1,721,032 research outputs found
Gevrey asymptotic properties of slow manifolds
No full textIn geometric singular perturbation theory, Fenichel manifolds are typically only finitely smooth. In this paper, we prove better local smoothness properties in the analytic setting, under the condition that no singularities in the slow flow are present. We also investigate cases where the slow flow has a node or focus, where summability results are obtained. Various techniques are being employed like formal power series methods, majorant equations, Gevreyasymptotics, and studies in the Borel plane.The authors acknowledge support from FWO-NAFOSTED grant G0E6618N
Intrinsic Determination of the Criticality of a Slow–Fast Hopf Bifurcation
No full textThe presence of slow-fast Hopf (or singular Hopf) points in slow-fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In De Maesschalck et al. (Canards from birth to transition, 2020), Wechselberger (Geometric singular perturbation theory beyond the standard form, Springer, New York, 2020) and Jelbart and Wechselberger (Nonlinearity 33(5):2364-2408, 2020) an intrinsic presentation of slow-fast vector fields is initiated, showing hands-on formulas to check for the presence of such singular contact points. We generalize the results in the sense that the criticality of the Hopf bifurcation can be checked with a single formula. We demonstrate the result on a slow-fast system given in non-standard form where slow and fast variables are not separated from each other. The formula is convenient since it does not require any parameterization of the critical curve.This work was supported by the bilateral research cooperation fund of the Research Foundation Flanders (FWO) under Grant No. G0E6618N and the Vietnam National Foundation for Science and Technology (NAFOSTED) under Grant No. FWO.101.2020.01.Wynen, J (corresponding author), Hasselt Univ, Dept Math, Hasselt, Belgium.
[email protected]; [email protected]
Limit cycles and critical periods with non-hyperbolic slow-fast systems
Full text linkBy considering planar slow-fast systems with a curve of double singular points, we obtain lower bounds on the number of limit cycles of polynomial systems surrounding a single singular point, as well as on the number of critical periods in one annulus of periodic orbits. In some circumstances, orbits of such slow-fast systems do not exhibit the typical slow-fast behavior but instead follow a hit-and-run pattern: they quickly move toward the critical curve, pause briefly there, and then continue their path. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.This work has been funded by Flanders FWO agency (G0F1822N grant); the Catalonia AGAUR agency (2021 SGR 00113 grant); the Spanish Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación (PID2022-136613NB-I00 grant), the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R& (CEX2020-001084-M grant). We would like to thank the kind hospitality of the research team of Universitat de les Illes Balears who gave us the opportunity to work together for this work. This project was mainly carried out during a research visit of both authors to UIB. We thank the anonymous referees for their careful review of our initial version. Their comments have helped improve the clarity of the manuscript
Critical periods in planar polynomial centers near a maximum number of cusps
No full textprovide the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. This lower bound doubles the previous one from the literature, and we end up with at least n2 - 2 (resp. n2 - 2n - 1) critical periods for planar polynomial systems of odd (resp. even) degree n. Key idea is the perturbation of a vector field with many cusp equilibria, whose construction is by itself a nontrivial construction that uses elements of catastrophe theory.(c) 2023 Elsevier Inc. All rights reserved.This work has been realized thanks to the Spanish AEI PID2019-104658GB-I00 grant; the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&M CEX2020-001084-M grant; and the Catalan AGAUR 2021SGR00113 grant. Support was also received through FWO-NSFC bilateral grant G0F1822N. We would like to thank the kind hospitality of the research team of Universitat de les Illes Balears who gave us the opportunity to work together for this work. This project was mainly carried out during a research visit of both authors to UIB. We would like to thank Armengol Gasull for his helpful discussions in some moments during the development of the work
Intrinsic determination of the criticality of a slow-fast Hopf bifurcation
No full textThe presence of slow-fast Hopf (or singular Hopf) points in slow-fast systems
in the plane is often deduced from the shape of a vector field brought into
normal form. It can however be quite cumbersome to put a system in normal form.
