Extending Slow Manifold Near Generic Transcritical Canard Point
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摘要
We consider the dynamics of planar fast-slow systems near generic transcritical type canard point.By using geometric singular perturbation theory combined with the recently developed blow-up technique, the existence of canard cycles, relaxation oscillations and solutions near the attracting branch of the critical manifold is established. The asymptotic expansion of the parameter for which canard exists is obtained by a version of the Melnikov method.
We consider the dynamics of planar fast-slow systems near generic transcritical type canard point.By using geometric singular perturbation theory combined with the recently developed blow-up technique, the existence of canard cycles, relaxation oscillations and solutions near the attracting branch of the critical manifold is established. The asymptotic expansion of the parameter for which canard exists is obtained by a version of the Melnikov method.
We consider the dynamics of planar fast-slow systems near generic transcritical type canard point.By using geometric singular perturbation theory combined with the recently developed blow-up technique, the existence of canard cycles, relaxation oscillations and solutions near the attracting branch of the critical manifold is established. The asymptotic expansion of the parameter for which canard exists is obtained by a version of the Melnikov method.
引文
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[3]Brφns,M.,Bar-Eli,K.Canard explosion and excitation in a model of the Belousov-Zhabotinskii.J.Phys.Chem.,1991
[4]Callot,J.L.Champs lents-rapides complexes`a une dimension lente.Annales scientifiques de l’Ecole Normale Sup′erieure,4,26:149-173(1993)
[5]Chiba,H.Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points.J.Differential Equations,250:112–160(2011)
[6]Dumortier,F.Singularities of vector field on the plane.J.Differential Equations,23(1):53–106(1977)
[7]Dumortier,F.Techniques in the theory of local bifurcations:Blow-up,normal forms,nilpotent bifurcations,singular perturbations.In:Bifurcations and Periodic Orbits of Vector Fields(D.Szlomiuk,Ed.),Kluwer,Dordrecht,1993
[8]Dumortier,F.,Roussarie,F.Canard cycles and center manifold.Men.Amer.Math.Soc.,121,1996
[9]Dumortier,F.,Roussarie,F.Geometric singular perturbation theory beyond normal hyperbolicity.In:Multiple-Time-Scale Dynamical Systems(C.K.R.T.Jones and A.Khibnik,Eds.)IMA Volumes in Mathematics and its Applications,Vol.122,29–64,Springer-Verlag,Berlin/New York,2001
[10]Dumortier,F.,Llibre,J.,Art′es,J.C.Qualitative theory of planar differential systems,Universitext.Springer-Verlag,Berlin,2006.
[11]Diener,M.Regularizing microscopes and rivers.SIAM J.Appl.Math.,25:148–173(1994)
[12]Eckhaus,W.Relaxation oscillations including a standard chase on French ducks.In:Asymptotic Analysis II,Lecture Notes in Mathematics,Vol.985,449–494,Springer-Verlag,Berlin/New York,1983
[13]Fenichel,N.Grometric singular perturbation theory for ordinary differential equations.J.Differential Equations,31:53–98(1979)
[14]Haberman,R.Slowly varying jump and transition phenomena associated with algebraic bifurcation problems.SIAM J.Appl.Math.,37:69–106(1979)
[15]Jones,C.K.R.T.Geometric singular perturbation theory.In:Dynamical Systems(Montencatini Terme,1994),in:Lecture Notes in Mathematics,vol.1609,Springer,1995,44–118
[16]Krupa,M.,Szmolyan,P.Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions.SIAM J.Math.Anal.,33(2):286–314(2001)
[17]Krupa,M.,Szmolyan,P.Relaxation oscillation and canard explosion.J.Differential Equations,174:312–368(2001)
[18]Krupa,M.,Szmolyan,P.Extending slow manifold near transcritical and pitchfork singularites.Nonlinearity,14(6):1473–1491(2001)
[19]van Gils,S.,Krupa,M.,Szmolyan,P.Asymptotic expansions using blow-up.Z.Angew.Math.Phys.,56(3):369–397(2005)
