Applied Mathematics E-Notes, 1(2001), 124-129 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Quadratic Systems with Homoclinic Cycles Described by Quintic Curves ^{∗}

Yan-hong Zheng and Xue-peng Li^{†}

Received 1 February 2001

Abstract

We find quadratic systems with homoclinic cycles described by quintic curves.

The determination of homoclinic bifurcations of quadratic systems is not known in general [1,5,6,7]. In [2-4], cubic or quartic homoclinic cycles are found. In this paper, we present quadratic systems with homoclinic cycles which are described by quintic curves.

Consider the quadratic system

\frac{dx}{dt} = P (x, y), \frac{dy}{dt} = Q(x, y),  (1)

where P and Q are second order bivariate polynomials with real coefficients. If the system (1) has a homoclinic cycle, then without loss of any generality, we may suppose that (1) the homoclinic cycle passes through the origin, which is a hyperbolic saddle;

(2) the stable and unstable manifold of the origin are tangent to the lines x^{2} − y^{2} = 0 and the homoclinic orbit is located in the region D = {(x, y)||y| < |x|, x > 0}; and (3) one of the infinite singular points is ‘in the y-axis direction’.

Under these assumptions, the corresponding normal form is

\frac{dx}{dt} = P̄(x, y), \frac{dy}{dt} = Q̄(x, y),  (2)

where P̄(x, y) = cx + y + a_{1}x^{2} + a_{2}xy, Q̄(x, y) = x + cy − x^{2} − b_{2}xy − b_{1}y^{2}, |c| < 1 and a_{1}, a_{2}, b_{1}, b_{2} are real. It is easy to see that the quintic algebraic homoclinic cycle through the origin of the system must take the following form:

F (x, y) = x^{2} − y^{2} + F_{3} + F_{4} + F_{5} = 0,  (3)

where F_{3}, F_{4} and F_{5} are homogeneous polynomials of degrees 3, 4 and 5 respectively.

By Batins’ formula, we are able to find, by means of the software Mathematica, an invariant quintic curve of the above system.

∗Mathematics Subject Classifications: 34C05, 34C37.

†Department of Mathematics, Fujian Normal University, Fuzhou, Fujian 350007, P. R. China

THEOREM 1. If the coefficients in the polynomials P̄(x, y) and Q̄(x, y) satisfy

a_{1} = −\frac{150c + 10c^{3} + 225cα + 93c^{3}α − 6c^{5}α}{5(75 + 29c^{2})},
a_{2} = −α,
b_{1} = 2α,
b_{2} = \frac{575c + 25c^{3} + 425cα + 214c^{3}α − 15c^{5}α}{5(75 + 29c^{2})},

where

α = \frac{5[−1125 − 270c^{2} + 43c^{4} + (75 + 29c^{2})\sqrt{225 − 18c^{2} + c^{4}}]}{2(−5625 − 2925c^{2} − 259c^{4} + 21c^{6})}.

Then (2) has an invariant quintic curve described by

F (x, y) = x^{2} − y^{2} + a_{30}x^{3} + a_{21}x^{2}y + a_{12}xy^{2} + a_{40}x^{4} + a_{31}x^{3}y + a_{50}x^{5} = 0, (4)

which satisfies

F_{x}P̄ + F_{y}Q̄ = (2c + Ax + By)F,  (5)

where A = 5a_{1}, B = 5a_{2}, a_{12} = α,

a_{30} = \frac{−250 − 30c^{2} − 625α − 225c^{2}α + 18c^{4}α}{5(75 + 29c^{2})},
a_{21} = −\frac{2(100c + 25cα + 27c^{3}α)}{5(75 + 29c^{2})},
a_{31} = \frac{2cα(375 + 5c^{2} + 55c^{2}α − 3c^{4}α)}{15(75 + 29c^{2})},
a_{40} = {27500c^{4} − 4800c^{6} + 180c^{8} + 3515625α + 1968750c^{2}α
−5000c^{4}α + 71250c^{6}α + 5391c^{8}α − 108c^{10}α}
/ {5(75 + 29c^{2})(5625 + 2925c^{2} + 259c^{4} − 21c^{6})} ,
a_{50} = − {α(−3093750c^{2} − 427500c^{4} − 8900c^{6} + 2500c^{8} − 30c^{10}
+10546875α + 6187500c^{2}α − 701250c^{4}α − 325750c^{6}α
+42317c^{8}α − 1614c^{10}α + 18c^{12}α)}
/ {75(75 + 29c^{2})(5625 + 2925c^{2} + 259c^{4} − 21c^{6})} .

Next, note that when c < 0, if we let x = x̄, y = −ȳ, t = −t̄, and c = −c̄ in (2), then the resulting system has the same form. Therefore, we will restrict our attention to the cases where c = 0 or 0 < c < 1 in the following discussions.

