Fundamental matrix solutions of piecewise smooth differential systems

Abstract: We consider the fundamental matrix solution associated to piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. We review the cases of transversal intersection and of sliding motion on one surface. We also consider the case when sliding motion takes place on the intersection of two or more surfaces. Numerical results are also given.

[1]  Ugo Galvanetto.  Some Discontinuous Bifurcations in a Two-Block STICK-SLIP System , 2001 .

[2]  K. Henrik,et al.  Limit Cycles with Chattering in Relay Feedback Systems , 1997 .

[3]  J. Alexander,et al.  SLIDING MODES IN INTERSECTING SWITCHING SURFACES , I : BLENDING , 2003 .

[4]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[5]  H. D. Jong,et al.  Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability , 2006, Journal of mathematical biology.

[6]  A. P. Ivanov,et al.  The stability of periodic solutions of discontinuous systems that intersect several surfaces of discontinuity , 1998 .

[7]  S. Campbell.  Linearization of DAEs along trajectories , 1995 .

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

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

[10]  S. Żak,et al.  Sliding Modes in Solving Convex Programming Problems , 1998 .

[11]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[12]  G. A. Leonov.  Pendulum with Positive and Negative Dry Friction.: Continuum of Homoclinic Orbits , 1995 .

[13]  Vadim I. Utkin,et al.  Sliding Modes and their Application in Variable Structure Systems , 1978 .

[14]  Peter Kukučka.  Jumps of the fundamental solution matrix in discontinuous systems and applications , 2007 .

[15]  Mario di Bernardo,et al.  Piecewise smooth dynamical systems , 2008, Scholarpedia.

[16]  Bernold Fiedler,et al.  Ergodic theory, analysis, and efficient simulation of dynamical systems , 2001 .

[17]  Zheng Hui.  On the Stability of Periodic Motion of an Unbalanced Elastic Rotor System Supported on Lubricated Bearings , 2002 .

[18]  H. D. Jong,et al.  Qualitative simulation of genetic regulatory networks using piecewise-linear models , 2004, Bulletin of mathematical biology.

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

[20]  H. Nijmeijer,et al.  Dynamics and Bifurcations ofNon - Smooth Mechanical Systems , 2006 .

[21]  Vadim I. Utkin,et al.  The simplex method for nonlinear sliding mode control , 1999 .

[22]  P. Müller.  Calculation of Lyapunov exponents for dynamic systems with discontinuities , 1995 .

[23]  M. Kunze,et al.  Non-Smooth Dynamical Systems: An Overview , 2001 .

[24]  Erik Plahte,et al.  Analysis and generic properties of gene regulatory networks with graded response functions , 2005 .

[25]  B. Brogliato,et al.  Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics , 2008 .

[26]  Luca Dieci,et al.  Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach , 2009, SIAM J. Numer. Anal..

[27]  J. Gouzé,et al.  A class of piecewise linear differential equations arising in biological models , 2002 .

[28]  M. A. Aizerman,et al.  DETERMINATION OF STABILITY BY LINEAR APPROXIMATION OF A PERIODIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES , 1958 .

[29]  Vincent Acary,et al.  Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics , 2008 .