Abstract
A class of cubic systems with two invariant straight lines is studied. It is obtained that the focal quantities of are, ; if , then ; if , then ; if , then is a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focal . This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).
1. Introduction
The study of the polynomial differential system attracts more and more researchers because of the Hilbert's 16th problem [1–5]. The major problem of the polynomial differential system is to calculate the highest order of focal quantities (also known as focal values, or Lyapunov exponents) at its focal and to decide how many limit cycles surrounding a singular point; the system generated at least under some perturbation of coefficients. All this problem is still open.
There are many papers to study the Kukles system, and many achievements are reached, which include the calculation of the focal quantities and decision of the maximum number or limit cycles of the system. Such as paper [5], Hill et al. had studied a class of cubic differential systems and also brought to our attention that a system used to model predator-prey interactions with intratrophic predation could be transformed so that it can be an example of a system of type (1.1).
In papers [6, 7], the authors consider a class of cubic Kukles systems : where . Paper [6] has proved that the focal quantities of in (1.1) are , if , is a center, and if , system (1.1) has at most one limit cycle surrounding .
Paper [8] considers a class of cubic systems: where , and has proved that the focal quantities of in (1.2) are, , if , is a center, and if , system (1.2) has at most one limit cycle surrounding . But in Paper [8], the case is not considered.
Recently, in paper [9], the authors consider a class of cubic systems: where . Paper [9] has proved that the focal quantities of in (1.3) are, . If , is a center, and if , system (1.3) has at most one limit cycle surrounding .
In this paper, we consider the following cubic system It comes from system (1.1) by adding some invariant straight line, so system (1.4) is said to be the accompany system of (1.1), and it is also said that system (1.1) and (1.4) is part of the accompany system. Paper [10] introduces the concept of accompany system, and studies the qualitative property of some accompany system.
Now without loss of generality, we may assume that (if , let , then change its sign), and may assume that (if , let , then change its sign, but does not change its sign). So we study the system (1.4) with .
System (1.4) has a critical point and , if , and other critical points (if have) lie on the invariant straight line . Now we transform (1.4) into Lienard equation; note that
Let then (1.5) can be reduced to where , , , and let System (1.4) can be reduced to
2. The Problem of the Center or Focal for Critical Point
In this section, since we will study the problem of center or focus for critical point , we take in system (1.8). In order to calculate the focal quantities of system (1.4) (or system (1.8)) in , we need to let , and only consider, , that is , so
So
We use method of paper [11], so need to be written in the power series as follows: We use mark in paper [11]; let , and , then If , then since , that is, , so From paper [11], if then is stable(unstable) weak focal of order one. So for system (1.4), ; if , then , and if , then is unstable (stable) weak focal of order one; If , that is, , then (note that and , and have opposite signs). If and , we will prove that is a center of system (1.4). Since , that is, , so or If then system (1.4) can be reduced to This system is symmetry about -axis because of , so is a center. If , then system (1.4) can be reduced to It is integrable system, so is a center, hence we have the following theorem.
Theorem 2.1. For system (1.4), let ; the focus quantities of are if , then if , then if then is a center. If , or , or then is an unstable (stable) critical point. If then system (1.4) has at least two limit cycles surrounding .
3. Nonexistence of Limit Cycle Surrounding
In this section, we study the nonexistence of limit cycle surrounding the weak focal . is weak focal if and only if so we let in system (1.4).
Lemma 3.1. If , system (1.4) has no limit cycles surrounding .
Proof. Since system (1.4) forms a generalized rotated vector field with respect to parameter (refer to paper [12, page 241]), and when , is a center, so when , system (1.4) has no limit cycles (refer to paper [12, page 244, th. ]).
By Lemma 3.1, we let in the following.
Now we change (1.4) to lienard equation by (1.8):
Since the limit cycle of (1.4) surrounding must lay in . Let then
where
Now we define the curve and as follows:
It easy to see that are continuously differentiable.
