The Contrast Structures for a Class of Singularly Perturbed Systems ...

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2016, Article ID 6405853, 8 pages http://dx.doi.org/10.1155/2016/6405853

Research Article The Contrast Structures for a Class of Singularly Perturbed Systems with Heteroclinic Orbits Han Xu and Yinlai Jin School of Science, Linyi University, Linyi, Shandong 276005, China Correspondence should be addressed to Yinlai Jin; [email protected] Received 16 November 2015; Accepted 13 January 2016 Academic Editor: Elvan Akın Copyright © 2016 H. Xu and Y. Jin. 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. Singularly perturbed problems are often used as the models of ecology and epidemiology. In this paper, a class of semilinear singularly perturbed systems with contrast structures are discussed. Firstly, we verify the existence of heteroclinic orbits connecting two equilibrium points about the associated systems for contrast structures in the corresponding phase space. Secondly, the asymptotic solutions of the contrast structures by the method of boundary layer functions and smooth connection are constructed. Finally, the uniform validity of the asymptotic expansion is defined and the existence of the smooth solutions is proved.

1. Introduction Contrast structures in singularly perturbed problems can be classified as step-type contrast structures or spike-type contrast structures [1–3]. The existence of contrast structures is relevant to the existence of homoclinic orbits and heteroclinic orbits of their associated systems in [4–6]. Recently, contrast structures in singularly perturbed problems are attached great importance. Ni and Wang study four-dimensional contrast structures in singularly perturbed problems with fast variables in [7]. In [8], Wang considers a kind of step-type contrast structure for singularly perturbed problems with slow and fast variables and proves the existence of a heteroclinic orbit of its associated system in the corresponding phase space. Since contrast structures can express the instantaneous transformation more accurately, we often use them in singularly perturbed problems as the models of the collision of cars and the transfer law of neurons. In [9], a kind of epidemical model with spike-type contrast structures is proposed. Chattoadhyay and Bairagi build an ecoepidemiological model in [10]:

𝑑𝑤 = 𝑑Δ𝑤 + 𝑓3 (𝑢, V) , 𝑑𝑡 𝜕𝑢 𝜕V 𝜕𝑤 = = = 0, 𝜕𝑛 𝜕𝑛 𝜕𝑛

(𝑡, 𝑥) ∈ (0, +∞) × Ω, (𝑡, 𝑥) ∈ (0, +∞) × 𝜕Ω,

𝑢 (𝑥, 0) ≥ 0, V (𝑥, 0) ≥ 0, 𝑤 (𝑥, 0) ≥ 0, 𝑥 ∈ Ω, (1) where 𝑑 is a diffusion coefficient, Ω is a population habitat, 𝑢 is susceptible prey, V is infected prey, 𝑤 is density of predators, and 𝑛 is a unit outer normal vector. By geometric methods and functional skills, Chattoadhyay and Bairagi study the existence of 𝑢(𝑥), which is a stable solution. Considering the complexity of the ecoepidemiological model and the small parameter 𝑑, we propose the following semilinear singularly perturbed system with contrast structures: 𝑑𝑧 = 𝑔 (𝑢, V, 𝑤, 𝑧, 𝑡) , 𝑑𝑡 𝑑𝑢 = 𝑓1 (𝑢, V, 𝑤, 𝑧, 𝑡) , 𝜇 𝑑𝑡

𝑑𝑢 = 𝑑Δ𝑢 + 𝑓1 (𝑢, V) , 𝑑𝑡

(𝑡, 𝑥) ∈ (0, +∞) × Ω,

𝜇

𝑑V = 𝑑ΔV + 𝑓2 (𝑢, V) , 𝑑𝑡

(𝑡, 𝑥) ∈ (0, +∞) × Ω,

𝜇

𝑑V = 𝑓2 (𝑢, V, 𝑤, 𝑧, 𝑡) , 𝑑𝑡

𝑑𝑤 = 𝑓3 (𝑢, V, 𝑤, 𝑧, 𝑡) , 𝑑𝑡

2

Discrete Dynamics in Nature and Society 𝑢 (0, 𝜇) = 𝑢0 , V (0, 𝜇) = V0 , 𝑤 (1, 𝜇) = 𝑤1 , 𝑧 (0, 𝜇) = 𝑧0 , (2)

where 0 < 𝜇 ≪ 1, 0 ≤ 𝑡 ≤ 1, the functions 𝑓1 , 𝑓2 , 𝑓3 , and 𝑔 are sufficiently smooth on the domain G = {(𝑢, V, 𝑤, 𝑧, 𝑡) | ‖𝑢‖ ≤ 𝑙1 , ‖V‖ ≤ 𝑙2 , ‖𝑤‖ ≤ 𝑙3 , ‖𝑧‖ ≤ 𝑙4 , 0 ≤ 𝑡 ≤ 1}, and 𝑙𝑖 (𝑖 = 1, 2, 3, 4) are given positive real numbers. Assuming that y = (𝑢, V, 𝑤)𝑇 and f = (𝑓1 , 𝑓2 , 𝑓3 )𝑇 , system (2) is equivalent to the following system: 𝑑𝑧 = 𝑔 (y, 𝑧, 𝑡) , 𝑑𝑡 𝜇

̃(𝜏))𝑇 and 𝑡 and 𝑡 are paramewhere ỹ(𝜏) = (̃ 𝑢(𝜏), ̃V(𝜏), 𝑤 ters. Obviously, system (7) exhibits two families of equilibrium points 𝑀1 (𝛼(𝑧(−) , 𝑡), 𝑧(−) , 𝑡) and 𝑀2 (𝛽(𝑧(+) , 𝑡), 𝑧(+) , 𝑡). Assuming that 𝐵𝑙 (𝑡) = 𝐷ỹ f(̃y, 𝑧, 𝑡)|𝑀𝑙 , 𝑙 = 1, 2, we can classify sixteen cases (can be seen in [11, 12]) on relations between 𝑀1 and 𝑀2 by the symbols of eigenvalues. There might exist interior layers satisfying one of the following cases: (1) 𝑀1 [−, −, +], 𝑀2 [−, −, +]; (2) 𝑀1 [−, +, +], 𝑀2 [−, +, +]. We will discuss case (1); case (2) can be debated similarly. (A2) Assume that 𝐵𝑙 (𝑡) has three real-valued eigenvalues and satisfies the inequality 𝜆 𝑙1 < 𝜆 𝑙2 < 0 < 𝜆 𝑙3 . By (A2), system (7) may exhibit a heteroclinic orbit connecting 𝑀1 to 𝑀2 . To give the necessary conditions about the existence of a heteroclinic orbit, we introduce the following hypothesis: (A3) The associated system (7) has two manifolds expressed by 𝛽𝑙 (̃y, 𝑧̃, 𝑡) = 𝐶𝑙 , 𝑙 = 1, 2. The manifold crossing through 𝑀1 is

𝑑y = f (y, 𝑧, 𝑡) , 𝑑𝑡

(−) (−) Φ𝑙 (̃y, 𝑧̃, 𝑡) = Φ(−) 𝑙 (𝛼 (𝑧 , 𝑡) , 𝑧 , 𝑡) .

