Ulam’s Type Stability and Fixed Points Methods 2016View this Special Issue
Research Article | Open Access
Existence of Generalized Homoclinic Solutions of Lotka-Volterra System under a Small Perturbation
This paper investigates Lotka-Volterra system under a small perturbation , . By the Fourier series expansion technique method, the fixed point theorem, the perturbation theorem, and the reversibility, we prove that near the system has a generalized homoclinic solution exponentially approaching a periodic solution.
The Lotka-Volterra system introduced by Lotka  and Volterra  is frequently used to describe the dynamics of biological systems and particularly plays an important role in studying population dynamics. Another reason for the importance of such system is that it has been successfully used in economic theory . Many authors have studied the existence and uniqueness of stationary solution for a variety of Lotka-Volterra equations (e.g., Dancer  and Gui and Lou ). In recent years, it has been broadly studied in many aspects. Stability was discussed in Sakthivel et al. . Travelling wave solutions were studied by Hosono  and Lin and Ruan . The multiple periodic solutions were obtained in Wang et al. . The bifurcation structures are given by Kan . Limit cycles bifurcations were discussed by Wu and Liang . The cyclicity of period annulus was analyzed in Li and Llibre .
In this paper, we will study the most well-known Lotka-Volterra systemwhere , , and are constants and satisfy , .
Motivated by [13, 14], we will study the existence of generalized homoclinic solutions of this Lotka-Volterra system. Since the real world always has some small noises or disturbances, here we are specially interested in the solutions of system (1) under small perturbations; that is, we will focus on the following system:where is a small parameter and the general nonlinear terms and are smooth real functions. When this system has four equilibriums , , , and . When the linearized operator of (2) at has two pairs of purely imaginary eigenvalues. When the linearized operator of the system at has double real eigenvalues 0 and a pair of purely imaginary eigenvalues. When the linearized operator at has a positive eigenvalue, a negative eigenvalue, and a pair of purely imaginary eigenvalues. This yields that the linearized operator of system (2) has a saddle-center near while , which means is the bifurcation point for system (2). We guess there might exist a homoclinic solution exponentially approaching a periodic solution (called generalized homoclinic solution) near . Using the dynamical system approach, we will rigorously prove the existence of a generalized homoclinic solution (homoclinic solution exponentially tending to a small periodic solution at infinity) for but close to 0, which corresponds to a generalized solitary wave solution of (2).
This paper is organized as follows. Section 2 derives the homoclinic solution of the dominant system of (2), which exponentially approaches zero at infinity. Section 3 shows that the system of (2) has a periodic solution. In Section 4, the fixed point theorem and the perturbation method yield that when higher order terms are added, the homoclinic solution of the dominant system deforms to a homoclinic solution exponentially approaching the periodic solution obtained in Section 3. This gives the existence of a generalized homoclinic solution of (2).
Let and , and we change (2) intoIn this paper, we assume that and satisfyUnder assumption (4), system (3) is reversible with a reverser defined bythat is, is also a solution whenever is a solution of (3). A solution is reversible if . This means that and are even functions and and are odd functions. The reversibility will play an important role in the existence of the generalized homoclinic solution.
Letand we have from (4)Note that (6) changes system (3) intoSystem (8) can be written aswhere andFrom (7), system (9) is still reversible with the reverser ifWe write the dominant system of (9) for small and aswhich has a homoclinic solution given byapproaching zero as . Moreoverand satisfies the following inequality:where is a positive constant.
3. Periodic Solutions
By using the Fourier series expansion technique and the fixed point theorem, we will prove that (9) has a periodic solution which determines the forms of the generalized homoclinic solutions at infinity. Letwhere is a small real constant to be determined later. Then system (8) is changed intowhere is a real function and is a purely imaginary function andFrom (7) and (11), we may definesuch that system (17) is reversible where we avoid again the introduction of a new notation.
Let be a space of periodic functions of with a period such that their derivatives up to order are in ; the norm is denoted by . Fix and define two spacesFor and , use (21) and we define a mapping from to itself bywhere . Assume that is a ball with a radius in the space ; we have the following lemma.
