0 and Im𝜆(𝑡)≠0 near 𝑡=0 and that (5) holds. Let every solution 𝐱(𝑡)=(𝑥(𝑡),𝑦(𝑡)) of system (1) satisfy (37) as well as the following assumption: let 𝑢(𝑡) be defined by 𝑢(𝑡)∶=𝑥(𝑡) or 𝑢(𝑡)∶=𝑦(𝑡) and let there be two constants 𝑐₁>0,𝑐₂>0, a number 𝑛₀∈ℕ, and a decreasing sequence 𝑠_𝑛∈(0,𝑡₀], 𝑠_𝑛→0 as 𝑛→∞, which all depend on function 𝑢(𝑡), such that for all 𝑛≥𝑛₀, one has 𝑢𝑠_𝑛𝑢𝑠_𝑛+1<0,𝑢𝑠_𝑛≥𝑐₁𝑒^{−∫𝑡0𝑠𝑛Re𝜆(𝑡)𝑑𝑡},_{𝑠𝑛𝑠𝑛+1}Im𝜆(𝑡)𝑑𝑡≤𝑐₂. (58) If Im𝜆(𝑡)𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}𝑑𝑡∉𝐿¹0,𝑡₀, (59) then the zero solution of (1) is nonrectifiable attractive as 𝑡→0.

Proof. At first, the attractivity of the zero solution of linear system (1) follows from the assumptions (5) and (37).
Next, let 𝜑(𝑡) and 𝑎(𝑡) denote two new functions defined by 𝜑(𝑡)=^𝑡0_𝑡Im𝜆(𝑠)𝑑𝑠,𝑎(𝑡)=𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}. (60) Let 𝑢(𝑡)∶=𝑥(𝑡) or 𝑢(𝑡)∶=𝑦(𝑡), where 𝐱(𝑡)=(𝑥(𝑡),𝑦(𝑡)) is a solution of system (1). With the help of notation (60) and assumption (58), it is easy to check that the function 𝑢(𝑡) satisfies all hypotheses (𝐻₁), (𝐻₂), and (𝐻₃) from Lemma 12. Indeed, the existence of a sequence 𝑠_𝑛∈(0,𝑡₀) satisfying (𝐻₁) follows from the first inequality in (58). Also, from (18), (58), and (60) we obtain 𝜑^′(𝑡)(𝑡)𝜑𝑠=−Im𝜆≠0near𝑡=0,_𝑛+1−𝜑𝑠_𝑛=^𝑠𝑛+1_𝑠𝑛Im𝜆(𝑠)𝑑𝑠≤𝑐₂∀𝑛≥𝑛₀,𝑢𝑠_𝑛≥𝑐₁𝑒^{−∫𝑡0𝑠𝑛Re𝜆(𝑠)𝑑𝑠}=𝑐₁𝑎𝑠_𝑛∀𝑛≥𝑛₀, (61) which show that (𝐻₂) and (𝐻₃) from Lemma 12 are fulfilled too. Moreover, the equality 𝑎(𝑡)𝜑^′(𝑡)=|Im𝜆(𝑡)|𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠} (62) together with assumption (59) ensure that the hypothesis (𝐻₄) from Lemma 12 is also fulfilled. Now, by Lemma 12 we obtain that the graph of 𝑢(𝑡)∶=𝑥(𝑡) or 𝑢(𝑡)∶=𝑦(𝑡) is a nonrectifiable curve in ℝ². In particular, it follows that the solution curve Γ_𝐱 is also a nonrectifiable curve in ℝ², where 𝐱(𝑡)=(𝑥(𝑡),𝑦(𝑡)).

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 𝐴(𝑡) near 𝑡=0 given in (4).

Theorem 14. Let (4) be satisfied, and let 𝜆(𝑡)=Re𝜆(𝑡)±𝑖Im𝜆(𝑡) be two complex conjugate eigenvalues of the matrix 𝐴(𝑡) for 𝑡∈(0,𝑡₀]. Assume that Re𝜆(𝑡)≥0 near 𝑡=0 and that (5) holds. If the condition (59) holds, then the zero solution of (1) is nonrectifiable attractive as 𝑡→0.