0

𝑢 (0, 𝜇) = 𝑢 ,

(3)

(8)

The manifold crossing through 𝑀2 is

V (0, 𝜇) = V0 ,

(+) (+) Φ𝑙 (̃y, 𝑧̃, 𝑡) = Φ(+) 𝑙 (𝛽 (𝑧 , 𝑡) , 𝑧 , 𝑡) .

𝑤 (1, 𝜇) = 𝑤1 ,

According to (8) and (9), the necessary conditions about the existence of heteroclinic orbits can be obtained by

𝑧 (0, 𝜇) = 𝑧0 .

(−) (−) (+) (+) (+) Φ(−) 𝑙 (𝛼 (𝑧 , 𝑡) , 𝑧 , 𝑡) = Φ𝑙 (𝛽 (𝑧 , 𝑡) , 𝑧 , 𝑡) .

The degenerate equations of (3) are

(9)

(10)

By (A3), (8) and (9) can be expressed by

f (y, 𝑧, 𝑡) = 0,

̃V(−) (𝜏) = Ψ2(−) (̃ 𝑢(−) , 𝑧(−) , 𝑡) ,

𝑑𝑧 = 𝑔 (y, 𝑧, 𝑡) . 𝑑𝑡

(4)

̃(−) (𝜏) = Ψ3(−) (̃ 𝑢(−) , 𝑧(−) , 𝑡) , 𝑤

In the following, we let (A1) the degenerate equation f(y, 𝑧, 𝑡) = 0 have two isolated smooth solutions y = 𝛼(𝑧, 𝑡) and y = 𝛽(𝑧, 𝑡) on 𝐷 = {(𝑧, 𝑡) | |𝑧| ≤ 𝑙5 , 0 ≤ 𝑡 ≤ 1}, where 𝛼 = (𝛼1 , 𝛼2 , 𝛼3 )𝑇 , 𝛽 = (𝛽1 , 𝛽2 , 𝛽3 )𝑇 , and 𝑙5 is a given positive real number.

̃(+) (𝜏) = Ψ3(+) (̃ 𝑢(+) , 𝑧(+) , 𝑡) . 𝑤 Supposing that 𝐻ℎ (𝑡) = Ψℎ(−) (̃ 𝑢(−) , 𝑧(−) , 𝑡) − Ψℎ(+) (̃ 𝑢(+) , 𝑧(+) , 𝑡) , ℎ = 2, 3;

According to (A1), the initial value problem 𝑑𝑧(−) = 𝑔 (𝛼 (𝑧(−) , 𝑡) , 𝑧(−) , 𝑡) , 𝑑𝑡

𝑧(−) (0) = 𝑧0 ,

(11)

̃V(+) (𝜏) = Ψ2(+) (̃ 𝑢(+) , 𝑧(+) , 𝑡) ,

(12)

we can give the following hypothesis: (5)

(A4) Assume that (12) has a solution of 𝑡 = 𝑡0 , and (𝜕𝐻2 /𝜕𝑡)|𝑡=𝑡0 , (𝜕𝐻3 /𝜕𝑡)|𝑡=𝑡0 are not simultaneously equal to zero.

𝑑𝑧(+) = 𝑔 (𝛽 (𝑧(+) , 𝑡) , 𝑧(+) , 𝑡) , 𝑧(+) (𝑡0 ) = 𝑧(−) (𝑡0 ) , (6) 𝑑𝑡

In accordance with (A4), we can realize that 𝐻ℎ (𝑡0 ) = 0, namely, there exists a heteroclinic orbit connecting 𝑀1 and 𝑀2 .

has the unique solution 𝑧(−) (𝑡) in [0, 1], 0 < 𝑡0 < 1, and

has the unique solution 𝑧(+) (𝑡) in [0,1]. The associated system of (3) is 𝑑̃y = f (̃y, 𝑧, 𝑡) , 𝑑𝜏

𝜏=

𝜇 ≥ 0, 𝑡

(7)

Lemma 1. Under conditions (A1)–(A4), the associated system (7) has a heteroclinic orbit connecting 𝑀1 and 𝑀2 , which is expressed by (10) and (12); therefore, system (3) has the solution with interior layer.

Discrete Dynamics in Nature and Society

3

2. The Construction of Asymptotic Expansion In accordance with (A1)–(A4), system (3) has a step-like solution from 𝛼(𝑡) to 𝛽(𝑡). Hence, we can suppose that 𝑡∗ = 𝑡0 + 𝜇𝑡1 + ⋅ ⋅ ⋅ + 𝜇𝑘 𝑡𝑘 + ⋅ ⋅ ⋅ , x (𝑡∗ , 𝜇) = x0∗ + 𝜇x1∗ + ⋅ ⋅ ⋅ + 𝜇𝑘 x𝑘∗ + ⋅ ⋅ ⋅ ,

(13)

where 𝑡∗ is the transit point from 𝛼(𝑡) to 𝛽(𝑡) and 𝑡𝑖 and x𝑖∗ , 𝑖 = 1, 2, . . ., are undetermined coefficients. The interior-layer solution of system (3) can be divided into two parts. The left problem is (0 ≤ 𝑡 ≤ 𝑡∗ ) (−)

𝑑𝑧 = 𝑔 (y(−) , 𝑧(−) , 𝑡) , 𝑑𝑡 𝜇 𝑢 V

(−)

(−)

(−) f (y(−) 0 , 𝑧0 , 𝑡) = 0, 󸀠

(−) (−) (𝑧(−) 0 ) = 𝑔 (y0 , 𝑧0 , 𝑡) , 0 𝑧(−) 0 (0) = 𝑧 ,

󸀠

𝑑y(−) = f (y(−) , 𝑧(−) , 𝑡) , 𝑑𝑡

(+) (+) (𝑧(+) 0 ) = 𝑔 (y0 , 𝑧0 , 𝑡) ,

0

(0, 𝜇) = 𝑢 ,

(14)

0

(0, 𝜇) = V ,

𝑧(−) (0, 𝜇) = 𝑧0 . 𝑑𝑧(+) = 𝑔 (y(+) , 𝑧(+) , 𝑡) , 𝑑𝑡

(−) 𝑧(+) 0 (𝑡0 ) = 𝑧0 (𝑡0 ) .

