c 2003 Nonlinear Phenomena in Complex Systems °
Abelian Integrals and Limit Cycles in Polynomial Dynamical Systems Valery A. Gaiko Belarusian State University of Informatics and Radioelectronics Minsk, BELARUS E-mail:
[email protected]
(Received 04 October 2002) Two-dimensional polynomial dynamical systems are considered. Earlier we suggested two different approaches to the second part of Hilbert’s Sixteenth Problem on a maximum number and relative position of limit cycles. The first approach is bifurcational, and it is based on the ideas of global qualitative investigation. The second one is algebraic, and it is based on the ideas 6f solving a so-called “inverse problem” of dynamical systems. In this paper, we discuss the third approach which is based on the method of Abelian integrals. Key words: Polynomial dynamical system, Abelian integral, Limit cycle PACS numbers: 02.30.Hq, 02.30.Ik
1
Introduction
We consider two-dimensional dynamical systems dx dy = P (x, y), = Q(x, y), (1) dt dt where P and Q are polynomials degree not greater than n of real variables x, y with real coefficients. Together with (1), we will consider a generated vector field ∂ ∂ + Q(x, y) . (2) X(x, y) = P (x, y) ∂x ∂y In [1, 2], we have suggested two different approaches to the second part of Hilbert’s Sixteenth Problem on the maximum number and relative position of limit cycles. The first approach is bifurcational, and it is based on the ideas of the global qualitative investigation [1]. The second one is algebraic, and it is based on the ideas solving a so-called “inverse problem” of the dynamical systems [3]. Unfortunately, Hilbert’s Sixteenth Problem has not been solved completely even for the simplest nonlinear case, na0e3y for the case of quadratic systems. We have developed in [1] a new global approach to the complete solution of the Problem, first
of all, for the quadratic systems. This approach can be applied also to the study of arbitrary polynomial systems and to global qualitative analysis of higher-dimensional dynamical systems. The idea of our approach is the following [1]. 1. To use five-parameter canonical systems with field-rotation (dynamic) parameters. 2. To divide the plane of two rest (static) parameters into domains corresponding to various number and character of finite singularities and to consider canonical systems separately in each of such domains, i. e., to reduce the study of limit cycle bifurcations to the analysis of three-parameter domains of dynamic parameters. 3. Using the monotonicity of one-parameter families of multiple limit cycles generated by fieldrotation parameters, to prove in every concrete case of finite singularities that the maximal oneparameter families of multiple limit cycles are not cyclic. 4. Using Bautin’s result on the cyclicity of a singular point which is equal to three and the Wintner– Perko termination principle stating that the multiplicity of limit cycles cannot be higher than the
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multiplicity (cyclicity) of the singular point in which they terminate, to prove by contradiction in every case the nonexistence of neither a multiplicity-four limit cycle nor four limit cycles around a singular point. 5. To control simultaneously bifurcations of limit cycles around different singular points and to prove that the maximum number of limit cycles in a quadratic system is equal to four and the only possible their distribution is (3 : 1). By means of algebraic methods similar to the methods developed in [3], in the work [4], it is proved that the maximum number of limit cycles of an arbitrary polynomial system is bounded by the number equals to the product of the degrees of right-hand side polynomials of the system. This result conforms to our conjecture for quadratic systems, but it contradicts to some results in the cubic case. For example, in [5] is proved that a cubic system can have at least eleven limit cycles (instead of nine). It is clear that the application of the algebraic methods developed in [3] for the qualitative investigation of dynamical systems could be very useful both for the complete solution of Hilbert’s Sixteenth Problem and for solving other open problems of qualitative theory of differential equations. For instance, the results on the construction of systems of two differential equations having a given integral curve are generalized in [3] to the case of three and higher number of equations. These results could be applied to the construction of higherdimensional dynamical systems with various limit periodic sets such as a “strange attractor”, for example. In [1] and earlier, we put forward an idea how to solve the “inverse problem” on the construction of a three-dimensional quadratic system with a “strange attractor”. Almost at the same time, a similar idea appeared independently in [6], but for a cubic system. Both problems (Hilbert’s Sixteenth Problem on limit cycles and the Problem related to the Lorenz-like attractor) have been included by S. Smale into his list of the most important mathematical problems for the XXI century [7].
2
Abelian integrals and limit cycles
There are, of course, alternative methods to that we have developed for the study of limit cycle bifurcations. Following L. Gavrilov [8], let us discuss the possibilities of application of the method of Abelian integrals. Let H(x, y) be a real cubic polynomial and suppose that the quadratic Hamiltonian vector field XH = Hy
∂ ∂ − Hx ∂x ∂y
(3)
has a center. Consider a small polynomial deformation Xε = XH + εYε of the field XH , where Yε (x, y) = Y1 (x, y, ε)
∂ ∂ + Y2 (x, y, ε) ∂x ∂y
is a quadratic vector field which depends analytically on ε. Without loss of generality we assume that the center is located at the origin, Yε (0, 0) ≡ (0, 0), and H(x, y) = (x2 + y 2 )/2 + . . .
