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 ℝ(3).

1. Introduction

It is well known that n-dimensional generalized Lotka-Volterra systems are widely used as the first approximation for a community of n 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. [2-4] 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 X (t) is the number of individuals in the ith population at time t and X (t) ≥ 0, β is the intrinsic growth rate of the ith population, the α are interaction coefficients measuring the extent to which the jth species affects the growth rate of the ith, β and α 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. [5-12]). In [13], Bobieński and Żołądek 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 ℝ3 with two small parameters λ 1 and λ 2 and other bounded parameters. In Section 3, we first change the system (1) into a system of the form where a , b , and c for i, j, k = 0,1, 2 are functions of the parameters β and α in system (1), u and v > 0 are bounded parameters, and 0 < ε ≪ 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.

2. Preliminary Theorems

In this section, we first consider a normal form system of degree two in ℝ3. 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.

Consider the 3-dimensional system where X 2(x) = O(|x|2) is C in x ∈ ℝ3, and By adding up the 2-parameter linear part diag⁡(λ 1, λ 1, λ 2)x we obtain where D(λ 1, λ 2) = diag⁡(A(λ 1), λ 2), with It can be verified that (5) has the following real normal form up to order 3 (see [15]): For convenience, we assume that a 1 b 1 c 1 ≠ 0 as in [15]. By the scaling (7) becomes where Then, by introducing polar coordinates (9) further becomes where S 1, S 2, and S 3 are 2π periodic in θ, and S 1, S 2, and S 3 = O(|p, x 3|4). By a further scaling of the form (12) becomes We obtain from (14) where Note that the functions and in (15) are 2π periodic in θ but may not be well defined at p = 0. Thus, we suppose p ≫ ε > 0 for (15).

The averaging system has a singular point (p 0, s 0) on the half plane p > 0 if where . By denoting we obtain , and the characteristic polynomial of B is . We define

According to Theorem 4.1.3 in [15], we can obtain the following theorem.

Theorem 1

Suppose that (18) holds. Then, (7) has a periodic orbit near the origin for 0 < ε ≪ 1. Further, the periodic orbit is stable (resp. unstable) if one (resp. none) of the following conditions holds:

Δ = 0 and ,

Δ < 0 and ,

Δ > 0, , and ,

where Δ is given by (20).

Then, by letting s = x 3 + δ 1 and θ → ε −1 θ and truncating the terms of order ε 2, we have from (15) where Thus, in order that (21) has a limit cycle, we necessarily suppose and , , that is, This yields , and hence (21) becomes where For small ε > 0, (25) has a focus A (p(ε), s(ε)) with We define By using the coefficients in (7), we have Then, in 1997, the following result was obtained in [15].

Theorem 2

Suppose that (24) holds and Δ0 ≠ 0. Then, for any given ε 1 > 0 there exist an ε 0 > 0 and a C 1 function ϕ 0(λ 1) = (2d 1 λ 1/a 1) + δ 0 λ 1 2 + O(λ 1 3) and ϕ 1(λ 1) = (2d 1 λ 1/a 1) + δ 0′λ 1 2 + O(λ 1 3) such that for 0 < λ 1 2 + λ 2 2 < ε 0, (7) has a unique invariant torus near the origin if Δ0 ϕ 1(λ 1) − ε 1 λ 1 2 < Δ0 λ 2 < Δ0 ϕ 0(λ 1) and has no invariant torus if Δ0 λ 2 > Δ0 ϕ 0(λ 1). Moreover, the torus, if it exists, is stable (resp. unstable) when Δ0 < 0 (resp. >0).

3. Normal Form of System (2)

In this section, we consider system (1) in the first octant ℝ+ 3, where ℝ+ = {x ∈ ℝ : x > 0}. We now look for the conditions for the existence of positive equilibria of system (1), which is equivalent to find the positive solutions of the following 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 Y = X − 1, i = 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 M(ε) = (α (ε))3×3, and we suppose M(ε) is similar to 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) becomes where Let by changing y = Tx, where y = (y 1, y 2, y 3), x = (x 1, x 2, x 3), and system (35) becomes a complex system of the form where

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).