= 𝑧(−) (𝑡), y(−) = By (A2), we can obtain that 𝑧(−) 0 0 (−) (+) (+) (+) (+) 𝛼(𝑧 (𝑡), 𝑡) = 𝛼(𝑡, 𝑡0 ), 𝑧0 = 𝑧 (𝑡), and y0 = 𝛽(𝑧 (𝑡), 𝑡) = 𝛽(𝑡, 𝑡0 ), where 𝑧(±) (𝑡) are functions about 𝑡 and 𝑡0 and 𝑡0 is an undetermined constant. For 𝑄0(−) (𝜏), we have

𝑑𝑄0(−) y (𝜏) 𝑑𝜏

𝑑y(+) = f (y(+) , 𝑧(+) , 𝑡) , 𝑑𝑡

(−) = f (𝛼 (𝑡0 ) + 𝑄0(−) y (𝜏) , 𝑧(−) 0 + 𝑄0 𝑧 (𝜏) , 𝑡0 ) ,

𝑢(+) (𝑡∗ , 𝜇) = 𝑢∗ ,

(15)

V(+) (𝑡∗ , 𝜇) = V∗ ,

𝛼1 (𝑡0 ) + 𝑄0(−) 𝑢 (0) = 𝑢∗ ,

By (20), we can solve 𝑄0(−) 𝑧(𝜏) ≡ 0. Assuming that ỹ𝑙 = 𝛼(𝑡0 )+ 𝑄0(−) y(𝜏), (20) can be rewritten as

𝑧(+) (𝑡∗ , 𝜇) = 𝑧(−) (𝑡∗ , 𝜇) .

𝑑̃y𝑙 = f (̃y𝑙 , 𝑧(−) 0 (𝑡0 ) , 𝑡0 ) , 𝑑𝜏

The step-like contrast structure of (3) can be regarded as the smooth connection at the point of 𝑡∗ by two solutions of (14) and (15). Let x = (𝑢, V, 𝑤, 𝑧)𝑇 , and by the method of boundary layer functions [8–10], the asymptotic expansion of (14) can be constructed as follows: ∞

(−) x(−) (𝑡, 𝜇) = ∑ 𝜇𝑘 [x(−) 𝑘 (𝑡) + 𝐿 𝑘 x (𝜏0 ) + 𝑄𝑘 x (𝜏)] , 𝑘=0

𝑡 𝑡 − 𝑡∗ 𝜏0 = , 𝜏 = . 𝜇 𝜇

x

(𝑡, 𝜇) = ∑ 𝜇 𝑘=0

[x(+) 𝑘

(𝑡) +

𝑄𝑘(+) x (𝜏)

(16)

𝑡−1 , 𝜇

(21)

ỹ𝑙 (−∞) = 𝛼 (𝑡0 ) . 𝑑𝑄0(+) 𝑧 (𝜏) = 0, 𝑑𝜏 𝑑𝑄0(+) y (𝜏) 𝑑𝜏 (+) = f (𝛽 (𝑡0 ) + 𝑄0(+) y (𝜏) , 𝑧(+) 0 + 𝑄0 𝑧 (𝜏) , 𝑡0 ) ,

+ 𝑅𝑘 x (𝜏1 )] , 𝜏1 =

̃ 𝑙 (0) = 𝑢∗ , 𝑢

For 𝑄0(+) (𝜏), we have

Also, the asymptotic expansion of (15) can be constructed as follows: 𝑘

(20)

𝑄0(−) x (−∞) = 0.

𝑤(+) (1, 𝜇) = 𝑤1 ,

∞

(19)

𝑑𝑄0(−) 𝑧 (𝜏) = 0, 𝑑𝜏

The right problem is (𝑡∗ ≤ 𝑡 ≤ 1)

(+)

(18)

(+) f (y(+) 0 , 𝑧0 , 𝑡) = 0,

𝑤(−) (𝑡∗ , 𝜇) = 𝑤∗ ,

𝜇

where x(±) 𝑘 (𝑡) are the coefficients of regular series, 𝐿 𝑘 x(𝜏0 ) are the coefficients of left boundary layer series, 𝑅𝑘 x(𝜏1 ) are the coefficients of right boundary layer series, and 𝑄𝑘(±) x(𝜏) are the coefficients of interior-layer series. Putting (16) and (17) into (14) and (15), and separating equations by scales 𝑡, 𝜏0 , 𝜏, and 𝜏1 , then x(±) 0 (𝑡) satisfies

(17)

𝛽1 (𝑡0 ) + 𝑄0(+) 𝑢 (0) = 𝑢∗ , 𝑄0(+) x (+∞) = 0.

(22)

4


By (22), we can solve 𝑄0(+) 𝑧(𝜏) ≡ 0. Supposing that ỹ𝑟 = 𝛽(𝑡0 ) + 𝑄0(+) y(𝜏), (22) can be rewritten as 𝑑̃y𝑟 = f (̃y𝑟 , 𝑧(+) 0 (𝑡0 ) , 𝑡0 ) , 𝑑𝜏 ̃ 𝑟 (0) = 𝑢∗ , 𝑢

(23)

𝑑𝑅0 𝑧 (𝜏1 ) = 0, 𝑑𝜏1 𝑑𝑅0 y (𝜏1 ) 𝑑𝜏1 = f (𝛽 (1) + 𝑅0 y (𝜏1 ) , 𝑧(+) 0 (1) + 𝑅0 𝑧 (𝜏1 ) , 1) ,

ỹ𝑟 (+∞) = 𝛽 (𝑡0 ) .

𝑅0 x (−∞) = 0. According to (27), we can solve 𝑅0 𝑧(𝜏1 ) ≡ 0. Supposing that 𝑟 ỹ̃ = 𝛽(1) + 𝑅0 y(𝜏1 ), (27) can be rewritten as

𝑑̃y = f (̃y, 𝑧0 (𝑡0 ) , 𝑡0 ) , 𝑑𝜏

𝑟

(24)

𝑟 𝑑ỹ̃ = f (ỹ̃ , 𝑧(+) 0 (1) , 1) , 𝑑𝜏1

ỹ (+∞) = 𝛽 (𝑡0 ) ,

̃ ̃ (0) = 𝑤1 , 𝑤

ỹ (−∞) = 𝛼 (𝑡0 ) .