From now on, we will suppose that the coefficients a_{1}, a_{2}, b_{1}, b_{2} and α satisfy the conditions of Theorem 1. To emphasize its dependence on the parameter c, the system (2) will also be denoted by E(c).

The curve (4) can be rewritten as

F (x, y) = (a_{12}x − 1)y^{2} + (a_{21} + a_{31}x)x^{2}y + x^{2}(1 + a_{30}x + a_{40}x^{2} + a_{50}x^{3}) = 0. (6)

Solving y from the equation F (x, y) = 0, we get

y_{±} = {−(a_{21} + a_{31}x)x^{2} ± x((a_{13}^{2} − 4a_{12}a_{50})x^{4}
+2(a_{21}a_{31} − 2a_{12}a_{40} + 2a_{50})x^{3} + (a_{21}^{2} − 4a_{12}a_{30} + 4a_{40})x^{2}
−4(a_{12} − a_{30})x + 4)^{1/2}} / {2(a_{12}x − 1)} .

It can be checked that when 0 < c < 1, the equation 1 + a_{30}x + a_{40}x^{2} + a_{50}x^{3} = 0 has only a real root, which indicates that the curve defined by (4) has, besides the origin, only one point of intersection with the x-axis. Moreover, x = 1/a_{12} = 1/α (note that α < 0) is the asymptote of the quintic curve (see the following figure).

LEMMA 1. The system E(c) has at most one singular point which is not on the curve defined by (4).

PROOF. According to (5), we know that a singular point A_{0} = (x_{0}, y_{0}), which is not on F = 0, must be on the straight line 2c + Ax + By = 0. Solving the system of equations

cx + y + a_{1}x^{2} − αxy = 0,
x + cy − x^{2} − b_{2}xy − 2αy^{2} = 0,
2c + 5a_{1}x + 5a_{2}y = 0,

we obtain the unique solution

x_{0} = \frac{75 + 29c^{2}}{75 + 5c^{2} + 3c^{2}α − 3c^{4}α},
y_{0} = −\frac{3c(75 + 29c^{2})}{5(75 + 5c^{2} + 3c^{2}α − 3c^{4}α)},

as required.

LEMMA 2. The system E(c) has two singular points A_{1} = (x_{1}, y_{1}) and A_{2} = (x_{2}, y_{2}) besides (0, 0) and A_{0} = (x_{0}, y_{0}), where

x_{1} = \frac{−5m + 5\sqrt{k}}{n} ,

y_{1} = \frac{cx_{1} + a_{1}x_{1}^{2}}{αx_{1} − 1} ,
x_{2} = \frac{−5m − 5\sqrt{k}}{n} ,
y_{2} = \frac{cx_{2} + a_{1}x_{2}^{2}}{αx_{2} − 1} ,

and

m = 115c^{2} − 375α − 3c^{6}α + c^{4}(5 + 66α),
n = α[550c^{2}(α − 1) + 1875α + 6c^{6}α − c^{4}(10 + 143α)],
k = m^{2} + (−75 + 46c^{2} + 29c^{4})n.

Indeed, the proof follows by solving P̄ = 0, Q̄ = 0.

From the expressions of the singular points A_{0}, A_{1} and A_{2}, it is easy to see that :

(i) when c = 0, A_{0} is on the x-axis, A_{1} and A_{2} “disappear to the infinity”; (ii) when 0 < c < 1, A_{0} is in the fourth quadrant, A_{1} is in the third quadrant and A_{2} is in the second quadrant; and (iii) both A_{1} and A_{2} are on the curve defined by (4).

LEMMA 3. (i) If c = 0, then A_{0} is a center; and (ii) if 0 < c < 1, then A_{0} is a focus and A_{1}, A_{2} are nodes.

PROOF. (i) If c = 0, then the system E(c) becomes dx/dt = y, dy/dt = x − x^{2}. It is evident that the system E(c) is a symmetrical integrable system such that (y^{2} − x^{2} + 2x^{3}/3) is constant. Thus A_{0}(1, 0) is a center. (ii) If 0 < c < 1, A_{0} is a focus, since

D = div(P̄, Q̄)|_{A_{0}} = \frac{c(c^{2} − 25)(−5 + 10α + 3c^{2}α)}{5(−75 − 5c^{2} − 3c^{2}α + 3c^{4}α)}

J = (P̄_{x}Q̄_{y} − P̄_{y}Q̄_{x})|_{A_{0}}
= \frac{(1875 − 600c^{2} − 75c^{4} − 1875α − 475c^{2}α + 17c^{4}α + 45c^{6}α)}{25(75 + 5c^{2} + 3c^{2}α − 3c^{4}α)}

and D^{2} −4J < 0. In addition, neither A_{1} nor A_{2} can be a focus or a center for they are on the curve (4). Furthermore, (0, 0) is a saddle, and the four points (0, 0), A_{0}, A_{1}, A_{2} are the corners of a concave quadrilateral. According to Berlinskii Theorem, A_{1} and A_{2} are nodes. The proof is complete.

