Abstract BIFURCATION SET AND LIMIT CYCLES FORMING COMPOUND EYES IN A PERTURBED HAMILTONIAN SYSTEM

BIFURCATION SET AND LIMIT CYCLES FORMING COMPOUND EYES IN A PERTURBED HAMILTONIAN SYSTEM JIBIN LI AND ZHENRONG LIU In this paper we consider a class of perturbation of a Hamiltonian cubic system with 9 finite critical points . Using detection functions, we present explicit formulas for the global and local bifurcations of the flow . We exhibit various patterns of compound eyes of limit cycles . These results are concerned with the weakened Hilbert's 16th problem posed by V.I . Arnold in 1977. 1 . Introduction The weakened Hilbert 16th problem, posed by V.I . Arnold in 1977 [1], is to determine the number of limit cycles that can be generated from a polynomial Hamiltonian system of degree n 1 with perturbed terms of a polynomial of degree m + 1 . The separatrixes and relative positions of the limit cycles for the Hamiltonian system with perturbations play an important role [2] . For a polynomial differential system of degree n, the results of [3] imply that, in order to get more limit cycles and various patterns of their distribution,, one efficient method is to perturb a Hamiltonian system with symmetry which has the maximal number of centers . Thus, to study the weakened Hilbert 16th problem, we should first investigate the property of unperturbed Hamiltonian systems, Le ., determine the global property of the family of planar algebraic curves . Then, by using proper perturbation techniques, we can obtain the global information of the perturbed non-integrable system . Only two particular examples were given in the paper [3] . In this paper we discuss the following system : dx dt =Y(, + x2 ay2) + ex(mx2 + ny 2 d = -x(1 cx2 + y2) +ey(mx2 + ny2 where a > c > 0, ac > 1, 0 < e « 1, m, n, A are parameters . Our object is to reveal the bifurcation set in the 5-parameter space . Since the vector field defined


. Introduction
The weakened Hilbert 16th problem, posed by V.I.Arnold in 1977 [1], is to determine the number of limit cycles that can be generated from a polynomial Hamiltonian system of degree n -1 with perturbed terms of a polynomial of degree m + 1 .The separatrixes and relative positions of the limit cycles for the Hamiltonian system with perturbations play an important role [2] .For a polynomial differential system of degree n, the results of [3] imply that, in order to get more limit cycles and various patterns of their distribution,, one efficient method is to perturb a Hamiltonian system with symmetry which has the maximal number of centers.Thus, to study the weakened Hilbert 16th problem, we should first investigate the property of unperturbed Hamiltonian systems, Le., determine the global property of the family of planar algebraic curves .Then, by using proper perturbation techniques, we can obtain the global information of the perturbed non-integrable system .
Only two particular examples were given in the paper [3] .In this paper we discuss the following system : where a > c > 0, ac > 1, 0 < e « 1, m, n, A are parameters .Our object is to reveal the bifurcation set in the 5-parameter space .Since the vector field defined by (1 .1)E-o is invariant under the rotation over 7r, the phase portrait of (1.1) .has a high degree of symmetry.By bifurcatiog limit cycles from homoclinic and heteroclinic orbits and centers, we obtain many interesting distributions of limit cycles which form various pattems of compound eyes.
It is well known that a point is defined to belong to the bifurcation set if, in any neighbourhood in the parameter values, there exist at least two topologically distinct phase portraits .By computing detection functions [5] [6], we can give a description of the bifurcation set in the five-parameter space of (1.1) E .For the fixed pair of a, c, the half parameter plane (n, m) with m >_ 0 can be partitioned into 19 angle regions .Hence, various possible phase portraits of (1 .1) .can be found .Especially, for a complex polynomial system, II'jasenko [7] has proved that with applications to real cases, the cubic system has 5 limit cycles with disjoint interiors .This paper shows that there exist a large region of parameters such that the Il'jasenko distribution of limit cycles can be realized by (1.1) E.
The first author has been supported by the C. C. Wu Cultural & Education Foundation fund Ltd. in Hong Kong.Moreover, he is indebted to Jack K. Hale and Shui-Nee Chow for helpful discussions .
(vi) {r4} : 1 < h < a~°1 2 .This is composed of four families of closed orbits surrounding respectively one critical point A°(i = 1 -4) .If we write ]112, k2 = (a 2 -q2 )/a 2 , a, /p are the same as in (iv), then one family of {rh} has the parametric representation : Note that as h increasing, the curve I'2 extends outside, the other curves con-2 inside.