𝑟 ỹ̃ (−∞) = 𝛽 (1) .

𝑑𝐿 0 𝑧 (𝜏0 ) = 0, 𝑑𝜏0 𝑑𝐿 0 y (𝜏0 ) 𝑑𝜏0 (25)

𝐿 0 𝑢 (0) = 𝑢∗ − 𝛼1 (0) , 𝐿 0 V (0) = V0 − 𝛼2 (0) ,

𝑙 By (25), we can solve 𝐿 0 𝑧(𝜏0 ) ≡ 0. Assuming that ỹ̃ = 𝛼(0) + 𝐿 0 y(𝜏0 ), (25) can be rewritten as 𝑙

𝑙 𝑑ỹ̃ = f (ỹ̃ , 𝑧(−) 0 (0) , 0) , 𝑑𝜏0

𝑙

̃̃V (0) = V0 , 𝑙 ỹ̃ (+∞) = 𝛼 (0) .

Equations (26) and (28) coincide with the associated system (7). By (A1), systems (26) and (28) have the equilibrium points ̃ 2 (𝜓(1), 𝑧(+) (1),1), respectively. We ̃ 1 (𝜑(0), 𝑧(−) (0), 0) and 𝑀 𝑀 0 0 will give the following hypothesis to obtain the solution of systems (26) and (28): 𝑙

𝑙

̃̃ (0) = 𝑢0 and ̃̃V (0) = V0 are (A5) The initial values 𝑢 intersected with the one-dimensional stable manifold ̃ 1 (0)) near the equilibrium point 𝑀 ̃ 1 , and the 𝑊𝑠 (𝑀 𝑟 ̃ ̃ (0) = 𝑤1 is intersected with the oneinitial value 𝑤 ̃ 2 (1)). dimensional unstable manifold 𝑊𝑢 (𝑀 By (A1)–(A4) and (20)–(24), 𝑄0(±) x(𝜏) is solved, which decays exponentially as 𝜏 → ±∞. Then, by (A5) and (25)– (28), we can solve 𝐿 0 x(𝜏0 ), which decays exponentially as 𝜏 → +∞, and 𝑅0 𝑧(𝜏1 ), which decays exponentially as 𝜏 → −∞. So, the following conclusion is obtained. Lemma 2. Under conditions (A1)–(A5) and (20)–(28), there exist the interior-layer functions 𝑄0(±) x(𝜏) and the boundary layer functions 𝐿 0 x(𝜏0 ), 𝑅0 x(𝜏1 ), which satisfy the following inequality:

𝐿 0 x (+∞) = 0.

̃̃ 𝑙 (0) = 𝑢0 , 𝑢

(28)

𝑟

By (A1), (A2), and (A3), system (24) has a solution, which is a heteroclinic orbit connecting 𝑀1 (𝛼(𝑡0 ), 𝑧(−) 0 (𝑡0 ), 𝑡0 ) with (𝑡 ), 𝑡 ). On the basis of (A4) and (10), the value 𝑀2 (𝛽(𝑡0 ), 𝑧(+) 0 0 0 (±) of 𝑡0 can be confirmed, so 𝑄0 y(𝜏) is determined completely. For 𝐿 0 x(𝜏0 ), we have

= f (𝛼 (0) + 𝐿 0 y (𝜏0 ) , 𝑧(−) 0 (0) + 𝐿 0 𝑧 (𝜏0 ) , 0) ,

(27)

𝑅0 𝑤 (0) = 𝑤1 − 𝛽3 (1) ,

Obviously, systems (21) and (23) coincide with the associated system (7), so we can consider their combined system

̃ (0) = 𝑢∗ , 𝑢

For 𝑅0 x(𝜏1 ), we have

(26)

󵄩 󵄩󵄩 (−) 󵄩󵄩𝑄0 x (𝜏)󵄩󵄩󵄩 ≤ 𝐶0 𝑒𝑘0 𝜏 , 󵄩 󵄩 󵄩󵄩 (+) 󵄩 󵄩󵄩𝑄0 x (𝜏)󵄩󵄩󵄩 ≤ 𝐶1 𝑒−𝑘1 𝜏 , 󵄩 󵄩 󵄩󵄩 󵄩󵄩 −𝑘 𝜏 󵄩󵄩𝐿 0 x (𝜏0 )󵄩󵄩 ≤ 𝐶2 𝑒 2 0 , 󵄩󵄩󵄩𝑅0 x (𝜏1 )󵄩󵄩󵄩 ≤ 𝐶3 𝑒𝑘3 𝜏1 , 󵄩 󵄩

(29)

where 𝐶𝑙 and 𝑘𝑙 (𝑙 = 0, 1, 2, 3) are all positive constants. Now, the coefficients of zero-order terms for (16) and (17) are completely determined. To determine functions y(±) 𝑖 (𝑡) (±) and 𝑧𝑖 (𝑡), we need the following hypothesis:


5

(±) (A6) The determinant of fy (y(±) 0 (𝑡), 𝑧0 , 𝑡) is not equal to zero all the time.

For x(±) 𝑖 (𝑡), we can obtain (±) (±) (y(±) 𝑖−1 (𝑡)) = f𝑦 y𝑖 (𝑡) + f𝑧 𝑧𝑖 (𝑡) + ℎ𝑖 (𝑡) ,

𝑖

𝑄𝑖(±) 𝑢 (0) = 𝑢𝑖∗ − ∑ [𝑢(±) 𝑖−𝑘 (𝑡0 )]

󸀠

(±) (±) (𝑧(±) 𝑖 (𝑡)) = 𝑔𝑦 y𝑖 (𝑡) + 𝑔𝑧 𝑧𝑖 (𝑡) + 𝑔𝑖 (𝑡) ,

𝑘=0

𝑧(−) 𝑖 (0) = −Π𝑖 𝑧 (0) , (𝑡0 ) +

𝑑𝑄𝑖(±) y ̃(±) (±) (±) = f 𝑦 𝑄𝑖 y + ̃f 𝑧 𝑄𝑖(±) 𝑧 + 𝐻𝑖(±) (𝜏) , 𝑑𝜏 𝑑𝑄𝑖(±) 𝑧 (±) ̃ (±) ̃ (±) =𝑔 𝑦 𝑄𝑖−1 y + 𝑔 𝑧 𝑄𝑖−1 𝑧 + 𝑁𝑖−1 (𝜏) , 𝑑𝜏