LEMMA 4. When 0 < c < 1, the system E(c) has three singular points in the infinity and one is a node, the others are saddles.

PROOF. Applying the Poincare transformation: x = v/z, y = 1/z, dt = zdτ, a_{1} + b_{2} = −β, and a_{2} + b_{1} = α, the system E(c) changes to

\{ \begin{matrix}dz/dτ = −z[−2α + cz + (a_{1} + β)v + vz − v^{2}] = P^{∗}(x, y) \\ dv/dτ = (1 − v^{2})z + v(α − βv + v^{2}) = Q^{∗}(x, y) \end{matrix} .  (7)

When 0 < c < 1, there are three singular points B_{0}(0, v_{0}), B_{1}(0, v_{1}), B_{2}(0, v_{2}) on the straight line z = 0, where

v_{0} = 0, v_{1} = \frac{β + \sqrt{(β)^{2} − 4α}}{2}, v_{2} = \frac{β − \sqrt{(β)^{2} − 4α}}{2}.

Now we analyze the properties of the points:

(P_{z}^{∗})|_{z=0} = 2α − (a_{1} + β)v + v^{2},
(P_{v}^{∗})|_{z=0} = 0,
(Q_{z}^{∗})|_{z=0} = 1 − v^{2},
(Q_{v}^{∗})|_{z=0} = α − 2βv + 3v^{2} = f′(v).

The roots of the characteristic equations of the singular points are (λ_{1})^{(i)} = α − a_{1}v_{i}, (λ_{2})^{(i)} = f′(v_{i}) for i = 1, 2, (λ_{1})^{(0)} = 2α and (λ_{2})^{(0)} = α. Since α < 0, α−a_{1}v_{2} < 0 and f′(v_{2}) > 0, thus B_{0}(0, 0) is a stable node and B_{2}(0, v_{2}) is a saddle. Finally, according to the Euler’s index-sum theorem, we know that B_{1}(0, v_{1}) is a saddle. The proof is complete.

By means of Lemmas 1-4, we know that, under the condition 0 < c < 1, there is no singular point other than the origin which lies on the non-isolated closed component of the quintic curve defined by (4). The following result now holds.

THEOREM 2. When 0 < c < 1, the non-isolated closed component of the quintic curve defined by (4) constitutes the homoclinic cycles of the system E(c) and its inner singular point is a focus.

To determine the global phase graphs of the system E(c), we need to discuss whether the system has limit cycles. By means of Dulac’s theorem, we obtain the following result.

THEOREM 3. The system E(c) has no limit cycles.

Indeed, the y-axis is non-tangent (besides the origin), therefore the y-axis and the invariant curve defined by (4) divide the plane into five regions. Taking Dulac function F^{−1}, then

div(F^{−1}P̄, F^{−1}Q̄) = F^{−1}[P̄_{x} + Q̄_{y} − (2c + Ax + By)]
= F^{−1}(−3a_{1} − b_{2})x
= F^{−1}(−2a_{1} + β)x.

The right hand side of the above equality is a continuous function and it keeps the same sign in each region. Thus by Dulac’s Theorem, there is no limit cycle in each region.

We remark that by letting y = xu, the system E(c) is changed into a cubic system

\{ \begin{matrix}dx/dt = x[c + u + (a_{1} + a_{2}u)x] \\ du/dt = 1 − u^{2} − (1 − βu + αu^{2})x \end{matrix} ,

while the homoclinic cycle defined by (4) changes into the heteroclinic cycle:

x^{2}[1 − u^{2} + (a_{30} + a_{21}u + a_{12}u^{2})x + (a_{40} + a_{31}u)x^{2} + a_{50}x^{3}] = 0.

By means of the informations obtained above, the global phase graphs of the system E(c) can easily be obtained (see the following figures).

References

[1] Y. Q. Ye, Qualitative Theory of Polynomial Differential Systems (in Chinese), Shanghai Science Technical Publisher, Shanghai, 1995.

[2] B. Q. Shen, The existential problem of cubic curve limit cycle and separatrix cycle in a quadratic system (in Chinese), Chinese Ann. Math. 12A(2)(1991), 382-389.

[3] X. P. Li, A kind of two-parameters-cubic-curve families defined by the quadratic systems (in Chinese), Chinese Ann. Math. 19A(6)(1998), 677-688.

[4] X. P. Li, The topological classification of the quartic curve homoclinic cycles of quadratic systems, preprint.

[5] B. Y. Feng, Survey of the current research on homoclinic orbits and heteroclinic orbits, Journal of Math. Research and Exposition, 14(2)1994, 299-311.

[6] Z. F. Zhang, T. R. Ding, W. Z. Huang, and Z. X. Dong, Qualitative Theory of Differential Equation, Chinese Science Press ,1985.

[7] S. N. Chow, J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin and New York, 1982.