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2007 | Buch

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Limit Cycles of Differential Equations

verfasst von: Colin Christopher, Chengzhi Li

Verlag: Birkhäuser Basel

Buchreihe : Advanced Courses in Mathematics - CRM Barcelona

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Around the Center-Focus Problem

Frontmatter

Chapter 1. Centers and Limit Cycles

Abstract

In this chapter I want to give a general background to the center-focus problem, and then to show why the problem is interesting: both in what it tells us about the distinctive algebraic features of polynomial vector fields, and also in the simple concrete estimates it gives of the number of limit cycles which can exist in these vector fields.

Chapter 2. Darboux Integrability

Abstract

In this chapter, we consider one of the two main mechanisms which seem to underlie the existence of centers in polynomial vector fields. We only hint at the historical side, which is covered in detail by Schlomiuk [57].

Chapter 3. Liouvillian Integrability

Abstract

In this chapter we want to prove that Darboux integrability corresponds to the notion of Liouvillian integrability, or “solution by quadratures”.

Chapter 4. Symmetry

Abstract

In this chapter we consider the second mechanism which gives rise to centers in polynomial systems: the existence of an algebraic symmetry.

Chapter 5. Cherkas’ Systems

Abstract

In this chapter we give an extended example of a non-trivial classification of centers which involves both Darboux and symmetry mechanisms for producing a center. Further details can be found in [26], which we follow closely.

Chapter 6. Monodromy

Abstract

In this chapter we begin the second part of these notes, looking at some ideas based on the concept of monodromy. Very roughly, this is the study of how objects depending on a parameter, and which are locally constant in some sense, change as the parameter moves around a non-trivial path. This idea is particulary appropriate for the center-focus problem, as the essence of this problem is about trying to make global extensions of local information. For example, we might naïvely hope to be able to extend the local first integral at the origin to a global first integral. This is not possible in general, but even if we could do so, the first integral would certainly ramify as a global object. Our desire would then be to read off some important information about the system from this global ramification.

Chapter 7. The Tangential Center-Focus Problem

Abstract

As is well known, the second part of Hilbert’s 16th problem is concerned with bounding the number of limit cycles in a polynomial system (1.2) of degree n in terms of n. This is a very hard problem, but Arnold has suggested a “Weak Hilbert’s 16th problem” which seems far more tractable: to find a bound on the number of limit cycles which can bifurcate from a first-order perturbation of a Hamiltonian system,

$$ \dot x = - \frac{{\partial H}} {{\partial y}} + \varepsilon P,\dot y = \frac{{\partial H}} {{\partial x}} + \varepsilon Q, $$

(7.1)

where the Hamiltonian, H, is a polynomial of degree n + 1 and the perturbation terms, P and Q are polynomials of degree m.

Chapter 8. Monodromy of Hyperelliptic Abelian Integrals

Abstract

We want to show that in the case of Hamiltonians of the form

$$ H(x,y) = y^2 + f(x), $$

where f(x) is a polynomial of degree n, the existence of a tangential center implies that either P dx — Qdy is relatively exact, or the polynomial f(x) is composite. That is, it can be expressed as a polynomial of a polynomial, f(x) = a(b(x)), in a non-trivial way.

Chapter 9. Holonomy and the Lotka-Volterra System

Abstract

In this section we give another idea related to monodromy. This is the holonomy of the foliation P dy—Qdx = 0 associated to the system (1.2) in the neighborhood of an invariant curve. This object, roughly speaking, is the nonlinear analog to the monodromy of the solutions of a linear differential equation as they turn around a singular point. Alternatively, it can be thought of as a kind of Poincaré return map for foliations.

Chapter 10. Other Approaches

Abstract

In this final chapter I want to mention briefly three other approaches to the general center-focus problem. In the first, we try to identify whole components of the center variety by finding their intersections with specific subsets of parameter space and then showing that the type of center is “rigid”. In the second approach, we try to see the consequences of a center on its bounding graphic. Monodromytype arguments play an implicit role in both of these approaches. The last section describes an experimental approach to the center-focus via intensive computations using modular arithmetic and an application of the Weil conjectures. It makes a fitting conclusion to our range of monodromy techniques, since the arithmetic analog of monodromy was an essential ingredient in Deligne’s proof of the Weil conjectures [27].

Backmatter

Abelian Integrals and Applications to the Weak Hilbert’s 16th Problem

Frontmatter

Chapter 1. Hilbert’s 16th Problem and Its Weak Form

Abstract

Consider the planar differential systems

$$ \dot x = P_n (x,y),\dot y = Q_n (x,y), $$

(1.1)

where P_n and Q_n are real polynomials in x, y and the maximum degree of P and Q is n. The second half of the famous Hilbert’s 16th problem, proposed in 1900, can be stated as follows (see [70]):

For a given integer n, what is the maximum number of limit cycles of system (1.1) for all possible P_n and Q_n ? And how about the possible relative positions of the limit cycles ?

Chapter 2. Abelian Integrals and Limit Cycles

Abstract

In this chapter we will explain the relation between the number of zeros of the Abelian integrals and the number of limit cycles of the corresponding planar polynomial differential systems.

Chapter 3. Estimate of the Number of Zeros of Abelian Integrals

Abstract

To study the weak Hilbert’s 16th problem by using Abelian integrals, it is crucial to estimate the number of zeros of the Abelian integral. In this chapter, we introduce several methods to study the number of zeros of the Abelian integral I(h) given in (1.10), which is related to the codimension 2 Bogdanov-Takens bifurcation problem, as we explained in subsection 1.2.2.

Chapter 4. A Unified Proof of the Weak Hilbert’s 16th Problem for n=2

Abstract

As we explained in Subsection 1.2.1, any cubic generic Hamiltonian, with at least one period annulus contained in its level curves, can be transformed into the normal form

$$ H(x,y) = \frac{1} {2}(x^2 + y^2 - )\frac{1} {3}x^3 + axy^2 + \frac{1} {3}by^3 , $$

(4.1)

where a, b are parameters lying in the open region

$$ G = \left\{ {(a,b): - \frac{1} {2} < a < 1,0 < b < (1 - a)\sqrt {1 + 2a} } \right\}. $$

(4.2)

Figure 1 (in Subsection 1.2.1) shows all five possible phase portraits of X_H in the generic cases. Here X_H is the Hamiltonian vector field corresponding to H, i.e.,

$$ X_H = H_y \frac{\partial } {{\partial x}} - H_x \frac{\partial } {{\partial y}}. $$

(4.3)

The vector field X_H has a center at the origin in the (x, y)-plane, and the continuous family of ovals, surrounding the center, is

$$ \{ \gamma h\} \subset \{ (x,y):H(x,y) = h,0 < h < 1/6\} . $$

(4.4)

The oval γ_h shrinks to the center as h → 0, and the oval γ_h terminates at the saddle loop of the saddle point (1, 0) when h → 1/6.

Backmatter

Titel: Limit Cycles of Differential Equations
verfasst von: Colin Christopher
Chengzhi Li
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8410-4
Print ISBN: 978-3-7643-8409-8
DOI: https://doi.org/10.1007/978-3-7643-8410-4