We denote (38) as , where F(0) = 0, and for simplicity, we write the nonlinear part of (38) as Θ(y). By doing the following transformation: where P(z) = (P 1(z), P 2(z), P 3(z)), which is to be determined, (38) becomes Then, by noting we can get from (41) where γ 1 = λ 1 + i and γ 2 = λ 1 − i, γ 3 = λ 2. In order to eliminate the quadratic homogeneous polynomial, we need We take P , i = 1,2, 3 as quadratic homogeneous polynomial, having the form where l , k = 1,…, 6, are real undermined coefficients. By inserting (45) into (44) and comparing the coefficients of similar items, we can obtain Note that |γ 3 | = |λ 2 | ≪1, |γ 1 + γ 2 − γ 3 | = |2λ 1 − λ 2 | ≪1. The terms with coefficients l 15, l 26, l 33, and l 34 that appeared above cannot be removed. Those terms are called the resonance terms. Then, we have and system (43) becomes Let L(z) denote the cubic terms in z of (48). Then, from (41) and (42) we have where P = ∂P /∂z , i, j = 1,2, 3, By substituting (42) into the above, we obtain

where

We make a further change z = w + Q(w) ≡ h 2(w), where Q = (Q 1, Q 2, Q 3) is homogeneous cubic polynomial, so that (48) becomes where Q = ∂Q /∂w , i, j = 1,2, 3 and L has the form as before. In order to eliminate some possibly cubic terms, we consider the equations below Suppose that for i = 1,2, 3, By inserting these representations into (54), we can solve as before Hence, system (53) becomes now where w and all of the coefficients are complex. Finally making the change w = Tx and then taking the real parts of x and the coefficients of all terms of the resulting system, we can get a cubic real normal form of the form (7) with

Then, by the equations in (36), we finally get the relationship between the coefficients of the system (2) and of the normal form (7).

4. Examples

4.1. 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.

We consider the following three-parameter Lotka-Volterra system in the first octant ℝ+ 3. Consider where 0 < ε ≪ 1, v > 0 and u are bounded parameters.

First of all, we need to change the system (59) to the form of system (2) as in [14]. It can be checked that the point (1,1, 1) is zero-Hopf equilibrium of system (59). We do the change of variables X = x − 1, Y = y − 1, and Z = z − 1 to obtain The Jacobian matrix of system (60) at (0,0, 0) has eigenvalues ε, εu + vi and εu − vi with v > 0. According to [14], in order to obtain the real Jordan normal form of system (60) at the origin, we do the linear transformation where p 11 = −(uε 2 + 3vε + 3v uε + 5v 2)/(v(v + ε)), p 12 = −(−v 2 + v uε + vε + uε 2)/(v(v + ε)), p 13 = 2v/(v + ε), p 21 = (3v + ε)/(v + ε), p 31 = −(6v 2 + 6v uε + u 2 ε 2)/2v 2, and p 32 = −uε(2v + uε)/2v 2. Then, in the new variables (U 1, V 1, and W 1) system (60) becomes where a , b , and c have the following expressions:

Next, we need to calculate the partial coefficients of the normal form of system (63). We can get , , and by (36), and then by the formulas of (58) we have By Theorem 1, we have the following conclusion.

Theorem 3

For any given ε 0 > 0, suppose that −7/(24 + 7ε 0) < u < 0, and then for 0 < ε ≪ ε 0, (59) has a periodic orbit near the origin, which is unstable.

Proof

In this example, it is easy to see that δ 1 = 1, δ 2 = 1/u. From (64) and (10) we can get , thus, in order to satisfy (18), we need For any given ε 0 > 0, suppose that −7/(24 + 7ε 0) < u < 0. It can be checked that for 0 < ε ≪ ε 0. Then, by Theorem 1, (59) has a periodic orbit near the origin. Next, we consider the stability of the periodic orbit.

From (64), we can also get when −7/(24 + 7ε 0) < u < 0 holds, where Δ is given by (20). So none of the conditions (a), (b), or (c) in Theorem 1 holds; further, we know that the periodic orbit is unstable.

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).

4.2. 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 < ε ≪ 1.

According to Section 3, we have Let A 1 = 1666324684/40625 and A 2 = 1731664/3125. Then, we have the following theorem.

Theorem 5

For any given 0 < ε 1 < A 1 − A 2, there exists an ε 0 > 0 such that for 0 < ε < ε 0, (68) has a unique invariant torus near the origin, which is unstable.

By (69) and (29), we can obtain Thus, for 0 < ε ≪ 1, Δ0 > 0. Further, we can get where ϕ 1(λ 1) and ϕ 0(λ 1) are defined in Theorem 2, and here λ 1 = ε and λ 2 = −(24/5)ε. By some easy calculations, we can obtain that for 0 < ε 1 < A 1 − A 2 inequality Δ0 ϕ 1(λ 1) − ε 1 λ 1 2 < Δ0 λ 2 < Δ0 ϕ 0(λ 1) holds. Thus, by Theorem 2 we can get the result in this theorem.