Proof. This proof is a consequence of Theorem 13 and the asymptotic formula (21). At first, according to Lemma 7, every solution 𝐱(𝑡)=(𝑥(𝑡),𝑦(𝑡)) of system (1) satisfies (21). If we denote by 𝑢(𝑡)∶=𝑥(𝑡) or 𝑢(𝑡)∶=𝑦(𝑡), then (21) ensures that 𝑢(𝑡) could be written in the form 𝑢(𝑡)=𝑐₁+𝜀₁(𝑡𝑎(𝑡)(𝑡)+𝑐)cos𝜑₂+𝜀₂(𝑡)𝑎(𝑡)sin𝜑(𝑡)near𝑡=0, (63) where 𝑐₁,𝑐₂∈ℝ with 𝑐²₁+𝑐²₂>0, 𝜀₁,𝜀₂∈𝐶(0,𝑡₀], lim_𝑡→0𝜀₁(𝑡)=lim_𝑡→0𝜀₂(𝑡)=0, and 𝑎(𝑡),𝜑(𝑡) are defined in (60).
Since 𝜆(𝑡) is a continuous complex-valued function on (0,𝑡₀], from (18), we conclude that eitherIm𝜆(𝑡)<0orIm𝜆(𝑡)>0near𝑡=0. (64) Therefore, there are two real constants 𝑐₃,𝑐₄∈ℝ and 𝛿∈𝐶(0,𝑡₀] such that 𝑐₃≠0, lim_𝑡→0𝛿(𝑡)=0 and 𝑢(𝑡)=𝑐₃+𝛿(𝑡)𝑒^{−∫𝑡0𝑡Re𝜆(𝜎)𝑑𝜎}×sin^𝑡0_𝑡|Im𝜆(𝜎)|𝑑𝜎+𝑐₄near𝑡=0. (65)
Because of (59), we have ∫^𝑡0_𝑡|Im𝜆(𝜎)|𝑑𝜎→∞ as 𝑡→0 and therefore there is a decreasing sequence 𝑠_𝑛∈(0,𝑡₀] and a number 𝑛₀∈ℕ such that ^𝑡0_𝑠𝑛|Im𝜆(𝜎)|𝑑𝜎+𝑐₄=𝜋2+𝑛𝜋∀𝑛≥𝑛₀. (66) Putting (66) into (65) and retaking 𝑛₀∈ℕ as sufficiently large, we get 𝑢𝑠_𝑛𝑢𝑠_𝑛+1𝑢𝑠<0,_𝑛≥𝑐₃2𝑒^{−∫𝑡0𝑠𝑛Re𝜆(𝑠)𝑑𝑠}∀𝑛≥𝑛₀, (67) and because of (64) and (66), Im𝜆(^𝑠𝑛_𝑠𝑛+1𝑠)𝑑𝑠=^𝑠𝑛_𝑠𝑛+1|(𝑠=Im𝜆)|𝑑𝑠_{𝑡0𝑠𝑛+1}|(𝑠−Im𝜆)|𝑑𝑠_{𝑡0𝑠𝑛}|Im𝜆(𝑠)|𝑑𝑠=𝜋. (68) Now, statements (67) and (68) show that the function 𝑢(𝑡) and sequence 𝑠_𝑛 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 𝐴(𝑡) be the matrix of the model system (7) given by 𝐴(𝑡)=𝑒(𝑡)𝑡^−𝑝𝑓(𝑡)𝑡⁻²−𝑔(𝑡)𝑡⁻²ℎ(𝑡)𝑡^−𝑞for𝑡∈0,𝑡₀, (69) where the real numbers 𝑝, 𝑞, and functions 𝑒, 𝑓, 𝑔, and ℎ satisfy (8), (9), and (10), respectively. We will show that such a matrix 𝐴(𝑡) together with its eigenvalues 𝜆(𝑡) satisfy the required conditions Re𝜆(𝑡)≥0 near 𝑡=0, (5), and (4).

At first, because of (9), we have Re𝜆(𝑡)=(𝑒(𝑡)𝑡^−𝑝+ℎ(𝑡)𝑡^−𝑞)/2>0.

