An efficient algorithm for solving piecewise-smooth dynamical systems

This article considers the numerical treatment of piecewise-smooth dynamical systems. Classical solutions as well as sliding modes up to codimension-2 are treated. An algorithm is presented that, in the case of non-uniqueness, selects a solution that is the formal limit solution of a regularized problem. The numerical solution of a regularized differential equation, which creates stiffness and often also high oscillations, is avoided.

[1]  Manuel Calvo,et al.  Algorithm 968 , 2016 .

[2]  Ernst Hairer,et al.  Solving Optimization-Constrained Differential Equations with Discontinuity Points, with Application to Atmospheric Chemistry , 2009, SIAM J. Sci. Comput..

[3]  Mike R. Jeffrey,et al.  Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding , 2014, SIAM J. Appl. Dyn. Syst..

[4]  M. B. Carver.  Efficient integration over discontinuities in ordinary differential equation simulations , 1978 .

[5]  Ernst Hairer,et al.  Computing breaking points in implicit delay differential equations , 2008, Adv. Comput. Math..

[6]  E. Hairer,et al.  Solutions leaving a codimension-$$\varvec{2}$$2 sliding , 2017 .

[7]  Panagiotis Kaklamanos,et al.  Regularization and Geometry of Piecewise Smooth Systems with Intersecting Discontinuity Sets , 2019, SIAM J. Appl. Dyn. Syst..

[8]  F. Difonzo.  A note on attractivity for the intersection of two discontinuity manifolds , 2020 .

[9]  Ernst Hairer,et al.  Examples of Stiff Equations , 1996 .

[10]  Mike R. Jeffrey,et al.  Hidden dynamics in models of discontinuity and switching , 2014 .

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

[12]  J. P. Den Hartog,et al.  LXXIII. Forced vibrations with combined viscous and coulomb damping , 1930 .

[13]  Guiyou Mao,et al.  Efficient integration over discontinuities for differential-algebraic systems , 2002 .

[14]  L. Glass,et al.  Combinatorial explosion in model gene networks. , 2000, Chaos.

[15]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[16]  Arcady Ponosov,et al.  Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks , 2011 .

[17]  Jaume Llibre,et al.  Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation , 2009, SIAM J. Appl. Dyn. Syst..

[18]  Ernst Hairer,et al.  Classification of Hidden Dynamics in Discontinuous Dynamical Systems , 2015, SIAM J. Appl. Dyn. Syst..

[19]  Yuri A. Kuznetsov,et al.  An event-driven method to simulate Filippov systems with accurate computing of sliding motions , 2008, TOMS.

[20]  Luca Dieci,et al.  Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field , 2011, Numerische Mathematik.

[21]  Luca Dieci,et al.  A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side , 2012, J. Comput. Appl. Math..

[22]  Ole Østerby,et al.  Solving Ordinary Differential Equations with Discontinuities , 1984, TOMS.

[23]  Reinhold Mannshardt.  One-step methods of any order for ordinary differential equations with discontinuous right-hand sides , 1978 .

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

[25]  Luca Dieci,et al.  Sliding motion on the intersection of two manifolds: Spirally attractive case , 2015, Commun. Nonlinear Sci. Numer. Simul..