The Existence of Periodic Orbits and Invariant Tori for Some 3-Dimensional Quadratic Systems

We use the normal form theory, averaging method, and integral manifold theorem to study the existence of limit cycles in Lotka-Volterra systems and the existence of invariant tori in quadratic systems in ℝ3.


Introduction
It is well known that -dimensional generalized Lotka-Volterra systems are widely used as the first approximation for a community of interacting species, each of which would exhibit logistic growth in the absence of other species in population dynamics. And this system is of wide interest in different branches of science, such as physics, chemistry, biology, evolutionary game theory, and economics. We refer the reader to the book of Hofbauer and Sigmund [1] for its applications. The existence of limit cycles and invariant tori for these models is interesting and significant in both mathematics and applications since the existence of stable limit cycles and invariant tori provided a satisfactory explanation for those species communities in which populations are observed to oscillate in a rather reproducible periodic manner (cf. [2][3][4] and references therein).
To study the bifurcation of Lotka-Volterra class, we consider three-dimensional generalized Lotka-Volterra systems which describes the interaction of three species in a constant and homogeneous environment, where ( ) is the number of individuals in the th population at time and ( ) ≥ 0, is the intrinsic growth rate of the th population, the are interaction coefficients measuring the extent to which the th species affects the growth rate of the th, and are parameters, and the values of these parameters are not very small usually.
Over the last several decades, many researchers have devoted their effort to study the existence and number of isolated periodic solutions for system (1). There have been a series of achievements and unprecedented challenges on the theme even if system (1) is a competitive system (cf. [5][6][7][8][9][10][11][12]). In [13], Bobieński andŻołądek gave four components of center variety in the three-dimensional Lotka-Volterra class and studied the existence and number of isolated periodic solutions by certain Poincaré-Melnikov integrals of a new type. In [14], Llibre and Xiao used the averaging method to study the existence of limit cycles of three-dimensional Lotka-Volterra systems. In this paper, we will use the normal form theory to study the same question. And furthermore, we will give the existence of invariant tori in a system of the form (2). This paper is organized as follows. In Section 2, we obtain some preliminary theorems about a normal form system of degree two in R 3 with two small parameters 1 and 2 and other bounded parameters. In Section 3, we first change the system (1) into a system of the form The Scientific World Journal where , , and for , , = 0, 1, 2 are functions of the parameters and in system (1), and V > 0 are bounded parameters, and 0 < ≪ 1 is perturbation parameter. And then we get the real normal form of the system (2) after a series of transformations. Two examples are provided to illustrate these results in the last section.

Preliminary Theorems
In this section, we first consider a normal form system of degree two in R 3 . Then, by a series of transformations we introduce some theorems for the normal form. The reader is referred to [15] for more details about the following content.

Normal Form of System
We suppose that there exists at least one positive solution of (30). Without loss of generality, we assume that the positive equilibrium is (1, 1, 1). Then, we move it to the origin by doing the change of variables = − 1, = 1, 2, 3. Then, system (1) can be written as Now, we shall investigate a special form of system (31) with a small parameter; we write the perturbed system as Denote ( ) = ( ( )) 3×3 , and we suppose ( ) is similar to (33) Then, system (32) can be changed into the system (2) by a linear transformation. In this section, our task is to change system (2) into the normal form of (7). Making the transformation system (2) becomeṡ by changing , and system (35) becomes a complex system of the forṁ1 The Scientific World Journal By the fundamental theory of normal form [16], we know that system (38) can be converted to the normal form by some transformations. So our following task is to find the transformations and work out the normal form of system (38).

An Example about the Existence of a Limit Cycle in
Three-Dimensional Lotka-Volterra Systems. In this section, we construct a concrete example of three-dimensional Lotka-Volterra systems according to Theorem 1. It is shown that this system undergoes nonisolated zero-Hopf bifurcation.
Remark 4. From (63), we can find out that system (59) does not satisfy the conditions mentioned in [14]. Thus, we cannot use the results in [14] to study the existence of a limit cycle in (59).

An Example about the Existence of an Invariant Torus.
For convenience, we give an example about the existence of an invariant torus in a system, which has the form of (2). We consider the following system in the first octant R 3 + : where 0 < ≪ 1. According to Section 3, we have