In the monograph "Canards from birth to transition", an intrinsic presentation
of slow-fast vector fields is initiated, showing hands-on formulas to check for
the presence of such singular contact points. We generalize the results in the
sense that the criticality of the Hopf bifurcation can be checked with a single
formula. We demonstrate the result on a slow-fast system given in non-standard
form where slow and fast variables are not separated from each other. The
formula is convenient since it does not require any parameterization of the
critical curve
Intrinsic determination of the criticality of a slow-fast Hopf bifurcation
No full textThe presence of slow-fast Hopf (or singular Hopf) points in slow-fast systems
in the plane is often deduced from the shape of a vector field brought into
normal form. It can however be quite cumbersome to put a system in normal form.
In the monograph "Canards from birth to transition", an intrinsic presentation
of slow-fast vector fields is initiated, showing hands-on formulas to check for
the presence of such singular contact points. We generalize the results in the
sense that the criticality of the Hopf bifurcation can be checked with a single
formula. We demonstrate the result on a slow-fast system given in non-standard
form where slow and fast variables are not separated from each other. The
formula is convenient since it does not require any parameterization of the
critical curve
Fractal codimension of nilpotent contact points in two-dimensional slow-fast systems
Full text linkIn this paper we introduce the notion of fractal codimension of a nilpotent
contact point , for , in smooth planar slowfast
systems when the contact order of
is even, the singularity order of is odd and has
finite slow divergence, i.e., . The
fractal codimension of is a generalization of the traditional codimension
of a slow-fast Hopf point of Li\'{e}nard type, introduced in (Dumortier and
Roussarie (2009)), and it is intrinsically defined, i.e., it can be directly
computed without the need to first bring the system into its normal form. The
intrinsic nature of the notion of fractal codimension stems from the Minkowski
dimension of fractal sequences of points, defined near using the
socalled entryexit relation, and slow divergence integral. We apply our
method to a slowfast Hopf point and read its degeneracy (i.e., the first
nonzero Lyapunov quantity) as well as the number of limit cycles near such a
Hopf point directly from its fractal codimension. We demonstrate our results
numerically on some interesting examples by using a simple formula for
computation of the fractal codimension. We demonstrate our results numerically
on some interesting examples by using a simple formula for computation of the
fractal codimension.Comment: 32 pages, 4 figure
Jump-induced mixed-mode oscillations through piecewise-affine maps
No full textMixed-mode oscillations (MMOs) are complex oscillatory patterns in which large-amplitude oscillations (LAOs) of relaxation type alternate with small-amplitude oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary differential equations of slow-fast type, and are typically related to the presence of so-called folded singularities and the corresponding canard trajectories in such systems. Here, we introduce a canonical family of three-dimensional slow-fast systems that exhibit MMOs which are induced by relaxation-type dynamics, and which are hence based on a "jump mechanism", rather than on a more standard canard mechanism. In particular, we establish a correspondence between that family and a class of associated one-dimensional piecewise affine maps (PAMs) which exhibit MMOs with the same signature. Finally, we give a preliminary classification of admissible mixed-mode signatures, verifying results of (Rajpathak, Pillai, and Bandyopahdyay (2012) [29]) in the process, and we illustrate our findings with numerical examples.The authors thank the School of Mathematics at the University of Edinburgh for its hospitality during several research visits. In particular, we are grateful to Panagiotis Kaklamanos for his fruitful and meticulous comments, as well as to the entire Edinburgh Dynamical Systems Study Group for general feedback on a draft version of the paper. We would also like to thank two anonymous reviewers whose comments improved the quality of the original manuscript
Relaxation oscillations and canards of a regulated two–gene model
No full textWe investigate a two–gene system with an autoregulatory feedback loop using geometric singular perturbation theory. We identify (coexisting) relaxation oscillations, singular Hopf bifurcations, homoclinic loops etc. We also demonstrate a new method to compute the criticality of the singular Hopf bifurcations
- …