Lemma 1. For and any small bounded and , is smooth in its arguments and satisfies
Take andwhere , , , and are fixed constants; Lemma 1 yields that is a contraction mapping on for small . Thus, has a unique fixed point which is a smooth function of ; write this fixed point aswhich satisfiesUsing the same argument we can show that (27) is in and satisfies (28) with -norm for any integer . We use to denoteNow we solve (22) for . Substitute (27) into (22) and obtainwhereis smooth when , , , , are near 0. According to (26) and (18) we get , so is a contraction mapping. By the fixed point theorem, has a unique fixed pointas a smooth real function for small satisfyingTherefore, (17) has a periodic solution in if and (26) holds where is a fixed small positive constant.
By the relation , we write the periodic solution as with the frequencyfor . Letting , we haveDefinewhich is smooth for and small with condition (26). Then, is a reversible periodic solution of (9) under the reserve with frequency which from (28) satisfies that for any integer The Sobolev embedding theorem gives that (38) holds in -norm, which is a space of continuously differentiable functions up to the order with a supreme norm.
4. Existence of Generalized Homoclinic Solutions
Theorem 2. Suppose that assumption (4) holds. There exist constants , , and such that, for , if the small parameters and , then (3) has a generalized homoclinic solution which is reversible and exponentially approaches the periodic solutions as and as , where the phase shift is a continuous function in and the operator is defined in (5).
We divide the proof into two steps. Using the relationship between (3) and (9), we will first prove that (9) has a solution for , which exponentially approaches the periodic solution for some phase shift as . Then we solve for as a function of , , and . This yields that this solution can be extended to by using the reversibility.
Step 1 (existence of generalized homoclinic solution for ). Assume that the solution of (9) has the following form:where and are defined in (13) and (37), respectively, the phase shift is a constant, and the cut-off function is in satisfying andwhere is a perturbation term to be determined, which exponentially tends to 0 as so that is a solution of (9) that approaches the periodic solution as . Plugging (39) into (9) yieldswhere and means taking the Fréchet derivative.
Lemma 3. If , , and are small and for some positive constant , then and satisfy for
Lemma 4. The solution of (39) that decays to zero at infinity can be found as
Proof. has four linearly independent solutionswhich satisfyfor , andThe adjoint equation of (45) has four linearly independent solutions given bywhich satisfyfor each ; we havewhere denotes the Euclidean inner product on . From the above conditions we can get the expression of in Lemma 4.
Lemma 5. The function satisfiesfor
For any fixed constant , we let andwhere are positive constants for . Thus, (26) is satisfied, and we can show from Lemma 4 that is a contraction on for small . Therefore, (44) has a unique solution satisfyingUsing the same argument as that for (56) and an extension of a contraction principle , we can show that is smooth in its argument. Thus the solution of (41) exists if is in a finite interval and an initial condition is given, so that defined in (39) exists for with any fixed .
Step 2 (reversible generalized homoclinic solution for ). Using (5), (39), and the relationship between and in (6), we may define where . In this step, we show that system (3) has a reversible generalized homoclinic solution for . Depending on the reversibility of systems, this problem is equivalent to solving the following equation:for . By (14), the definition of the cut-off function in (40), and , it is easy to check that (58) is equivalent toUsing (48) and (44) we know that (59) holds automatically. Thus we only need study (60) which can be transformed to
Lemma 6. Under the assumptions of Theorem 2, (61) can be transformed towhere is differentiable with respect to its arguments and and its derivative with respect to are uniformly bounded for small bounded , , and .
The proof of Lemma 6 is given in the appendix.
Using the fixed point theorem and Lemma 3, we can solve (62) for as a smooth function of , so (60) is true. Uniqueness of the solution for an initial value problem implies that is a generalized homoclinic solution of (3) and , which exponentially approaches the periodic solution as and the periodic solution as . This completes the proof of Theorem 2.
In this section, we will give the proof for Lemma 6. Letwhere , , and are given in (32) and (37), respectively, which yieldsThus, is a -periodic solution of the following system:where and are given in (18). We can express aswhereWe know that the coefficient of is . Thus,