󸀠

𝑧(−) 𝑖

For 𝑄𝑖(±) x(𝜏) (𝑖 = 1, 2, . . .), we can obtain

𝑄𝑖(−) 𝑧 (0)

𝑖

+∑ 𝑘=1

(30) [𝑧(−) 𝑖−𝑘

(𝑡0 )]

(𝑘)

𝑡𝑘

𝑖

(+) (+) = 𝑧(+) 𝑖 (𝑡0 ) + 𝑄𝑖 𝑧 (0) + ∑ [𝑧𝑖−𝑘 (𝑡0 )] 𝑘=1

(𝑘)

(34)

𝑡𝑘 ,

𝑄𝑖(−) x (−∞) = 𝑄𝑖(+) x (+∞) = 0, (−) (−) (−) ̃ (±) where ̃f 𝑦 , ̃f 𝑧 , and 𝑔 𝑦 take value at (𝛼(𝑡0 ) + 𝑄0 y(𝜏), (−) ̃(+) ̃(+) ̃ 𝑧 take value at 𝑧(−) 0 (𝑡0 ) + 𝑄0 𝑧(𝜏), 𝑡0 ) and f 𝑦 , f 𝑧 , and 𝑔

(𝑘)

𝑡𝑘 ,

(±) where functions f𝑦 , f𝑧 , 𝑔𝑦 , and 𝑔𝑧 take value at (y(±) 0 (𝑡), 𝑧0 , 𝑡), while ℎ𝑖 (𝑡) and 𝑔𝑖 (𝑡) are known functions about y𝑚 (𝑡), 𝑧𝑚 (𝑡) (𝑚 = 0, 1, 2, . . . , 𝑖 − 1). On the basis of (A6) and the first equation of (30), we can ascertain 󸀠

−1 (±) (±) y(±) 𝑖 (𝑡) = f𝑦 [(y𝑖−1 (𝑡)) − f𝑧 𝑧𝑖 (𝑡) − ℎ𝑖 (𝑡)] .

(31)

(±) Inserting y(±) 𝑖 (𝑡) into the second equations of (30), 𝑧𝑖 (𝑡, 𝑐) is solved, where 𝑐 is an undetermined coefficient. For 𝐿 𝑖 x(𝜏0 ) (𝑖 = 1, 2, . . .), we can obtain

𝑑𝐿 𝑖 y ̃ = f 𝑦 𝐿 𝑖 y + ̃f 𝑧 𝐿 𝑖 𝑧 + 𝐺𝑖 (𝜏0 ) , 𝑑𝜏0 𝑑𝐿 𝑖 𝑧 ̃ 𝑦 𝐿 𝑖−1 y + 𝑔 ̃ 𝑧 𝐿 𝑖−1 𝑧 + 𝑀𝑖−1 (𝜏0 ) . =𝑔 𝑑𝜏0

(+) (±) (𝛽(𝑡0 ) + 𝑄0(+) y(𝜏), 𝑧(+) 0 (𝑡0 ) + 𝑄0 𝑧(𝜏), 𝑡0 ). 𝐻𝑖 (𝜏) is a known (±) (±) vector function about x𝑖−𝑛 (𝑡), 𝑄𝑖−𝑛 x(𝜏), and 𝑡𝑖−𝑛 (0 ≤ 𝑛 ≤ 𝑖 − 1). By the condition 𝑄𝑖(−) 𝑧(−∞) = 0 and the second 𝜏 (−) 𝑔(𝑠)𝑑𝑠, equation of (34), we can solve 𝑄𝑖(−) 𝑧(𝜏) = ∫−∞ 𝑄𝑖−1 (±) (±) (±) (±) ̃ 𝑦 𝑄𝑖−1 y + 𝑔 ̃ 𝑧 𝑄𝑖−1 𝑧 + 𝑁𝑖−1 (𝜏). We can where 𝑄𝑖−1 𝑔(𝜏) = 𝑔 know that 0

𝑄𝑖(−) 𝑧 (0) = ∫

−∞

(−) 𝑄𝑖−1 𝑔 (𝑠) 𝑑𝑠.

By (35) and the first equation of (34), 𝑄𝑖(−) x(𝜏) is solved, which decays exponentially as 𝜏 → −∞. Then, we can solve 𝑄𝑖(+) x(𝜏) which decays exponentially as 𝜏 → +∞. Inserting (35) into the second condition of (30), 𝑡𝑖 is solved. The boundary value function 𝑅𝑖 x(𝜏1 ) satisfies the following equations: 𝑑𝑅𝑖 y ̃ = f 𝑦 𝑅𝑖 y + ̃f 𝑧 𝑅𝑖 𝑧 + 𝐺𝑖 (𝜏1 ) , 𝑑𝜏1 𝑑𝑅𝑖 𝑧 ̃ 𝑦 𝑅𝑖−1 y + 𝑔 ̃ 𝑧 𝑅𝑖−1 𝑧 + 𝑀𝑖−1 (𝜏1 ) . =𝑔 𝑑𝜏1

(32)

(35)

(36)

𝑅𝑖 𝑤 (0) = 𝑤1 − 𝑤𝑖 (1) , 𝑅𝑖 x (−∞) = 0.

𝐿 𝑖 𝑢 (0) = −𝑢𝑖 (0) , 𝐿 𝑖 x (+∞) = 0, ̃ 𝑦 , and 𝑔 ̃ 𝑧 take value at (𝛼(0) + 𝐿 0 y(𝜏0 ), 𝑧0 (0) + where ̃f 𝑦 , ̃f 𝑧 , 𝑔 𝐿 0 𝑧(𝜏0 ), 0), 𝐺𝑖 (𝜏) is a known vector function about x𝑖−1 (𝑡) and 𝐿 𝑖−1 x(𝜏0 ), and 𝑀𝑖−1 (𝜏0 ) is a known vector function about x𝑖−2 (𝑡) and 𝐿 𝑖−2 x(𝜏0 ). By the second equation of (32), we can 𝜏0 ̃ 𝑦 𝐿 𝑖−1 y + 𝐿 𝑖−1 𝑔(𝑠)𝑑𝑠, where 𝐿 𝑖−1 𝑔(𝜏0 ) = 𝑔 solve 𝐿 𝑖 𝑧(𝜏0 ) = ∫+∞ 0

̃ 𝑧 𝐿 𝑖−1 𝑧 + 𝑀𝑖−1 (𝜏0 ), so 𝐿 𝑖 𝑧(0) = ∫+∞ 𝐿 𝑖−1 𝑔(𝑠)𝑑𝑠. By (30), we 𝑔 can know +∞

𝑧(−) 𝑖 (𝑡) = −𝐿 𝑖 𝑧 (0) = ∫

0

𝐿 𝑖−1 𝑔 (𝑠) 𝑑𝑠.