(4)
Consider the continuous family of ovals γ(h) ⊂ {(x, y) ∈ R2 : H(x, y) = h} which tend to the origin in R2 as h → 0, and are ˜ Let l be defined on a maximal open interval (0, h). a closed arc contained in the open periodic annulus [ γ(h) (5) ˜ h∈(0,h)
and transversal to the family of ovals γ(h). For sufficiently small |ε|, the arc l is still transversal to the vector field Xε , and can be parameterized by h = H(x, y)|l . Therefore we can define, on a suitable open set of l, the first return map h → Pε (h) associated to the vector field Xε and the arc l. The
Nonlinear Phenomena in Complex Systems Vol. 6, No. 1, 2003
Valery A. Gaiko: Abelian Integrals and Limit . . . limit cycles of the perturbed vector field Xε correspond to the fixed points of the analytic map Pε . It is well known [9] that Pε − h = −ε IY0 (h) + o(ε),
(6)
where the Pontryagin function IY0 (h) is given by ZZ IY0 (h) = div(Y0 ) dx ∧ dy, (7) {H≤h}
div(Y0 ) = (Y1 (x, y, 0))x + (Y2 (x, y, 0))y and lim o(ε)/ε = 0
ε→0
˜ uniformly in h on any compact subset of [0, h). In contrast to the first return map Pε (h), the Pontryagin function IY0 (h) does not depend on the choice of the arc l. If in addition H(x, y) has distinct critical values and Pε (h) 6= h, then IY0 (h) 6= 0 [10]. It is easy to see in this case that the num˜ provides ber of zeros of IY0 (h) on the interval [0, h) an upper bound for the number of limit cycles of Xε which bifurcate from the open period annulus (5). As Y0 (x, y) is a quadratic vector field, then the function IY0 (h) can be written in the form ZZ IY0 (h) = Iαβγ (h) = (αx + βy + γ) dx ∧ dy. {H≤h}
It is a complete elliptic integral and its qualitative behavior as a function of the complex variable h, which is studied by means of the Abelian integral d2 Iαβγ (h) dh2
{(x, y) ∈ R2 : H(x, y) = 0},
however, very incomplete. So, according to Harnack’s theorem, the number of the connected components of a smooth real planar projective curve is less or equal to (n − 1)(n − 2)/2 + 1 (the bound is exact). The analogue of Harnack’s theorem for differential equations would be to find an exact upper bound H(n) for the number of limit cycles of any planar polynomial vector field of degree at most n. D. Hilbert suggested that the problem “may be attacked by the method of continuous variation of the coefficients”. The latter was successfully used in the study of ovals of algebraic curves, and is actually known as the Hilbert–Rohn method. The study of bifurcations of limit cycles leads naturally to the notations of limit periodic set and cyclicity of such sets with respect to a given family of vector fields [1]. The limit periodic sets can be classified according to their co-dimension in the space of all planar polynomial vector fields. If a polynomial vector field has a limit periodic set of infinite co-dimension, then it has a period annulus which is an infinite union of periodic orbits, each of them being a limit periodic set. Therefore, in this case we have to study rather the cyclicity of the whole period annulus, that the cyclicity of an individual limit periodic set. An important example is the following. Consider a real polynomial H(x, y) of degree n+1 which is generic in the sense that it has n2 distinct critical values. We will also suppose that the differential equation dH = 0 has a center at the origin. Consider also the perturbed Hamiltonian system dH + ε ω = 0,
in a complex domain, can solve Hilbert’s Sixteenth Problem on the number of limit cycles for the perturbed vector field Xε [8]. Recall that the first part of this Problem asks for a (projective) classification of the ovals of a real planar algebraic curve (8)
where H(x, y) is an arbitrary real polynomial of degree n. The analogy between limit cycles (which are transcendental curves in general) and ovals remains,
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ω = P dx + Q dy,
where P = P (x, y) and Q = Q(x, y) are real polynomials of degree n, and ε is a “small” parameter. Denote by γ(h) ⊂ {H(x, y) = h} the continuous family of ovals of {H(x, y) = h} which tend to the origin as h → 0. Then, as was proved by Yu. S. Ilyashenko [10], either d ω ≡ 0 (in which case the perturbed Hamiltonian system dH + ε ω = 0 is Hamiltonian), or the Pontryagin function Z I(h) = ω (9) γ(h)
does not vanish identically, and in this situation
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Valery A. Gaiko: Abelian Integrals and Limit . . .