Now Let
If and intersect in , then in , , that is,
so in the intersecting point of and ,
that is,
Theorem 3.2. If then system (1.4) or (3.1) has no limit cycle surrounding .
Proof. Since , and we have supposed that , so . Now we will prove that system (3.1) has no limit cycles under the conditions of .
First supposing , we will prove that and do not intersect ( intersect means that from one side of astride to another side at intersect point).
() If does not change its sign when , then (equal sign only for some , the same as below), so , for any , it means that does not exist, therefore, and do not intersect.
() If change its sign when , and , then ()=0 have one or two real roots in (If real roots do not exist, then similar to (), does not exist); then the curve is shown in Figure 1, and the relative position of curve and is shown in Figure 2 (If only part of exists, it does not influence the proof, the same as below). If and have an intersection point (the first intersecting point from ), then by Figure 2
but since (see Figure 1), , so
It is a contradiction, so and do not intersect.
() If change its sign when , and , then have one or two real roots in , then the curve is shown in Figure 3, and the relative position of curves and is shown in Figure 4. If and have an intersection point (the first intersecting point from ), then by Figure 4
Since , and from the fact that is a intersection point late to , we have , so
It is a contradiction, so and do not intersect.
Note that if change its sign in , then , because if , then
Secondly, supposing , we will prove that and do not intersect.
If does not change its sign in , then , so , for any , it means that does not exist, hence and do not intersect.
If change its sign when , and , then have one or two real roots in ; then the curve is shown in Figure 5, and the relative position of curves and is shown in Figure 2. If and have an intersection point (the first intersecting point from ), then by Figure 2
But since (see Figure 5), , so
It is a contradiction, so and do not intersect.
If change its sign in , and , then the curve is shown in Figure 6; then the relative position of curves and is similar to Figure 4. If and have an intersecting point (the first intersecting point from ), then by Figure 4
but since (see Figure 6), and , so
It is a contradiction, so and do not intersect.
Note that if change its sign in , then . Because if , then
As far, we have proved that and have no intersection if . According to the proof that and has no intersection to has no intersection in paper [13]; it can be extended to that if and has no intersection, then has no intersection. Furthermore, based on the proof of paper [14], if has no intersection, then the corresponding Lienard equation has no limit cycle.
Now, by [12, 15] we have proved that when system (1.4) or (3.1) has no limit cycles surrounding .
Finaly, we consider the case: , that is . If , then is a center of (1.4); If , or , then (If , then by (3.8), ). If when , or , and have an intersecting point, then when or this intersecting point also exists; it is a contradiction to ()–() and , so when , system (1.4) or (3.1) has no limit cycle surrounding ; this completes the proof of Theorem 3.2.
Lemma 3.3. If , system (1.4) or (3.1) has no limit cycle surrounding .
Proof. We consider five cases in the following:
() If then by (3.3), so
hence , for any ; it means that does not exist; therefore, and do not intersect.
() If , by (3.3), has at most one positive real root, then the curve is similar to Figure 6 (but has at most one extremal point, and part of exist; it does not influence the proof, the same as below), and the relative position of curve and is similar to Figure 4 (only below half of exists, the same as below). If and have an intersection point (the first intersecting point from ), then by Figure 4
but since (see Figure 6), , so
It is a contradiction, so and do not intersect.
() The case is similar to (); it is easy to prove that and do not intersect.
() If , then has at most one negative real root, then the curve is similar to Figure 1, and the relative position of curves and is similar to Figure 2. If and have an intersection point (the first intersecting point from ), then by Figure 2
but since , so
It is a contradiction, so and do not intersect.
() The case is similar to (); it is easy to prove that and do not intersect. This completes the proof of Lemma 3.3.
Lemma 3.4. If is a center of system (1.4).
Proof. If when is a weak focal. We let , and change its stability; then there is a limit cycle surrounding ; this means when , system (1.4) or (3.1) has a limit cycle surrounding ; it is a contradiction to Lemma 3.3. It is follows that is a center of system (1.4). By Lemma 3.4, Theorem 2.1 can be recension in the following.