(33)

By the above condition and (28), x(±) 𝑖 (𝑡) is confirmed; similarly, we can solve (32).

The solution of (36) is similar to the solution of (32). From (36), we can verify that there exists the right boundary value function 𝑅𝑖 x(𝜏1 ), which decays exponentially as 𝜏 → −∞, and the following conclusion is obtained. Lemma 3. Systems (32)–(36) have the solutions 𝐿 𝑖 x(𝜏0 ), 𝑄𝑖(±) x(𝜏), and 𝑅𝑖 x(𝜏1 ), respectively, satisfying the following inequality: 󵄩 󵄩󵄩 −𝑘 𝜏 󵄩󵄩𝐿 𝑖 x (𝜏0 )󵄩󵄩󵄩 ≤ 𝐶0 𝑒 0 0 , 󵄩󵄩 (−) 󵄩 󵄩󵄩𝑄𝑖 x (𝜏)󵄩󵄩󵄩 ≤ 𝐶1 𝑒𝑘1 𝜏 , 󵄩 󵄩 󵄩 󵄩󵄩 (+) 󵄩󵄩𝑄𝑖 x (𝜏)󵄩󵄩󵄩 ≤ 𝐶2 𝑒−𝑘2 𝜏 , 󵄩 󵄩 󵄩󵄩 󵄩 𝑘 𝜏 󵄩󵄩𝑅𝑖 x (𝜏1 )󵄩󵄩󵄩 ≤ 𝐶3 𝑒 3 1 , where 𝐶𝑙 and 𝑘𝑙 (𝑙 = 0, 1, 2, 3) are all positive constants.

(37)

6


3. The Existence of Asymptotic Expansion

satisfying the boundary value conditions as follows: 𝑢 (0, 𝜇) = 𝑢0 ,

There are many methods to prove the existence of the steplike contrast structure for system (3) and we will prove the existence with implicit functions theorems [11]. The solutions of left and right problems can be expressed by (15) and (16) and we will prove that (15) and (16) are connected smoothly at the point of 𝑡∗ , which is on the neighborhood of 𝑡0 . At the point of 𝑡∗ , (15) and (16) can be expressed by ∗ (−) x(−) (𝑡∗ , 𝜇) = x(−) 0 (𝑡 ) + 𝑄0 x (0) + 𝑂 (𝜇) , ∗ (+) x(+) (𝑡∗ , 𝜇) = x(+) 0 (𝑡 ) + 𝑄0 x (0) + 𝑂 (𝜇) .

(38)

V (0, 𝜇) = V0 , 𝑤 (1, 𝜇) = 𝑤1 , 𝑧 (0, 𝜇) = 𝑧0 .

Assuming that 𝜇 = 0, we can solve three groups of isolated solutions: 3 3 𝑇 y = 𝛼 (𝑡) = (− , 0, − ) , 2 2

As 𝐿 0 x(𝜏0 ) and 𝑅0 x(𝜏1 ) decay exponentially as 𝑡 = 𝑡∗ , they can be neglected. Because the first two components of the solution for the left and right problems are equal correspondingly at the point of 𝑡∗ , we can solve 𝑡∗ by the third component 𝑤(±) . Supposing that 𝑈(𝑡∗ , 𝜇) = 𝑢(−) (𝑡∗ , 𝜇)− 𝑢(+) (𝑡∗ , 𝜇), we have

1 1 𝑇 y = 𝜒 (𝑡) = (𝑡 − , 0, 𝑡 − ) , 2 2 3 3 𝑇 y = 𝛽 (𝑡) = ( , 0, ) . 2 2

(39)

𝑑̃ 𝑢 = ̃V, 𝑑𝜏

∗

= 𝐻3 (𝑡 ) + 𝑂 (𝜇) . According to (A4), (𝜕𝐻2 /𝜕𝑡)|𝑡=𝑡0 and (𝜕𝐻3 /𝜕𝑡)|𝑡=𝑡0 are not simultaneously equal to zero. If (𝜕𝐻2 /𝜕𝑡)|𝑡=𝑡0 = 0, we can ascertain that (𝜕𝐻3 /𝜕𝑡)|𝑡=𝑡0 ≠ 0. On the basis of the implicit functions theorem, there exists 𝑡∗ (𝜇) = 𝑡0 + 𝑂(𝜇) causing 𝑈(𝑡∗ , 𝜇) = 0. So there exists a step-like contrast structure at the point of 𝑡∗ . If (𝜕𝐻2 /𝜕𝑡)|𝑡=𝑡0 ≠ 0, (𝜕𝐻3 /𝜕𝑡)|𝑡=𝑡0 = 0 and 𝑡∗ can be confirmed by the second component V. By the above discussion, the following theorem is obtained. Theorem 4. By (A1)–(A6) and Lemmas 1–3, there exists the step-like solution of (3) as follows:

1 9 𝑑̃V = (̃ 𝑢 − 𝑡 + ) (̃ 𝑢2 − ) , 𝑑𝜏 2 4

(−) (−) ∗ {x0 (𝑡) + 𝐿 0 x (𝜏0 ) + 𝑄0 x (𝜏) + 𝑂 (𝜇) , 0 ≤ 𝑡 ≤ 𝑡 , (40) = { (+) x + 𝑄0(+) x (𝜏) + 𝑅0 x (𝜏1 ) + 𝑂 (𝜇) , 𝑡∗ ≤ 𝑡 ≤ 1. { 0 (𝑡)

4. Example The equations are given by

The corresponding characteristic equation is given by (𝜆 + 1) [𝜆2 − 𝐹𝑦1 (𝑀1,2 )] = 0.