Next, since max{𝑝,𝑞}≥1, let, for instance, max{𝑝,𝑞}=𝑝≥1, and 𝑡₀∈(0,1]; from (9) we obtain a constant 𝑐₀>0 such that ^𝑡0_𝑡(𝑠)1Re𝜆𝑑𝑠=2^𝑡0_𝑡𝑒(𝑠)𝑠^−𝑝+ℎ(𝑠)𝑠^−𝑞≥1𝑑𝑠2^𝑡0_𝑡𝑒(𝑠)𝑠^−𝑝𝑑𝑠≥𝑐₀^𝑡0_𝑡𝑠⁻¹𝑑𝑠=𝑐₀𝑡ln⁰𝑡⟶∞as𝑡⟶0. (70) Now, it is clear that (70) proves the desired statement (5).

Finally, since 𝑝<2 and 𝑞<2, lim_𝑡→0𝑡²𝐴(𝑡)=lim_𝑡→0𝑒(𝑡)𝑡^2−𝑝𝑓(𝑡)−𝑔(𝑡)ℎ(𝑡)𝑡^2−𝑞=(0)(0)00𝑓−𝑔=∶𝐴₀, (71) where 𝐴₀ is a constant matrix. Since 𝑓(0)𝑔(0)>0, matrix 𝐴₀ has two different eigenvalues. So, the condition (4) is fulfilled too.

Thus, matrix 𝐴(𝑡) together with its eigenvalues satisfy all required conditions so that Theorem 3 can be applied to the model system (7) in all cases of 𝑝 and 𝑞 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 Φ(𝑡)=|Im𝜆(𝑡)|𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}for𝑡∈0,𝑡₀, (72) we can conclude that the zero solutions of (7) is rectifiable attractive if Φ∈𝐿¹(0,𝑡₀) and nonrectifiable attractive if Φ∉𝐿¹(0,𝑡₀). Let us remark that in Lemma 6, it has been proved that |Im𝜆(𝑡)|∼𝑡⁻²as𝑡⟶0. (73)

(i) Assume that max{𝑝,𝑞}>1. For instance, we assume 𝑝>1. Since 𝑒(𝑡)>0 on [0,𝑡₀], we have 𝑒(𝑡)𝑡^−𝑝+1→∞ as 𝑡→0. In particular, there exist 𝑡₁∈(0,𝑡₀] and 𝛿>0 such that 𝑒(𝑡)𝑡^−𝑝+1≥2+𝛿for𝑡∈0,𝑡₁. (74) Since 𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}=𝑒^{(−1/2)∫𝑡0𝑡(𝑒(𝑠)𝑠−𝑝+ℎ(𝑠)𝑠−𝑞)𝑑𝑠}, (75) we have 𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}≤𝑒^{(−1/2)∫𝑡0𝑡𝑒(𝑠)𝑠−𝑝𝑑𝑠}=𝑒^{(−1/2)∫𝑡0𝑡𝑒(𝑠)𝑠−𝑝+1𝑠−1𝑑𝑠}≤𝑒^{−(1+(𝛿/2))∫𝑡0𝑡𝑠−1𝑑𝑠}=𝑐𝑡^1+(𝛿/2), (76) where the constant 𝑐=𝑡₀^{−1−(𝛿/2)}. Now from (72), (73), and (76), we observe that Φ(𝑡)≤𝑐₁𝑡⁻²𝑡^1+(𝛿/2)=𝑐₁𝑡^{−1+(𝛿/2)}∈𝐿¹0,𝑡₀. (77) Thus, it is shown that Φ∈𝐿¹(0,𝑡₀).

(ii) Assume that (11) is fulfilled. Since 𝑒(𝑡)>2 on [0,𝑡₀], there exists 𝛿>0 such that 𝑒(𝑡)≥2+𝛿 on [0,𝑡₀]. Since 𝑝=1 and ℎ(𝑡)>0 on (0,𝑡₀], from (75) we have 𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}≤𝑒^{(−1/2)∫𝑡0𝑡𝑒(𝑠)𝑠−1𝑑𝑠}≤𝑒^{−(1+(𝛿/2))∫𝑡1𝑡𝑠−1𝑑𝑠}=𝑐𝑡^1+(𝛿/2), (78) where the constant 𝑐=𝑡₀^{−1−(𝛿/2)}. Now from (72), (73), and (78), we obtain Φ∈𝐿¹(0,𝑡₀). When (12) is fulfilled, we obtain Φ∈𝐿¹(0,𝑡₀) by the similar argument.

