Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center ()
1. Introduction
In the International Congress of Mathematics held in Paris in 1900, Hilbert made a list of 23 problems. The second part of Hilbert’s 16th problem is still an open and difficult question: to find a upper bound of the number of limit cycles and their relative locations in polynomial vector fields of order n.
If the singular point of the system is a non-saddle, nor nilpotent, the related Hopf bifurcations are elementary, see [1-3] and their references. Hopf bifurcations from the elementary focus type of singularities have found broad and important applications in biology, chemistry and physics and engineering, see [4-7] for examples. Yet for the bifurcation of limit cycles from a non-elementary center in a more general planar vector field, its intrinsic dynamics is still far away from understanding due to the complexity and technical difficulties in dealing with such bifurcations.
Then it was natural to restrict the study of the nilpotent center by assuming the system is a perturbation of a Hamiltonian system. Consider the following system
(1.1)
where
,
and
are
functions,
is small and
with D a compact set.
When
, system (1.1) becomes
(1.2)
which is Hamiltonian system. Now suppose that the Hamiltonian system (1.2) has a nilpotent center at the origin, namely the function H satisfies the following conditions:
(H1)
is a
function, satisfying
;
(H2)
, the equation
defines a closed curve Lh surrounding the origin and Lh approaches the origin as h goes to zero;
(H3)
,
.
It follows that the expansion of H at the origin has the form
![](https://www.scirp.org/html/15-31777\74119c01-4920-48d7-894d-7dbedacce3cc.jpg)
Assume that the equation
intersects the positive x-axis at
. Let
denote the first intersection point of the positive orbit of (1.1) starting at
with the positive x-axis. Then, we have
(1.3)
where
(1.4)
The Abelian integral M above is called the first order Melnikov function of system (1.1). From Han [8], we have a general theorem as follows.
Theorem 1.1. Suppose that the origin is nilpotent singular point
and that
approaches the origin as h goes to zero. If there exist an integer
and
such that
![](https://www.scirp.org/html/15-31777\146242d5-8884-43f2-bf5a-b6d89dc8fbb2.jpg)
and
![](https://www.scirp.org/html/15-31777\3042bebf-7f67-4b19-bf0f-5338c6dfc90c.jpg)
then we have 1)
has at most k zeros near
for
and all
near
, and k zeros can appear for some
near
.
2) System (1.1) has at least k limit cycles near the origin for some
near
.
2. Main Results and Proof
Consider the following near-Hamiltonian system:
(2.1)
where
and p and q are cubic polynomials. We can write
(2.2)
Then unperturbed system
is a Hamiltonian system with Hamiltonian
(2.3)
system
has a nilpotent center at the origin. Let
be the closed curve defined by
. Then it can be presented as
(2.4)
Assume that the positive solution of the above equation in y is
(2.5)
where
and
. Then by (2.4) and (2.5) we obtain
![](https://www.scirp.org/html/15-31777\beaf9980-9b01-4ae6-a40b-8d25b851f110.jpg)
![](https://www.scirp.org/html/15-31777\d1cabc66-0bce-4878-a159-61393df2a5e5.jpg)
![](https://www.scirp.org/html/15-31777\99ae392a-acf1-4e7b-8162-9ec580b08246.jpg)
![](https://www.scirp.org/html/15-31777\cf48317a-c978-4e27-8bba-a1bfe1121baf.jpg)
