4.7 Article

The number of limit cycles for regularized piecewise polynomial systems is unbounded

Related references

Note: Only part of the references are listed.
Article Mathematics, Applied

Relaxation oscillations in substrate-depletion oscillators close to the nonsmooth limit

K. Uldall Kristiansen et al.

Summary: In this paper, a novel type of relaxation oscillations in substrate-depletion oscillators is described using geometric singular perturbation theory and blow-up as a key technical tool. The oscillations in the planar model are shown to be produced by a complicated interplay between two stable nodes and a discontinuity set in the singular limit. This interplay introduces a new mechanism for generating relaxation-type oscillations, which is also discussed in a more general setting.

NONLINEARITY (2021)

Article Mathematics, Applied

The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

M. Esteban et al.

Summary: This paper deals with discontinuous piecewise differential systems formed by linear centers and quadratic centers, providing upper bounds for the maximum number of limit cycles they can exhibit. The study has solved the 16th Hilbert problem for such differential systems, with seven classes reaching the obtained upper bounds on the maximum number of limit cycles.

CHAOS (2021)

Article Mathematics

Singularly perturbed boundary-focus bifurcations

Samuel Jelbart et al.

Summary: We study the transition of smooth systems to piecewise-smooth systems with a boundary-focus bifurcation as epsilon -> 0, and identify different bifurcation structures. We uncover the evolution characteristics of cycles associated with BF bifurcations in the smooth system, and prove the existence of a family of stable limit cycles.

JOURNAL OF DIFFERENTIAL EQUATIONS (2021)

Article Mathematics, Applied

Singularly perturbed boundary-equilibrium bifurcations

S. Jelbart et al.

Summary: This work completes a classification of codimension-1 singularly perturbed boundary equilibria bifurcation (BEB) in the plane, utilizing tools from PWS theory, geometric singular perturbation theory, and the method of geometric desingularization known as blow-up. Local normal forms for generating all 12 singularly perturbed BEBs are derived, and the unfolding in each case is described. Detailed quantitative results on various bifurcations involved in the unfoldings and classification are presented, including saddle-node, Andronov-Hopf, homoclinic, and codimension-2 Bogdanov-Takens bifurcations.

NONLINEARITY (2021)

Article Mathematics

Asymptotic lower bounds on Hilbert numbers using canard cycles

M. J. Alvarez et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2020)

Article Mathematics, Applied

The Regularized Visible Fold Revisited

K. Uldall Kristiansen

JOURNAL OF NONLINEAR SCIENCE (2020)

Article Mathematics, Applied

Resolution of the Piecewise Smooth Visible-Invisible Two-Fold Singularity in R3 Using Regularization and Blowup

K. Uldall Kristiansen et al.

JOURNAL OF NONLINEAR SCIENCE (2019)

Article Mathematics

Regularization around a generic codimension one fold-fold singularity

Carles Bonet-Reves et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2018)

Article Mathematics, Applied

Stability Loss Delay and Smoothness of the Return Map in Slow-Fast Systems

Cheng Wang et al.

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS (2018)

Article Mathematics, Applied

Predator-prey systems with small predator's death rate

Renato Huzak

ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS (2018)

Article Mathematics, Applied

CYCLICITY OF DEGENERATE GRAPHIC DF2a OF DUMORTIER-ROUSSARIE-ROUSSEAU PROGRAM

Renato Huzak

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS (2018)

Article Mathematics, Applied

Singular limit analysis of a model for earthquake faulting

Elena Bossolini et al.

NONLINEARITY (2017)

Article Biology

Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle

Ilona Kosiuk et al.

JOURNAL OF MATHEMATICAL BIOLOGY (2016)

Article Mathematics, Applied

FINITE CYCLICITY OF SLOW-FAST DARBOUX SYSTEMS WITH A TWO-SADDLE LOOP

Marcin Bobienski et al.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2016)

Article Mathematics, Applied

Regularizations of Two-Fold Bifurcations in Planar Piecewise Smooth Systems Using Blowup

K. Uldall Kristiansen et al.

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS (2015)

Article Mathematics

Slow divergence integrals in generalized Liénard equations near centers

Renato Huzak et al.

Electronic Journal of Qualitative Theory of Differential Equations (2015)

Article Mathematics, Applied

Slow Divergence Integrals in Classical Li,nard Equations Near Centers

Peter De Maesschalck et al.

JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS (2015)

Article Mathematics

PSEUDO-ABELIAN INTEGRALS ON SLOW-FAST DARBOUX SYSTEMS

Marcin Bobienski et al.

ANNALES DE L INSTITUT FOURIER (2013)

Article Mathematics, Interdisciplinary Applications

LOWER BOUNDS FOR THE MAXIMUM NUMBER OF LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH A STRAIGHT LINE OF SEPARATION

J. Llibre et al.

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS (2013)

Article Mathematics

Canard cycles for predator-prey systems with Holling types of functional response

Chengzhi Li et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2013)

Article Mathematics, Applied

On the number of limit cycles of polynomial Lienard systems

Maoan Han et al.

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS (2013)

Article Mathematics, Applied

Small-amplitude limit cycles of polynomial Li,nard systems

Han MaoAn et al.

SCIENCE CHINA-MATHEMATICS (2013)

Article Mathematics, Applied

Numerical Continuation Techniques for Planar Slow-Fast Systems

P. De Maesschalck et al.

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS (2013)

Article Mathematics

Uniqueness of limit cycles for Lienard differential equations of degree four

Chengzhi Li et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2012)

Article Mathematics, Applied

Hilbert's Sixteenth Problem for Polynomial Lienard Equations

Magdalena Caubergh

QUALITATIVE THEORY OF DYNAMICAL SYSTEMS (2012)

Article Mathematics

Classical Lienard equations of degree n ≥ 6 can have [n-1/2]+2 limit cycles

P. De Maesschalck et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2011)

Article Mathematics, Applied

Slow Divergence Integral and Balanced Canard Solutions

Freddy Dumortier

QUALITATIVE THEORY OF DYNAMICAL SYSTEMS (2011)

Article Mathematics

Limit cycles of the generalized polynomial Lienard differential equations

Jaume Llibre et al.

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY (2010)

Article Mathematics, Applied

STUDY OF THE CYCLICITY OF SOME DEGENERATE GRAPHICS INSIDE QUADRATIC SYSTEMS

Freddy Dumortier et al.

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS (2009)

Article Mathematics

Transition maps at non-resonant hyperbolic singularities are o-minimal

T. Kaiser et al.

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK (2009)

Article Mathematics

Limit cycles near hyperbolas in quadratic systems

Joan C. Artes et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2009)

Article Mathematics

Canard cycles in the presence of slow dynamics with singularities

P. De Maesschalck et al.

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS (2008)

Article Mathematics, Applied

More limit cycles than expected in Lienard equations

Freddy Dumortier et al.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2007)

Article Mathematics

Time analysis and entry-exit relation near planar turning points

P De Maesschalck et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2005)

Review Mathematics, Interdisciplinary Applications

One-parameter bifurcations in planar filippov systems

YA Kuznetsov et al.

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS (2003)

Review Mathematics, Interdisciplinary Applications

Hilbert's 16th problem and bifurcations of planar polynomial vector fields

JB Li

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS (2003)

Article Mathematics

Canards in R3

P Szmolyan et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2001)

Article Mathematics

Relaxation oscillation and canard explosion

M Krupa et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2001)

Article Mathematics

Multiple canard cycles in generalized Lienard equations

F Dumortier et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2001)