Theorem 3.5. For system (1.4), the focal quantities of are if , then if , then ; if is a center. If , , or is an unstable (stable) critical point. If then system (1.4) has at least two limit cycles surrounding .
4. Uniqueness of Limit Cycle Surrounding
In this section, we study the uniqueness of limit cycle surrounding the weak focal ; is weak focal if and only if so we let in system (1.4), we also suppose .
Lemma 4.1. If , and have at most one intersecting point.
Proof. Since , so we consider the two care of and .Care A. Let , so have two real roots . Let such that (see Figure 7)
Without loss of generality, we suppose that and (otherwise, only part of exist; it does not influence the proof); then the graph of is shown in Figure 7, and the relative position of and is shown in Figure 8, where . Now we suppose and intersect in (the first intersecting point from , the same as below, and if does not exist, then and do not intersect, so the system has no limit cycle surrounding ). We denoted the curve of from to by , and () (also define to see the Figure 8), then the following occurs.
() If , and If and have a second intersecting point , then from Figure 8
Since , so and , hence from (3.8) . Since , so and , hence . From , it is follows that , this is a contradiction, so and have no second intersecting point .
() If , since in , , so and have only one intersecting point.
() If , and If and have a second intersecting point , then from Figure 8
Since , so and ; hence from (3.8), . Since , so and , hence . From , it is follows that ; this is a contradiction, so and have no second interesting point .Care B. Let , so have two real roots . Let such that (see Figure 9)
Without loss of generality, we suppose that and , then the graph of is shown in Figure 9, and the relative position of and is shown in Figure 10, where . Now we suppose that and interest in (the first interesting point from , and if does not exist, then and have no intersecting point, so the system has no limit cycle surrounding ), then the following occurs.
If , and If and have a second intersecting point , then from Figure 10,
Since , so and , hence from (3.8), . Since , so and , so . From , it follows that ; this is a contradiction, so and have no second intersecting point .
If , since in , , so and have only one intersecting point.
If , and If and have a second intersecting point , then from Figure 10,
Since , so and ; hence . Since , so and , so . From , it follows that ; this is a contradiction, so and have no second intersecting point .
From ()–() and , we have proved that and have at most one intersecting point under the conditions .
Lemma 4.2. Let , if , then ; if , then .
Proof.
Now we only have to prove that
as (or , as , since
Care A. , according to , we have . Since , and , so ; hence when , , that is, , so , it follows that as .Care B. , according to , we have , since ; hence when , , that is, , so , it follows that as .
Since as , from Lemmas 4.1-4.2 and paper [1], we have the following theorem.
Theorem 4.3. If , then system (1.4) has at most one limit cycle surrounding .
5. The Remaining Issues in Paper [8]
In this section we will study the remaining issues in paper [8]. Paper [8] considers a class of cubic system (1.2), where , and has proved that the focal quantities of in (1.2) are , if , is a center, and if , then system (1.2) has at most one limit cycle surrounding . But in paper [8], the case is not considered.
Lemma 5.1. If , system (1.2) has no limit cycle surrounding .
Proof. Paper [8] has proved that when and in paper [8] do not intersect. Since , so (to see (1.8) in paper [8]), hence (if , then ). Now if when , and have an intersecting point , then under condition: , this intersecting point also exists, this is a contradiction to above. Hence under condition of Lemma 5.1, and do not intersect, and system (1.2) has no limit cycle surrounding .
Theorem 5.2. For system (1.2),the focus quantities of are if , then if , then if then is a center. If , , or then is unstable (stable) critical point.
Proof. By the theorem 1 of paper [8], we only need to prove that under conditions: , is a center of system (1.2).
If when , is a weak focus (not a center). We let , and change its stability, then there is a limit cycle surrounding ; this means that when , system (1.2) has a limit cycle surrounding ; this is a contradiction to Lemma 5.1, it follows that is a center of system (1.2).
Acknowledgments
This work is supported by the Natural Science Foundation of Fujian Province (2010J01010), the Technology Innovation Platform Project of Fujian Province (2009J1007).