The paper deals with differential equations of the formẋ = y,ẏ = −g(x) − f (x)y, with f and g pol... more The paper deals with differential equations of the formẋ = y,ẏ = −g(x) − f (x)y, with f and g polynomials of degree respectively two and three. They are called cubic Liénard equations with quadratic damping. Attention goes to the case in which the equations have three singular points of which one is a saddle and two are antisaddles. A lot of results are given on the upper-bound for the number of limit cycles, depending on the relative position of the zeros of f and g on the x-axis.
This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds... more This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. "Canard solutions" are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a "turning point" and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the "control curve".
In this paper we prove that several elementary graphics surrounding a focus or center in quadrati... more In this paper we prove that several elementary graphics surrounding a focus or center in quadratic systems have finite cyclicity. This paper represents an additional step in the large program to prove the existence of a uniform bound for the number of limit cycles of a quadratic vector field which we can call the finiteness part of Hilbert's 16th problem for quadratic vector fields. It nearly finishes the part of the program concerned with elementary graphics. In [3] this problem was reduced to the proof that 121 graphics have finite cyclicity. The graphics considered here are the hemicycles (H 3 4), (H 3 5) and (H 3 6) together with (I 2 14a), (I 2 15a), (I 2 15b) and (I 2 27) in the notation of [3] (Figure 1).
The paper deals with generic perturbations from a Hamiltonian planar vector field and more precis... more The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.
We study the limit cycles of planar slow-fast vector fields, appearing near a given slow-fast cyc... more We study the limit cycles of planar slow-fast vector fields, appearing near a given slow-fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow-fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow-fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow-fast cycles. We also provide examples of attracting common slow-fast cycles out of which more than one limit cycle can perturb, one of which is repelling. c
The paper deals with differential equations of the formẋ = y,ẏ = −g(x) − f (x)y, with f and g pol... more The paper deals with differential equations of the formẋ = y,ẏ = −g(x) − f (x)y, with f and g polynomials of degree respectively two and three. They are called cubic Liénard equations with quadratic damping. Attention goes to the case in which the equations have three singular points of which one is a saddle and two are antisaddles. A lot of results are given on the upper-bound for the number of limit cycles, depending on the relative position of the zeros of f and g on the x-axis.
This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds... more This paper deals with singular perturbation problems for vector fields on 2-dimensional manifolds. "Canard solutions" are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a "turning point" and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the "control curve".
In this paper we prove that several elementary graphics surrounding a focus or center in quadrati... more In this paper we prove that several elementary graphics surrounding a focus or center in quadratic systems have finite cyclicity. This paper represents an additional step in the large program to prove the existence of a uniform bound for the number of limit cycles of a quadratic vector field which we can call the finiteness part of Hilbert's 16th problem for quadratic vector fields. It nearly finishes the part of the program concerned with elementary graphics. In [3] this problem was reduced to the proof that 121 graphics have finite cyclicity. The graphics considered here are the hemicycles (H 3 4), (H 3 5) and (H 3 6) together with (I 2 14a), (I 2 15a), (I 2 15b) and (I 2 27) in the notation of [3] (Figure 1).
The paper deals with generic perturbations from a Hamiltonian planar vector field and more precis... more The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.
We study the limit cycles of planar slow-fast vector fields, appearing near a given slow-fast cyc... more We study the limit cycles of planar slow-fast vector fields, appearing near a given slow-fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow-fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow-fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow-fast cycles. We also provide examples of attracting common slow-fast cycles out of which more than one limit cycle can perturb, one of which is repelling. c
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Papers by F. Dumortier