Gevrey normal forms for nilpotent contact points of order two
P. De Maesschalck
Discrete & Continuous Dynamical Systems - A 2014, 34(2): 677-688 doi: 10.3934/dcds.2014.34.677
This paper deals with normal forms about contact points (`turning points') of nilpotent type that one frequently encounters in the study of planar slow-fast systems. In case the contact point of an analytic slow-fast vector field is of order two, we prove that the slow-fast vector field can locally be written as a slow-fast Liénard equation up to exponentially small error. The proof is based on the use of Gevrey asymptotics. Furthermore, for slow-fast jump points, we eliminate the exponentially small remainder.
keywords: slow-fast vector field Liénard system. Gevrey asymptotics Singular perturbations normal forms
Ackerberg-O'Malley resonance in boundary value problems with a turning point of any order
P. De Maesschalck
Communications on Pure & Applied Analysis 2007, 6(2): 311-333 doi: 10.3934/cpaa.2007.6.311
We consider an Ackerberg-O'Malley singular perturbation problem $\epsilon y'' + f(x,\epsilon)y' + g(x,\epsilon)y=0, y(a)=A, y(b)=B$ with a single turning point and study the nature of resonant solutions $y=\varphi(x,\epsilon)$, i.e. solutions for which $\varphi(x,\epsilon)$ tends to a nontrivial solution of $f(x,0)y'+ g(x,0)y=0$ as $\epsilon\to 0$. Many techniques have been applied to the study of this problem (WKBJ, invariant manifolds, asymptotic methods, spectral methods, variational techniques) and they have been successful in characterizing these resonant solutions when $f(x,0)$ has a simple zero at the origin. When the order of zero is higher the increase in complexity of the problem is significant. The existence of a nonzero formal power series solution is no longer necessary for resonance and resonant solutions are in general not smooth at the origin. We apply the method of blow up to study the nature of resonant solutions in this setting, using techniques from invariant manifold theory and planar singular perturbation theory. The main result is the sufficiency of the Matkowsky condition for turning points of arbitrary order (based on Gevrey-asymptotics), but we also give a characterization of the location of the boundary layer in resonant solutions.
keywords: Ackerberg-O'Malley resonance turning point Matkowsky condition. Gevrey analysis
Detectable canard cycles with singular slow dynamics of any order at the turning point
P. De Maesschalck Freddy Dumortier
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 109-140 doi: 10.3934/dcds.2011.29.109
This paper deals with the study of limit cycles that appear in a class of planar slow-fast systems, near a "canard'' limit periodic set of FSTS-type. Limit periodic sets of FSTS-type are closed orbits, composed of a Fast branch, an attracting Slow branch, a Turning point, and a repelling Slow branch. Techniques to bound the number of limit cycles near a FSTS-l.p.s. are based on the study of the so-called slow divergence integral, calculated along the slow branches. In this paper, we extend the technique to the case where the slow dynamics has singularities of any (finite) order that accumulate to the turning point, and in which case the slow divergence integral becomes unbounded. Bounds on the number of limit cycles near the FSTS-l.p.s. are derived by examining appropriate derivatives of the slow divergence integral.
keywords: singular perturbations blow-up slow-fast cycle generalized Liénard equation. turning point canards
Gevrey and analytic local models for families of vector fields
Patrick Bonckaert P. De Maesschalck
Discrete & Continuous Dynamical Systems - B 2008, 10(2&3, September): 377-400 doi: 10.3934/dcdsb.2008.10.377
We give sufficient conditions on the spectrum at the equilibrium point such that a Gevrey-$s$ family can be Gevrey-$s$ conjugated to a simplified form, for $0\le s\le 1$. Local analytic results (i.e. $s=0$) are obtained as a special case, including the classical Poincaré theorems and the analytic stable and unstable manifold theorem. As another special case we show that certain center manifolds of analytic vector fields are of Gevrey-$1$ type. We finally study the asymptotic properties of the conjugacy on a polysector with opening angles smaller than $s\pi$ by considering a Borel-Laplace summation.
keywords: Normal Forms resonance Gevrey series summation.
P. De Maesschalck Freddy Dumortier Martin Wechselberger
Discrete & Continuous Dynamical Systems - S 2009, 2(4): i-iii doi: 10.3934/dcdss.2009.2.4i
The theory of slow-fast systems is a challenging field both from the viewpoint of theory and applications. Advances made over the last decade led to remarkable new insights and we therefore decided that it is worthwhile to gather snapshots of results and achievements in this field through invited experts. We believe that this volume of DCDS-S contains a varied and interesting overview of different aspects of slow-fast systems with emphasis on 'bifurcation delay' phenomena. Unfortunately, as could be expected, not all invitees were able to sent a contribution due to their loaded agenda, or the strict deadlines we had to impose.
   Slow-fast systems deal with problems and models in which different (time- or space-) scales play an important role. From a dynamical systems point of view we can think of studying dynamics expressed by differential equations in the presence of curves, surfaces or more general varieties of singularities. Such sets of singularities are said to be critical. Perturbing such equations by adding an $\varepsilon$-small movement that destroys most of the singularities can create complex dynamics. These perturbation problems are also called singular perturbations and can often be presented as differential equations in which the highest order derivatives are multiplied by a parameter $\varepsilon$, reducing the order of the equation when $\varepsilon\to 0$.

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Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations
Renato Huzak P. De Maesschalck Freddy Dumortier
Communications on Pure & Applied Analysis 2014, 13(6): 2641-2673 doi: 10.3934/cpaa.2014.13.2641
In this paper we continue the study of ``large" small-amplitude limit cycles in slow-fast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blow-up and good normal forms.
keywords: cyclicity Blow-up center manifold normal form singular perturbations

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