A Stiction Oscillator with Canards: On Piecewise Smooth Nonuniqueness and Its Resolution by Regularization Using Geometric Singular Perturbation Theory

In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to ...

[1]  J. Dieterich Time-dependent friction and the mechanics of stick-slip , 1978 .

[2]  Gábor Csernák,et al.  On the chaotic behaviour of a simple dry-friction oscillator , 2014, Math. Comput. Simul..

[3]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[4]  K. U. Kristiansen,et al.  Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in Using Regularization and Blowup , 2018 .

[5]  Stephen Schecter,et al.  Composite Waves in the Dafermos Regularization , 2004 .

[6]  Neil Fenichel,et al.  Asymptotic stability with rate conditions for dynamical systems , 1974 .

[7]  Stephen John Hogan,et al.  On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in ℝ3 , 2014, SIAM J. Appl. Dyn. Syst..

[8]  M. Nakatani Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated rheology , 2001 .

[9]  Mike R. Jeffrey,et al.  Hidden Dynamics: The Mathematics of Switches, Decisions and Other Discontinuous Behaviour , 2018 .

[10]  Petri T. Piiroinen,et al.  Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator , 2008 .

[11]  Eusebius J. Doedel,et al.  Lecture Notes on Numerical Analysis of Nonlinear Equations , 2007 .

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

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

[14]  M. Wiercigroch,et al.  Hysteretic effects of dry friction: modelling and experimental studies , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  Ettore Pennestrì,et al.  Review and comparison of dry friction force models , 2016 .

[16]  Jaume Llibre,et al.  Sliding Vector Fields via Slow--Fast Systems , 2008 .

[17]  Karl Popp,et al.  ON THE MODELLING OF FRICTION OSCILLATORS , 1998 .

[18]  Peter Szmolyan,et al.  Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle , 2016, Journal of mathematical biology.

[19]  Kristian Uldall Kristiansen,et al.  The Regularized Visible Fold Revisited , 2019, Journal of Nonlinear Science.

[20]  T. K. Pratt,et al.  Non-linear analysis of stick/slip motion , 1981 .

[21]  Jim Woodhouse,et al.  Are there reliable constitutive laws for dynamic friction? , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[22]  G. Stépán,et al.  On the periodic response of a harmonically excited dry friction oscillator , 2006 .

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

[24]  K. Uldall Kristiansen,et al.  On the regularization of impact without collision: the Painlevé paradox and compliance , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  Karl Popp,et al.  A Historical Review on Dry Friction and Stick-Slip Phenomena , 1998 .

[26]  Martin Wechselberger,et al.  Canards of Folded Saddle-Node Type I , 2015, SIAM J. Math. Anal..

[27]  Gábor Stépán,et al.  Sub-harmonic resonant solutions of a harmonically excited dry friction oscillator , 2007 .

[28]  J. Cortés Discontinuous dynamical systems , 2008, IEEE Control Systems.

[29]  R. Haiduc Horseshoes in the forced van der Pol system , 2008 .

[30]  A. Akay Acoustics of friction. , 2002, The Journal of the Acoustical Society of America.

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

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

[33]  John Guckenheimer,et al.  Bifurcations of Relaxation oscillations Near Folded saddles , 2005, Int. J. Bifurc. Chaos.

[34]  Karl Johan Åström,et al.  Friction generated limit cycles , 1996, Proceeding of the 1996 IEEE International Conference on Control Applications IEEE International Conference on Control Applications held together with IEEE International Symposium on Intelligent Contro.

[35]  Peter Szmolyan,et al.  Singularly Perturbed Oscillators with Exponential Nonlinearities , 2019, Journal of Dynamics and Differential Equations.

[36]  K. K.,et al.  Stick-slip vibrations and chaos , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[37]  M. Krupa,et al.  Relaxation Oscillation and Canard Explosion , 2001 .

[38]  K. U. Kristiansen,et al.  Singular limit analysis of a model for earthquake faulting , 2016, 1603.02448.

[39]  K. U. Kristiansen,et al.  A new type of relaxation oscillation in a model with rate-and-state friction , 2019, Nonlinearity.

[40]  I. D. Abrahams,et al.  Curve squeal of train wheels, Part 1: mathematical model for its generation , 2000 .

[41]  Edgar Knobloch,et al.  Transonic canards and stellar wind , 2017 .

[42]  U. Galvanetto,et al.  Dynamics of a Simple Damped Oscillator Undergoing Stick-Slip Vibrations , 1999 .

[43]  Carlos Canudas de Wit,et al.  Friction Models and Friction Compensation , 1998, Eur. J. Control.

[44]  J. Dieterich Modeling of rock friction: 1. Experimental results and constitutive equations , 1979 .

[45]  Tere M. Seara,et al.  An Analytical Approach to Codimension-2 Sliding Bifurcations in the Dry-Friction Oscillator , 2010, SIAM J. Appl. Dyn. Syst..

[46]  P. Cox,et al.  Excitability in ramped systems: the compost-bomb instability , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[47]  Gianpiero Rocca,et al.  Brake comfort – a review , 2009 .

[48]  E. Rabinowicz The Nature of the Static and Kinetic Coefficients of Friction , 1951 .

[49]  A. Ruina Slip instability and state variable friction laws , 1983 .

[50]  Steven W. Shaw,et al.  On the dynamic response of a system with dry friction , 1986 .

[51]  Peter Szmolyan,et al.  Relaxation oscillations in R3 , 2004 .

[52]  J. Willis,et al.  Regimes of frictional sliding of a spring-block system , 2010 .

[53]  A. Champneys,et al.  A phase-plane analysis of localized frictional waves , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[56]  P. Casini,et al.  Dynamics of friction oscillators excited by a moving base and/or driving force , 2001 .