9 𝑑V 1 = (𝑢 − 𝑡 + ) (𝑢2 − ) , 𝑑𝑡 2 4

𝑑𝑤 3 𝜇 = (𝑢 − 𝑤) , √2 𝑑𝑡 𝑑𝑧 = 𝑢 + 𝑧, 𝑑𝑡

(41)

(45)

By (45), we can solve the following characteristic roots: 𝜆 1,2 = ±√𝐹𝑦1 (𝑀1,2 ) ,

(46)

𝜆 3 = −1, where 𝐹𝑦1 (𝑀1 ) = 3𝑡 + 3 > 0 and 𝐹𝑦1 (𝑀2 ) = 3(2 − 𝑡) > 0. So the two equilibrium points 𝑀1 (−3/2, 0, −3/2) and 𝑀2 (3/2, 0, 3/2) are all hyperbolic saddle points of (43). To determine 𝑡0 , we can discuss the equations 𝑄0(±) y(𝜏) as follows: 𝑑𝑄0(−) 𝑢 = 𝑄0(−) V, 𝑑𝜏

𝑑𝑢 𝜇 = V, 𝑑𝑡

(44)

𝑑̃ 𝑤 3 ̃) . = 𝑢−𝑤 (̃ 𝑑𝜏 √2

x (𝑡, 𝜇)

𝜇

(43)

According to (43), the solution of 𝑑𝑧/𝑑𝑡 = 𝑢 + 𝑧 exists. Considering the associated system,

𝑈 (𝑡∗ , 𝜇) = [𝛼3 (𝑡∗ ) + 𝑄0(−) 𝑤 (0)] − [𝛽3 (𝑡∗ ) + 𝑄0(+) 𝑤 (0)] + 𝑂 (𝜇)

(42)

𝑑𝑄0(−) V 𝑑𝜏 2 1 9 3 3 = (− + 𝑄0(−) 𝑢 − 𝑡0 + ) [(− + 𝑄0(−) 𝑢) − ] , 2 2 2 4

𝑑𝑄0(−) 𝑤 3 (𝑄(−) 𝑢 − 𝑄0(−) 𝑤) , = √2 0 𝑑𝜏


7

1 3 + 𝑄0(−) 𝑢 = 𝑡0 − , 2 2

−

Substituting 𝑡0 = 1/2 into (49), as 𝜏 ≤ 0, we have

𝑄0(−) y (−∞) = 0,

̃ (−) (𝜏) = 𝑢 (47)

𝑑𝑄0(+) 𝑢 = 𝑄0(+) V, 𝑑𝜏

̃V(−) (𝜏) =

𝑑𝑄0(+) V 𝑑𝜏

(−)

̃ 𝑤

2 1 9 3 3 = ( + 𝑄0(+) 𝑢 − 𝑡0 + ) [( + 𝑄0(+) 𝑢) − ] , 2 2 2 4

(48)

3 1 + 𝑄0(+) 𝑢 = 𝑡0 − , 2 2

̃V(+) (𝜏) =

𝑄0(+) y (+∞) = 0.

−2

(+)

̃ 𝑤

√2)𝜏

3𝐶 + 3𝑒−(3/

2𝐶 − 2𝑒−(3/√2)𝜏

2

4 (1 − 𝐶𝑒−(3/√2)𝜏 )

1 9 (𝑠 − 𝑡0 + ) (𝑠2 − ) 𝑑𝑠 (𝜏)] = 2 ∫ 2 4 −(3/2)(3/2)

𝑄0(+) V (𝜏) =

= 0. Since the solutions of (41) are connected smoothly at the point ̃ (−) (0) = 𝑢 ̃ (+) (0). Assuming that of 𝑡0 , we have 𝑢 ̃ (−) (0) − 𝑢 ̃ (+) (0) = 0, 𝐻 (𝑡0 ) = 𝑢

,

√ 9√2𝐶𝑒−3/ 2 𝜏

,

2

4 (1 − 𝐶𝑒(3/√2)𝜏 )

√2)𝜏

3𝑒−(3/

𝐶 − 𝑒−(3/√2)𝜏

(55)

,

√ 9√2𝐶𝑒−(3/ 2) 𝜏 2

4 (1 − 𝐶𝑒−(3/√2)𝜏 )

,

󵄨󵄨 󵄨󵄨 √ 2 ln 󵄨󵄨󵄨𝐶𝑒−(3/ 2)𝜏 − 1󵄨󵄨󵄨 󵄨. 󵄨 = 𝐶𝑒−(3/√2)𝜏

Similarly, the boundary layer function 𝐿 0 y(𝜏0 ) can be expressed by 𝐿 0 𝑢 (𝜏0 ) =

3/2

1 9 (𝑠 − 𝑡0 + ) (𝑠2 − ) 𝑑𝑠 = 0. 2 4 −3/2

𝐶𝑒(3/√2)𝜏 − 1

(51)

substituting (50) into (51), we can obtain ∫

𝑄0(+) 𝑤 (𝜏)

√2)𝜏

3𝐶𝑒(3/

󵄨󵄨 󵄨󵄨 √ 2 ln 󵄨󵄨󵄨𝐶𝑒(3/ 2)𝜏 − 1󵄨󵄨󵄨 󵄨, 󵄨 =3+ 𝐶𝑒(3/√2)𝜏

𝑄0(+) 𝑢 (𝜏) =

(50)

(54)

So we can solve the interior-layer functions as follows:

𝑄0(−) 𝑤 (𝜏)

̃ (∓) 𝑢

2

,

󵄨󵄨 −(3/√2)𝜏 󵄨󵄨 󵄨 − 1󵄨󵄨󵄨 3 2 ln 󵄨󵄨󵄨𝐶𝑒 󵄨. (𝜏) = + 2 𝐶𝑒−(3/√2)𝜏

𝑄0(−) V (𝜏) =

The first integral of (49) passing through 𝑀1 and 𝑀2 is

(53)

,

√ 9√2𝐶𝑒−3/ 2 𝜏

2 𝑑̃V(±) 1 9 = (̃ 𝑢(±) − 𝑡0 + ) [(̃ 𝑢(±) ) − ] , 𝑑𝜏 2 4

1 ̃ (±) (0) = 𝑡0 − . 𝑢 2

,

2

4 (1 − 𝐶𝑒(3/√2)𝜏 )

𝑄0(−) 𝑢 (𝜏) =

(49)

,

√ 9√2𝐶𝑒−3/ 2 𝜏

𝑑̃ 𝑢(±) = ̃V(±) , 𝑑𝜏

𝑑̃ 𝑤(±) 3 ̃(±) ) , (̃ 𝑢(±) − 𝑤 = √2 𝑑𝜏

[̃V

+3

2𝐶𝑒(3/√2)𝜏

As 𝜏 ≥ 0, we can solve ̃ (+) (𝜏) = 𝑢

(±)

√2)𝜏

󵄨󵄨 󵄨󵄨 (3/√2)𝜏 󵄨 − 1󵄨󵄨󵄨 3 2 ln 󵄨󵄨󵄨𝐶𝑒 󵄨. (𝜏) = + √2)𝜏 (3/ 2 𝐶𝑒

𝑑𝑄0(+) 𝑤 3 (𝑄(+) 𝑢 − 𝑄0(+) 𝑤) , = √2 0 𝑑𝜏

̃(±) = ̃ (±) = ±3/2 + 𝑄0(±) 𝑢, ̃V(±) = 𝑄0(±) V, and 𝑤 Assuming that 𝑢 ±3/2 + 𝑄0(±) 𝑤, (47) and (48) can be expressed by

3𝐶𝑒(3/

(52)

On the basis of (52), we can solve 𝑡0 = 1/2 and 𝜕𝐻(𝑡0 )/𝜕𝑡0 = 21/4 ≠ 0. So the solutions of (41) transfer at the point of 𝑡0 = 1/2.

