One Switch in the Plane: A Primer

  • Mike R. Jeffrey


This chapter presents a short course on dynamical systems with two variables and one switch,
$$\displaystyle{(\dot{x}_{1},\dot{x}_{2}) = \left (f(x_{1},x_{2};\lambda ),\;f(x_{1},x_{2};\lambda )\right )\;,\qquad \lambda =\mathop{ \mathrm{sign}}\nolimits (\sigma (x_{1},x_{2}))\;.}$$
These represent the simplest case of the problems we cover more generally in the rest of the book.

They are the most studied and most easily understood piecewise-smooth problems, in comparison with systems on the real line which are trivial, and higher-dimensional systems which are orders more challenging. Filippov in particular covered planar systems in great detail in [71], so it is a good place to begin summarizing the state of the art and setting off in search of something more.


  1. 19.
    C. M. Bender and S. A. Orszag. Advanced mathematical methods for scientists and engineers I. Asymptotic methods and perturbation theory. Springer-Verlag, New York, 1999.Google Scholar
  2. 25.
    B. Brogliato. Nonsmooth mechanics – models, dynamics and control. Springer-Verlag (New York), 1999.Google Scholar
  3. 29.
    C. A. Buzzi, T. de Carvalho, and M. A. Teixeira. Birth of limit cycles bifurcating from a nonsmooth center. J. Math. Pure Appl., 102:36–47, 2014.MathSciNetzbMATHGoogle Scholar
  4. 48.
    M. di Bernardo, C. J. Budd, A. R. Champneys, and P. Kowalczyk. Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, 2008.Google Scholar
  5. 49.
    M. di Bernardo, P. Kowalczyk, and A. Nordmark. Bifurcations of dynamical systems with sliding: derivation of normal-form mappings. Physica D, 170:175–205, 2002.MathSciNetzbMATHGoogle Scholar
  6. 70.
    A. F. Filippov. Differential equations with discontinuous right-hand side. American Mathematical Society Translations, Series 2, 42:19–231, 1964.Google Scholar
  7. 71.
    A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publ. Dortrecht, 1988 (Russian 1985).Google Scholar
  8. 82.
    P. Glendinning, S. J. Hogan, M. E. Homer, M. R. Jeffrey, and R. Szalai. Uncountably many cases of Filippov’s sewed focus. submitted, 2016.Google Scholar
  9. 92.
    J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences 42. Springer, 2002.Google Scholar
  10. 98.
    P. Hartman. Ordinary differential equations. Wiley: New York, 1964.zbMATHGoogle Scholar
  11. 100.
    D. Hilbert. Mathematische probleme. Göttinger Nachrichten, pages 253–297, 1900.Google Scholar
  12. 101.
    D. Hilbert. Mathematische Probleme. Archiv der Mathematik und Physik, 3(1):44–6, 213–237, 1901.Google Scholar
  13. 102.
    D. Hilbert. Mathematical problems. Bulletin of the American Mathematical Society, 8(10):437–479, 1902.MathSciNetzbMATHGoogle Scholar
  14. 104.
    E. J. Hinch. Perturbation Methods. Cambridge University Press, 1991.Google Scholar
  15. 114.
    M. R. Jeffrey. Hidden dynamics in models of discontinuity and switching. Physica D, 273–274:34–45, 2014.MathSciNetzbMATHGoogle Scholar
  16. 124.
    C. K. R. T. Jones. Geometric singular perturbation theory, volume 1609 of Lecture Notes in Math. pp. 44–120. Springer-Verlag (New York), 1995.Google Scholar
  17. 135.
    C. Kuehn. Multiple time scale dynamics. Springer, 2015.Google Scholar
  18. 137.
    M. Kunze. Non-Smooth Dynamical Systems. Springer, 2000.Google Scholar
  19. 140.
    Yu. A. Kuznetsov, S. Rinaldi, and A. Gragnani. One-parameter bifurcations in planar Filippov systems. Int. J. Bif. Chaos, 13:2157–2188, 2003.MathSciNetCrossRefGoogle Scholar
  20. 152.
    J. Llibre, D. N. Novaes, and M. A. Teixeira. Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dynamics, 82(3):1159–1175, 2015.MathSciNetzbMATHGoogle Scholar
  21. 162.
    J. D. Meiss. Differential Dynamical Systems. SIAM, 2007.Google Scholar
  22. 170.
    D. N. Novaes and E. Ponce. A simple solution to the braga–mello conjecture. IJBC, 25(1):1550009:1–7, 2015.Google Scholar
  23. 176.
    T. Poston and I. N. Stewart. Catastrophe theory and its applications. Dover, 1996.Google Scholar
  24. 199.
    S. Strogatz. Nonlinear Dynamics and Chaos. Westview Press, 2nd edition, 2014.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mike R. Jeffrey
    • 1
  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK

Personalised recommendations