Assume that (13) is fulfilled. Since 𝑒(𝑡)+ℎ(𝑡)>2 on [0,𝑡₀], there exists 𝛿>0 such that 𝑒(𝑡)+ℎ(𝑡)≥2+𝛿 on [0,𝑡₀]. From (75) we have 𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}=𝑒^{(−1/2)∫𝑡0𝑡𝑒(𝑠)+ℎ(𝑠)𝑠−1𝑑𝑠}≤𝑒^{−(1+(𝛿/2))∫𝑡0𝑡𝑠−1𝑑𝑠}=𝑐𝑡^1+(𝛿/2), (79) where the constant 𝑐=𝑡₀^{−1−(𝛿/2)}. Thus we obtain Φ∈𝐿¹(0,𝑡₀) by the argument above.

(iii) Let (14) be fulfilled. Since ℎ∈𝐶([0,𝑡₀]), there is a constant 𝑚₀>0 such that ℎ(𝑡)≤𝑚₀ on [0,𝑡₀], and since 𝑞<1 there is a constant 𝑐₀>0 such that 𝑒^{(−1/2)∫𝑡0𝑡ℎ(𝑠)𝑠−𝑞𝑑𝑠}≥𝑒^{−(𝑚0/2)∫𝑡0𝑡𝑠−𝑞𝑑𝑠}≥𝑐₀as𝑡⟶0. (80)

Hence, 𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}=𝑒^{(−1/2)∫𝑡0𝑡(𝑒(𝑠)𝑠−1+ℎ(𝑠)𝑠−𝑞)𝑑𝑠}=𝑒^{−∫𝑡0𝑡𝑠−1𝑑𝑠}𝑒^{(−1/2)∫𝑡0𝑡ℎ(𝑠)𝑠−𝑞𝑑𝑠}≥𝑐₀𝑒^{−∫𝑡0𝑡𝑠−1𝑑𝑠}=𝑐₀𝑡𝑡₀. (81) Now, by combining (72), (73), and (81), we observe ^𝑡0_𝑡Φ(𝑠)𝑐𝑑𝑠≥¹𝑡₀^𝑡0_𝑡𝑠⁻²𝑐𝑠𝑑𝑠=¹𝑡₀𝑡ln⁰𝑡⟶∞as𝑡→0. (82) Thus, Φ∉𝐿¹(0,𝑡₀). When (15) is fulfilled, we similarly obtain Φ∉𝐿¹(0,𝑡₀).

Let (16) be fulfilled. Then, from (75) we have 𝑒^{−∫𝑡0𝑡Re𝜆(𝑠)𝑑𝑠}=𝑒^{(−1/2)∫𝑡0𝑡(𝑒(𝑠)+ℎ(𝑠))𝑠−1𝑑𝑠}≥𝑒^{−∫𝑡0𝑡𝑠−1𝑑𝑠}=𝑡𝑡₀. (83) By combining (72), (73), and (83), we obtainΦ∉𝐿¹(0,𝑡₀). 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 𝐱 of linear system (1).

Proof. Putting 𝜏=1/𝑡 into system (1) and denoting 𝐰(𝜏)=𝐱(1/𝜏), we obtain the equivalent system 𝑑𝐰(𝜏)𝜏𝑑𝜏=𝐵𝐰for𝜏∈₀,∞,𝐰𝜏₀=𝐱₀, (A.1) where 𝜏₀=1/𝑡₀ and the matrix 𝐵(𝑡) 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 𝐴=𝐴(𝑡) such that the solution curve Γ_𝐱 of every nontrivial solution 𝐱(𝑡) of linear system (1) (as well as of (7)) is a Jordan curve.