![](https://www.scirp.org/html/15-31777\206ab8ff-d8f3-48f3-9dac-52313c1be7ba.jpg)
By [8] the negative solution of (2.4) in y satisfies
. Thus, two solutions of (2.4) are
(2.6)
On the other hand, the intersection points of Lh and xaxis have the x-coordinates
and
. Then by (2.2) we can write
(2.7)
where
(2.8)
Here,
![](https://www.scirp.org/html/15-31777\12692aa3-04f6-43e8-8ba7-256c04f3c0eb.jpg)
Introduce
(2.9)
Then, similar to the method of Han [8] we have
(2.10)
Therefore, in turn by (2.6)-(2.10) we have
![](https://www.scirp.org/html/15-31777\61782ce8-4929-4238-bb25-ff760228e72b.jpg)
![](https://www.scirp.org/html/15-31777\3848b566-344b-4ee0-af60-c13bc7f628cc.jpg)
![](https://www.scirp.org/html/15-31777\eafac5d7-2fb5-433e-8ffc-d3fe035796c7.jpg)
Noting that
, then similarly we have
![](https://www.scirp.org/html/15-31777\c7bd9364-e33d-4cbc-9844-8d496f857dc5.jpg)
![](https://www.scirp.org/html/15-31777\298fb5f4-27ad-4b48-a466-e8df348a592d.jpg)
In the same way, using
, we have
![](https://www.scirp.org/html/15-31777\c40bb88a-3921-4351-ad61-4e3b175e10eb.jpg)
Hence, we have
![](https://www.scirp.org/html/15-31777\ede644e4-3365-4c62-b298-b00bc91b5acb.jpg)
where
And
![](https://www.scirp.org/html/15-31777\77183b3b-d2af-4a4b-a37d-05daded9fc85.jpg)
![](https://www.scirp.org/html/15-31777\92ae910c-459c-493c-ba9f-27b83a4bb82e.jpg)
![](https://www.scirp.org/html/15-31777\031af091-e553-410e-8839-20dddc0d6c9d.jpg)
![](https://www.scirp.org/html/15-31777\83d9c693-4cd2-482e-a613-4c775c53332b.jpg)
![](https://www.scirp.org/html/15-31777\807878ec-b4f1-49b8-a4ea-e7a1700a8640.jpg)
![](https://www.scirp.org/html/15-31777\da4fda6b-063c-43bf-9633-3f3dd9fd14fc.jpg)
Now it is direct that
![](https://www.scirp.org/html/15-31777\513c5485-ad0c-40b0-a1d4-3bde73588639.jpg)
![](https://www.scirp.org/html/15-31777\7367e600-b338-4e25-a032-078d7361be26.jpg)
Here, if let
,
,
, then for some cubic system (2.1) we can obtain the above determinant is not zero, then the function M can have 5 simple zeros in h > 0 near h = 0 for some
near
. For example, let
,
,
, we obtain from the above formula
![](https://www.scirp.org/html/15-31777\b4b92a95-72e8-48a6-b18b-d2015c636404.jpg)
Here,
![](https://www.scirp.org/html/15-31777\30c1dbe8-90c4-425d-8782-6ba89d0e7fb4.jpg)
![](https://www.scirp.org/html/15-31777\7b2ed31e-364c-4a4f-8002-e3b9772ac3ce.jpg)
![](https://www.scirp.org/html/15-31777\03aa4fe4-d9c5-4a90-b439-96597fcf79d5.jpg)
![](https://www.scirp.org/html/15-31777\d3572220-fb34-4bf4-b1e6-8d9856e54927.jpg)
![](https://www.scirp.org/html/15-31777\aca7f428-4940-4b6e-9f34-9ad2d6398abf.jpg)
![](https://www.scirp.org/html/15-31777\263a5e8d-de01-4bd4-a318-1fdca70fc4b4.jpg)
![](https://www.scirp.org/html/15-31777\b836bd7d-384c-4568-b206-f896f70d0fcd.jpg)
![](https://www.scirp.org/html/15-31777\015c5f2b-3ccf-4cb9-a1da-8a9d654c5b34.jpg)
![](https://www.scirp.org/html/15-31777\105bce3e-bf7c-4a8a-87d5-d0e33e26fb69.jpg)
![](https://www.scirp.org/html/15-31777\39e5ff0e-ada0-4d88-b3a4-4bac9ea90f7d.jpg)
![](https://www.scirp.org/html/15-31777\c9e21f58-f50e-4e56-bf13-79ef96d249de.jpg)
![](https://www.scirp.org/html/15-31777\7db21a32-57e1-4ecb-a389-d81656e5c0a4.jpg)
then we can obtain
(2.11)
By Theorem 1.1 we have:
Theorem 2.1. The function
has at most 5 zeros in
near
, and for
small the cubic system (2.1) can have 5 limit cycles near the origin.
NOTES