𝐿 0 V (𝜏0 ) =

√2)𝜏0

3𝑒−(3/

𝑒−(3/√2)𝜏0 − 𝐴

,

√ 9√2𝐶𝑒−(3/ 2) 𝜏0

4 (1 − 𝐴𝑒−(3/√2)𝜏0 )

2

,

󵄨󵄨 󵄨󵄨 √ 2 ln 󵄨󵄨󵄨𝐴𝑒−(3/ 2)𝜏0 − 1󵄨󵄨󵄨 󵄨 󵄨. 𝐿 0 𝑤 (𝜏0 ) = √2)𝜏0 −(3/ 𝐴𝑒

(56)

8


The boundary layer function 𝑅0 y(𝜏1 ) can be expressed by 𝑅0 𝑢 (𝜏1 ) = 𝑅0 V (𝜏1 ) =

√2)𝜏1

3𝑒(3/

𝐵 − 𝑒(3/√2)𝜏1

,

√ 9√2𝐶𝑒3/ 2 𝜏1

√2)𝜏 2

4 (1 − 𝐵𝑒1(3/

)

,

(57)

󵄨󵄨 󵄨󵄨 √ 2 ln 󵄨󵄨󵄨𝐴𝑒(3/ 2)𝜏1 − 1󵄨󵄨󵄨 󵄨 󵄨, 𝑅0 𝑤 (𝜏1 ) = −3 + 𝐴𝑒(3/√2)𝜏1 so we can construct a zero-order asymptotic solution of (41) and (42) as follows: x (𝑡, 𝜇) (−) (−) {x0 (𝑡) + 𝐿 0 x (𝜏) + 𝑄0 x (𝜏) + 𝑂 (𝜇) , = { (+) x + 𝑄0(+) x (𝜏) + 𝑅0 x (𝜏1 ) + 𝑂 (𝜇) , { 0 (𝑡)

0 ≤ 𝑡 ≤ 𝑡∗ , (58) 𝑡∗ ≤ 𝑡 ≤ 1,

𝑇 where 𝑡∗ = 1/2 + 𝑂(𝜇), x(−) 0 (𝑡) = (−3/2, 0, −3/2) , and 𝑇 x(+) 0 (𝑡) = (3/2, 0, 3/2) .

5. Conclusive Remarks By the boundary layer function method and smooth connection, we study the contrast structure for a class of semilinear singularly perturbed systems. Under some assumptions, the existence of a step-like contrast structure of system (3) and a heteroclinic orbit connecting two equilibrium points of the corresponding associated systems is determined. Then, we obtain the asymptotic solution of system (3). In comparison with [8, 9], the system we study is more general.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution Han Xu completed the main part of this paper, and Yinlai Jin corrected the main theorems.

Acknowledgments This work is supported by the National Natural Science Funds (no. 11201211), Shandong Province Higher Educational Science and Technology Program (no. J13LI56), Shandong Province Higher Educational Youth Backbone Teachers Domestic Visiting Scholars Project, and Applied Mathematics Enhancement Program of Linyi University.

References [1] V. F. Butuzov, A. B. Vasileva, and N. N. Nefedov, “The asymptotic theory of contrast structures,” Automation and Remote Control, vol. 3, pp. 4–32, 1997.

[2] A. B. Vasileva, V. F. Butuzov, and N. N. Nefedov, “Contrast structures in singularly perturbed problems,” Fundamentalnaya i Prikladnaya Matematika, vol. 4, no. 3, pp. 799–851, 1998. [3] A. F. Wang, “The spike-type contrast structure for a secondorder semi-linear singularly perturbed differential equation with integral boundary condition,” Mathematica Applicata, vol. 25, no. 2, pp. 363–368, 2012. [4] X.-B. Lin, “Heteroclinic bifurcation and singularly perturbed boundary value problems,” Journal of Differential Equations, vol. 84, no. 2, pp. 319–382, 1990. [5] Y. L. Jin, F. Li, H. Xu, J. Li, L. Zhang, and B. Ding, “Bifurcations and stability of nondegenerated homoclinic loops for higher dimensional systems,” Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 582820, 9 pages, 2013. [6] Y. L. Jin, X. W. Zhu, Z. Guo, H. Xu, L. Zhang, and B. Ding, “Bifurcations of nontwisted heteroclinic loop with resonant eigenvalues,” The Scientific World Journal, vol. 2014, Article ID 716082, 8 pages, 2014. [7] M. K. Ni and Z. M. Wang, “On higher-dimensional contrast structure of singularly perturbed Dirichlet problem,” Science China Mathematics, vol. 55, no. 3, pp. 495–507, 2012. [8] A. F. Wang, “The step-type contrast structure for a singularly perturbed system with slow and fast variables,” Journal of Shandong University, vol. 48, no. 2, pp. 98–104, 2013. [9] Z. N. Ma and Y. C. Zhou, The Mathematics Modeling about Epidemic Dynamics, Science Press, Beijing, China, 2004. [10] J. Chattoadhyay and N. Bairagi, “Pelicans at risk in salton sea an ecoepidemiological model,” Ecological Modelling, vol. 136, no. 2-3, pp. 103–112, 2001. [11] H. Xu, “The two-order quasi-linear singular perturbed problems with infinite initial conditions,” Advances in Civil and Industrial Engineering, vol. 1, pp. 3248–3250, 2013. [12] H. Xu and Y. L. Jin, “The asymptotic solutions for a class of nonlinear singular perturbed differential systems with time delays,” The Scientific World Journal, vol. 2014, Article ID 965376, 7 pages, 2014.

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