limit cycles bifurcate from ovals γ(h), such that I(h) = 0. Therefore, the cyclicity of an open period annulus is bounded by the maximal number of zeros which an Abelian integral I(h) can have on a suitable interval (as was proved by Roussarie [11], this holds true even for the closed period annulus). Based on this, V. I. Arnold formulated in [12] a weakened (or rather infinitesimal) version of Hilbert’s Sixteenth Problem, which asks for the maximum number of zeros of Abelian integrals of the form (9). It should be stressed, however, that if the polynomial H(x, y) is not generic, or the degree of the polynomial one-form ω is strictly greater than deg (H) − 1, the problem of finding the limit cycles of dH + ε ω = 0 is not equivalent to a problem on the zeros of Abelian integrals. The reason is that I(h) ≡ 0 does not imply in general that the return map is equal to the identity map. The higher order Pontryagin functions have to be computed in this case, and they are not always Abelian integrals (see [8]). On the contrary, when the conditions of the above Ilyashenko’s statement are satisfied, then the problem of finding the limit cycles becomes a problem in algebraic geometry. Because of its importance, we will formulate this infinitesimal Hilbert’s Sixteenth Problem in more detail. Let Xλ , λ ∈ RΛ , be the space of all planar vector fields of degree n. Suppose that X 0 = X H = Hy
∂ ∂ − Hx , ∂x ∂y
where H(x, y) is a real polynomial of degree n + 1. ¯ Xλ ) be the maximal cycliciLet Z(n, H) = Cycl (Π, ¯ of XH can have ty which a closed period annulus Π with respect to Xλ (of course X0 can have several period annuli). We will say that a real polynomial H(x, y) of degree n+1 is generic, if it has n2 distinct critical values in a complex domain. Then we can state the following problem. Infinitesimal Hilbert’s Sixteenth Problem. For generic H(x, y), find the numbers Z(n) = sup {Z(n, H) : deg H ≤ n + 1}.
As the dimension of the vector space of Abelian integrals Z An = {I(h) : I(h) = ω }, (10) γ(h)
where ω = P (x, y) dx + Q(x, y) dy, deg (P ), deg (Q) ≤ n, equals to n(n + 1)/2 [10], then n(n + 1) Z(n) ≥ −1 2 and, by a theorem of Khovanskii–Varchenko (see [8]), Z(n) < ∞. L. Gavrilov suggests in [8] that in general Z(n) =
n(n + 1) − 1. 2
(11)
This is equivalent to say that the space of Abelian integrals (10) is a Chebyshev space (i. e., the number of zeros of a function which belongs to the space is less than its dimension), and (11) could be considered as a (partial) analogue of Harnack’s theorem.
3
Conclusion
It seems that most perspective approach would be an approach combining bifurcation methods with the theory of Abelian integrals. As is shown in [13], where integrable systems under small parameter perturbations are considered, it is possible even to solve the Problem on a maximum number and relative positions of limit cycles in some special cases of quadratic systems by means of the Abelian integrals. In [13], for example, a quadratic system in the case of two anti-saddles in a finite part of the plane and three singular points (a node and two saddles) at infinity is studied. The system is close to an integrable one in the case of symmetry, and in [13] is proved that it can have at most three limit cycles around each of the finite singularities. Combining the method of Abelian integrals with the Wintner– Perko termination principle [1], we could obtain a global result (for large parameters) in this case of singular points.
Nonlinear Phenomena in Complex Systems Vol. 6, No. 1, 2003
Valery A. Gaiko: Abelian Integrals and Limit . . . References [8] [1] V. A. Gaiko, Global Bifurcations of Limit Cycles and Hilbert’s Sixteenth Problem, Universitetskoje, Minsk, 2000. (Russian). [2] V. A. Gaiko, On a work by N. P. Erugin, Mat. Comput. Obraz. 9 (2002), to appear. (Russian). [3] N. P. Erugin, Construction of the complete set of systems of differential equations having a given integral curve, Prikl. Mat. Mekh. 16 (1952), 659–670. (Russian). [4] S. Pingxing, Hilbert’s sixteenth problem, in Analysis and Topology, World Scientific, Singapore, 1998, pp. 621–646. ˙ la¸dek, Eleven small limit cycles in a cubic vec[5] H. Zoà tor field, Nonlinear. 8 (1995), 843–860. [6] R. Bamon, A family of n-dimensional differential equations with Lorenz-like attractors, Contemp. Math. 240 (1999), 13–23. [7] S. Smale, Mathematical problems for the next century, in Mathematics: Frontiers and Perspectives,
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2000, pp. 271–294. L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math. 143 (2001), 449–497. L. S. Pontryagin, On dynamical systems close to Hamiltonian systems, Zh. Eksp. Teor. Fiz. 4 (1934), 234–238. (Russian). V. I. Arnold and Yu. S. Ilyashenko, Ordinary differential equations, in Modern Problems of Mathematics. Fundamental directions, VINITI Press, Moscow, Vol. 1, 1985, pp. 7–149. (Russian). R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Bras. Mat. 17 (1986), 67– 101.
[12] V. I. Arnold, Geometric Methods in Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1988. [13] F. Dumortier, C. Li, and Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Diff. Equat. 139 (1997), 146–193.
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