Definition A.2. A solution 𝐱 of linear system (1) is said to be trivial solution of (1) near 𝑡=0 if there is 𝑡₁∈(0,𝑡₀] such that 𝐱(𝑡)=0 for all 𝑡∈(0,𝑡₁]. Otherwise, 𝐱(𝑡) is said to be nontrivial solution of (1) near 𝑡=0.

Theorem A.3. Let the matrix elements 𝑎_𝑖,𝑗(𝑡) of 𝐴(𝑡) satisfy min𝑎_1,1(𝑡),𝑎_2,2(𝑡>1)2𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑡∈0,𝑡₀. (A.2) If 𝐱 is a nontrivial solution of linear system (1) near 𝑡=0, then ‖𝐱(𝑡)‖₂ is strictly positive and decreasing on (0,𝑡₀]. In particular, the solution curve Γ_𝐱 is a Jordan curve.

Remark A.4. It is clear that if 𝑎_1,1(𝑡)>0, 𝑎_2,2(𝑡)>0 and 𝑎_1,2(𝑡)=−𝑎_2,1(𝑡) on (0,𝑡₀], then condition (A.2) is fulfilled.

Proof of Theorem A.3. Let 𝐱(𝑡)=(𝑥(𝑡),𝑦(𝑡)) be a nontrivial solution of system (1) near 𝑡=0. The assumption (A.2) can be rewritten as 𝑎_1,1(𝑡)>12𝑎_1,2(𝑡)+𝑎_2,1(𝑡,𝑎_2,2)(𝑡)>12𝑎_1,2(𝑡)+𝑎_2,1(𝑡)on0,𝑡₀. (A.3)
Let 𝑣(𝑡) be a function defined by 𝑣(𝑡)=‖𝐱(𝑡)‖²=𝑥²(𝑡)+𝑦²(𝑡)for𝑡∈0,𝑡₀. (A.4) Taking the derivative 𝑑/𝑑𝑡 in (A.4), it follows that 𝑑𝑣(𝑡)𝑥𝑑𝑡=2𝑥^′(𝑡)+2𝑦(𝑡)𝑦^′(𝑡)for𝑡∈0,𝑡₀. (A.5) Multiplying the first and second equation in system (1), respectively, by 𝑥(𝑡) and 𝑦(𝑡), we obtain 𝑥(𝑡)𝑥^′(𝑡)+𝑦(𝑡)𝑦^′(𝑡)=𝑎_1,1(𝑡)𝑥²(𝑡)+𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑥(𝑡)𝑦(𝑡)+𝑎_2,2(𝑡)𝑦²(𝑡). (A.6) Because of inequality 𝑎𝑏≤(𝑎²+𝑏²)/2, we have 𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑥(𝑡)𝑦(𝑡≤1)2𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑥²(𝑡)+𝑦²(𝑡), (A.7) which together with (A.6) derives that 𝑥(𝑡)𝑥^′(𝑡)+𝑦(𝑡)𝑦^′(𝑡)≥𝑎_1,1(𝑡)𝑥²(𝑡)−12𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑥²(𝑡)+𝑦²(𝑡)+𝑎_2,2(𝑡)𝑦²(𝑡)=𝐷(𝑡), (A.8) where 𝐷(𝑡) denotes the function 𝐷(𝑡)=𝑎_1,1(𝑡)−12𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑥²(𝑡)+𝑎_2,2(𝑡)−12𝑎_1,2(𝑡)+𝑎_2,1(𝑡)𝑦²(𝑡). (A.9) Putting (A.8) into (A.5), we obtain 𝑑𝑣(𝑡)𝑑𝑡≥2𝐷for𝑡∈0,𝑡₀. (A.10) Because of (A.3), we have 𝐷(𝑡)≥0 on (0,𝑡₀]. Hence, from (A.10), we observe 𝑑𝑣/𝑑𝑡≥0 on (0,𝑡₀]. Integrating this inequality over the interval [𝑠,𝑡] for all 0<𝑠≤𝑡≤𝑡₀, we get 𝑣(𝑠)≤𝑣(𝑡). If we suppose for a moment that 𝑣(𝑡)=0, then from previous inequality we get 𝑣(𝑠)=0 for all 𝑠∈(0,𝑡], which is not possible because of Definition A.2. Therefore, 𝑣(𝑡)>0 for all 𝑡∈(0,𝑡₀], which implies ‖𝐱(𝑡)‖₂>0 for all 𝑡∈(0,𝑡₀] because of (A.4). It proves the first statement of this theorem.
Next, with the help of (A.3) and ‖𝐱(𝑡)‖₂>0 for all 𝑡∈(0,𝑡₀], we deduce that 𝐷(𝑡)>0 on (0,𝑡₀], which together with (A.10) gives 𝑣^′(𝑡)>0 on (0,𝑡₀]. Integrating this inequality over the interval [𝑠,𝑡] for all 0<𝑠<𝑡≤𝑡₀, we get 𝑣(𝑠)<𝑣(𝑡)∀𝑠,𝑡suchthat0<𝑠<𝑡≤𝑡₀. (A.11) If, for instance, there exist 𝑠,𝑡 such that 0<𝑠<𝑡≤𝑡₀ and ‖𝐱(𝑠)‖₂=‖𝐱(𝑡)‖₂, then involving this equality into (A.4), we obtain 𝑣(𝑠)=𝑣(𝑡), which is not possible because of (A.11). Therefore, ‖𝐱(𝑡)‖₂≠‖𝐱(𝑠)‖₂. Thus, it is shown that Γ_𝐱 is a Jordan curve near 𝑡=0. In the same way, ‖𝐱(𝑠)‖₂<‖𝐱(𝑡)‖₂ for all 𝑠,𝑡 such that 0<𝑠<𝑡≤𝑡₀.