Detection functions and bifurcation parameter of the perturbed system
In the paper [5], we have considered the perturbed Hamiltonian system where H(x, y) = h is a first integral of (3.0),0.Assume (1, 77) is a critical point of (3 .0),o,and there exists a family {I'h} of closéd orbits surrounding (1, 77) when 0 < h < h.We call the function a detection function, where f (x, y) = 77 + q + x 2E + y?£ . .Obviously, if (3.0) E is a polynomial system, then A(h) is the ratio of two Abelian integrals [8] .By using A(h), we can determine the existente and stability of limit cycles created by {rh} .
(ii) If (3 .0),has a homoclinic orbit I'h at h = h, which connects a hyperbolic critical point then the parametric value of the homoclinic bifurcation is A(h) = lim_A(h) .h-h (iii) The sign of A'(h) is determined by the sign of saddle value of (a, /.i) .
3 .The values of saddle points Sj -the detection values of direction of homoclinic and heteroclinic bifurcations .Under the condition that the unperturbed vector field has some symmetries, the sign of the values of a saddle point can be used to determine the stability of a bifurcating closed orbit from a homoclinic or heteroclinic loop, and give the signs of Ai(1/a), A í (1/c) (á = 1, 2, 3, j = 3,4) .
At the saddle points S1 and 52, when the parameter A talces the values of Aj (1/c) (j = 3, 4), we have Proof.-Sincewe see from (3 .4) that This integral formula can be written as a double integral [4] : When h tends to 1/a or The signs -and + on the right hand of, (4 .3)are respectively corresponding to the cases of i = 1 and i = 2. Thus, (4 .3)gives the conclusion of Lemma 4.2 .
Lemma 4.3 .[5] For h E (0, ho), assume that the functions O(h) and 1P(h) are suficiently smooth, nonnegative and monotone increasing, the function V)'(h)l0'(h) is nonnegative and monotone increasing (decreasing) .Then the function O(h)/o(h) must satisfy one of the following propertáes : (i) monotone increasing after it decreases to a minimum (monotone decreasing after it increases to a maximum) ; (ii) monotone increasing; (iii) monotone decreasing.
In (n, m) half plane, we compute all 18 half straight lines to get the partition of the parameter plane shown as Fig .5 .1.Corresponding to every angle region Ri (i = 1 -19) of (n, m) half plane, the sketches of the detection curves have been drawn in the table 5.1 .In order to understand the phenomenon of bifurcation of limit cycles, we give a group of phase portraits when the points (n, m) are inside the angle region R3 .They are shown as Fig .5 .2 .Using table 5 .1,we see that there exist many interesting distributions of limit cycles and homoclinic or heteroclinic loops, for the sake of brevity, we omit them.
Let C., denote a nest of k limit cycles which encloses m singular points .
The sign C is used to shown enclosing relations between limit cycles .And the sign + is used to divide limit cycles enclosing different critical points .Denote simply that C. + Ck ,,, = 2Ck ,,,, etc .
On the basis of the invariance of vector field of (1 .1)under a rotation over 7r, by the property of detection curves and theorems of Ref.
[5], we have the following two theorems.
. 1)E has two heteroclinic loops which are surrounding respectively 3 critical points; outside there loops, there is one limit cycle C9 .By using Table 5.1, we also see that the following result is true.Theorem 5 .2.For a fixed e, 0 < e << 1, we have (i) If (n, m) is inside the angle region R12, then the distributions of limit cycles of (1 .1),are (ii) If (n, m) is inside the angle region Rlo, then there are 5 limit cycles of (1 .1),with the distribution 5Ci, when A1(1/a) < A < 0. This is the Il'jasenko distribution .