On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in ℝ3

In this paper we use the blowup method of Dumortier and Roussarie, in the formulation due to Krupa and Szmolyan, to study the regularization of singularities of piecewise smooth dynamical systems in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira, we first demonstrate the power of our approach by considering the case of a fold line. We quickly extend a main result of Reves and Seara in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain nonresonance condition holds. Finally, we provide numerical evidence for the existence of secondary ...

[1]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[2]  S. John Hogan,et al.  Canards in piecewise-linear systems: explosions and super-explosions , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  Jaume Llibre,et al.  Regularization Of Discontinuous Vector Fields In Dimension Three , 1997 .

[4]  V. Utkin.  Variable structure systems with sliding modes , 1977 .

[5]  O. Makarenkov,et al.  Preface: Dynamics and Bifurcations of Nonsmooth Systems , 2012 .

[6]  Xavier Jarque,et al.  A Survey on the blow up Technique , 2011, Int. J. Bifurc. Chaos.

[7]  J. Callot,et al.  Chasse au canard , 1977 .

[8]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[9]  Marco Antonio Teixeira,et al.  Generic Bifurcation of Sliding Vector Fields , 1993 .

[10]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[11]  M. Krupa,et al.  Local analysis near a folded saddle-node singularity , 2010 .

[12]  Christian Kuehn,et al.  Multiscale Geometry of the Olsen Model and Non-classical Relaxation Oscillations , 2014, J. Nonlinear Sci..

[13]  Freddy Dumortier,et al.  Techniques in the Theory of Local Bifurcations: Blow-Up, Normal Forms, Nilpotent Bifurcations, Singular Perturbations , 1993 .

[14]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[15]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[16]  P. Szmolyan,et al.  Canards in R3 , 2001 .

[17]  Peter Szmolyan,et al.  Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points - Fold and Canard Points in Two Dimensions , 2001, SIAM J. Math. Anal..

[18]  Martin Krupa,et al.  Mixed Mode Oscillations due to the Generalized Canard Phenomenon , 2006 .

[19]  Martin Wechselberger,et al.  Existence and Bifurcation of Canards in ℝ3 in the Case of a Folded Node , 2005, SIAM J. Appl. Dyn. Syst..

[20]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[21]  Peter Szmolyan,et al.  Extending slow manifolds near transcritical and pitchfork singularities , 2001 .

[22]  S. McNamara.  Rigid and Quasi-Rigid Theories of Granular Media , 2007 .

[23]  F. Dumortier,et al.  Local Study of Planar Vector Fields: Singularities and Their Unfoldings , 1991 .

[24]  Mathieu Desroches,et al.  Canards and curvature: nonsmooth approximation by pinching , 2011, 1204.4036.

[25]  Freddy Dumortier,et al.  Canard Cycles and Center Manifolds , 1996 .

[26]  D Michelson,et al.  Steady solutions of the Kuramoto-Sivashinsky equation , 1986 .

[27]  Antonio E. Teruel,et al.  Canard trajectories in 3D piecewise linear systems , 2013 .

[28]  Karl Schweizerhof,et al.  FE-regularization of non-smooth vibrations due to friction and impacts , 2002 .

[29]  Stephen John Hogan,et al.  The Geometry of Generic Sliding Bifurcations , 2011, SIAM Rev..

[30]  A. V. Pukhlikov,et al.  Discontinuous Dynamical Systems in Optimal Control Theory , 2002 .

[31]  Peter Szmolyan,et al.  Scaling in Singular Perturbation Problems: Blowing Up a Relaxation Oscillator , 2011, SIAM J. Appl. Dyn. Syst..

[32]  Regularization of 2D Frictional Contacts for Rigid Body Dynamics , 2006 .

[33]  T. M. Seara,et al.  Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems , 2014, 1402.5237.