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

Corollary A.5. Let 𝑒,ℎ∈𝐶((0,𝑡₀]), 𝑒(𝑡)>0, ℎ(𝑡)>0, and 𝑓(𝑡)=𝑔(𝑡) near 𝑡=0. If 𝐱(𝑡) is a nontrivial solution of linear system (7) near 𝑡=0, then ‖𝐱(𝑡)‖₂ is strictly positive and decreasing on (0,𝑡₀]. In particular, the solution curve Γ_𝐱 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 ℎ(𝑡)>0 and 𝑓(𝑡)>0 on (0,𝑡₀]. The fundamental system of all solutions of linear differential system 𝑥^′(𝑡)=ℎ(𝑡)𝑡^−𝑝𝑥(𝑡)+𝑓(𝑡)𝑡⁻²𝑦(𝑡),𝑦^′(𝑡)=−𝑓(𝑡)𝑡⁻²𝑥(𝑡)+ℎ(𝑡)𝑡^−𝑝𝑦(𝑡),𝑡∈0,𝑡₀ (A.12) is explicitly given by the formula 𝑥(𝑡)=𝑒^{−∫𝑡0𝑡ℎ(𝑠)𝑠−𝑝𝑑𝑠}×−𝑐₁sin^𝑡0_𝑡𝑓(𝑠)𝑠⁻²𝑑𝑠+𝑐₂cos^𝑡0_𝑡𝑓(𝑠)𝑠⁻²,𝑦(𝑡)𝑑𝑠=𝑒^{−∫𝑡0𝑡ℎ(𝑠)𝑠−𝑝𝑑𝑠}×𝑐₁cos^𝑡0_𝑡𝑓(𝑠)𝑠⁻²𝑑𝑠+𝑐₂sin^𝑡0_𝑡𝑓(𝑠)𝑠⁻²𝑑𝑠. (A.13)

Obviously for 𝑝=𝑞, 𝑒(𝑡)=ℎ(𝑡)>0 and 𝑔(𝑡)=𝑓(𝑡)>0 on (0,𝑡₀], 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 𝑥=0 for the system (A.12) as follows.

Corollary A.7. Let ℎ(𝑡)>0 and 𝑓(𝑡)>0 on (0,𝑡₀].(i)If 𝑝>1, then the zero solution of (A.12) is rectifiable attractive as 𝑡→0.(ii)If 𝑝=1, then the zero solution of (A.12) is rectifiable attractive as 𝑡→0 provided thatℎ(𝑡)>1.(iii)If 𝑝=1, then the zero solution of (A.12) is nonrectifiable attractive as 𝑡→0 provided thatℎ(𝑡)≤1.

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.