[20]Lebovitz,N.R.,Schaar,R.J.Exchange of stabilities in autonomous systems.Stud.Appl.Math.,54:229–260(1975)
[21]Lebovitz,N.R.,Schaar,R.J.Exchange of stabilities in autonomous systems I.Vertical bifurcations Stud.Appl.Math.,56:1–50(1977)
[22]Mishchenko,E.F.,Kolesov,Yu.S.,Kolesov,A.Yu.,Rozov,N.Kh.Asymptotic Methods in Singularly Perturbed Systems.Consultants Bureau,New York,London,1994
[23]Szmolyan,P.,Wechselberger,M.Canards in R3.J.Differential Equations,177:419–453(2001)
[24]Wiggins,S.Normally Hyperbolic Invariant Manifolds in Dynamical Systems.Springer-Verlag,New York,1994
[25]Xie,F.,Han,M.A.Existence of canards under non-generic conditions.Chin.Ann.Math.,30B(3):239–250(2009)
[2]Benoit,E.Perturbation singuli re en dimension trois:Canards en un point pseudo-singulier neoud.Bulletin de la Soci′et′e Math′ematique de France,129-1:91–113(2001)
[3]Brφns,M.,Bar-Eli,K.Canard explosion and excitation in a model of the Belousov-Zhabotinskii.J.Phys.Chem.,1991
[4]Callot,J.L.Champs lents-rapides complexes`a une dimension lente.Annales scientifiques de l’Ecole Normale Sup′erieure,4,26:149-173(1993)
[5]Chiba,H.Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points.J.Differential Equations,250:112–160(2011)
[6]Dumortier,F.Singularities of vector field on the plane.J.Differential Equations,23(1):53–106(1977)
[7]Dumortier,F.Techniques in the theory of local bifurcations:Blow-up,normal forms,nilpotent bifurcations,singular perturbations.In:Bifurcations and Periodic Orbits of Vector Fields(D.Szlomiuk,Ed.),Kluwer,Dordrecht,1993
[8]Dumortier,F.,Roussarie,F.Canard cycles and center manifold.Men.Amer.Math.Soc.,121,1996
[9]Dumortier,F.,Roussarie,F.Geometric singular perturbation theory beyond normal hyperbolicity.In:Multiple-Time-Scale Dynamical Systems(C.K.R.T.Jones and A.Khibnik,Eds.)IMA Volumes in Mathematics and its Applications,Vol.122,29–64,Springer-Verlag,Berlin/New York,2001
[10]Dumortier,F.,Llibre,J.,Art′es,J.C.Qualitative theory of planar differential systems,Universitext.Springer-Verlag,Berlin,2006.
[11]Diener,M.Regularizing microscopes and rivers.SIAM J.Appl.Math.,25:148–173(1994)
[12]Eckhaus,W.Relaxation oscillations including a standard chase on French ducks.In:Asymptotic Analysis II,Lecture Notes in Mathematics,Vol.985,449–494,Springer-Verlag,Berlin/New York,1983
[13]Fenichel,N.Grometric singular perturbation theory for ordinary differential equations.J.Differential Equations,31:53–98(1979)
[14]Haberman,R.Slowly varying jump and transition phenomena associated with algebraic bifurcation problems.SIAM J.Appl.Math.,37:69–106(1979)
[15]Jones,C.K.R.T.Geometric singular perturbation theory.In:Dynamical Systems(Montencatini Terme,1994),in:Lecture Notes in Mathematics,vol.1609,Springer,1995,44–118
[16]Krupa,M.,Szmolyan,P.Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions.SIAM J.Math.Anal.,33(2):286–314(2001)
[17]Krupa,M.,Szmolyan,P.Relaxation oscillation and canard explosion.J.Differential Equations,174:312–368(2001)
[18]Krupa,M.,Szmolyan,P.Extending slow manifold near transcritical and pitchfork singularites.Nonlinearity,14(6):1473–1491(2001)
[19]van Gils,S.,Krupa,M.,Szmolyan,P.Asymptotic expansions using blow-up.Z.Angew.Math.Phys.,56(3):369–397(2005)
[20]Lebovitz,N.R.,Schaar,R.J.Exchange of stabilities in autonomous systems.Stud.Appl.Math.,54:229–260(1975)
[21]Lebovitz,N.R.,Schaar,R.J.Exchange of stabilities in autonomous systems I.Vertical bifurcations Stud.Appl.Math.,56:1–50(1977)
[22]Mishchenko,E.F.,Kolesov,Yu.S.,Kolesov,A.Yu.,Rozov,N.Kh.Asymptotic Methods in Singularly Perturbed Systems.Consultants Bureau,New York,London,1994
[23]Szmolyan,P.,Wechselberger,M.Canards in R3.J.Differential Equations,177:419–453(2001)
[24]Wiggins,S.Normally Hyperbolic Invariant Manifolds in Dynamical Systems.Springer-Verlag,New York,1994
[25]Xie,F.,Han,M.A.Existence of canards under non-generic conditions.Chin.Ann.Math.,30B(3):239–250(2009)