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One Switch in the Plane: A Primer

In: Hidden Dynamics

Author

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  • Mike R. Jeffrey

    (University of Bristol, Department of Engineering Mathematics)

Abstract

This chapter presents a short course on dynamical systems with two variables and one switch, ( ẋ 1 , ẋ 2 ) = f ( x 1 , x 2 ; λ ) , f ( x 1 , x 2 ; λ ) , λ = sign ( σ ( x 1 , x 2 ) ) . $$\displaystyle{(\dot{x}_{1},\dot{x}_{2}) = \left (f(x_{1},x_{2};\lambda ),\;f(x_{1},x_{2};\lambda )\right )\;,\qquad \lambda =\mathop{ \mathrm{sign}}\nolimits (\sigma (x_{1},x_{2}))\;.}$$ 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 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.

Suggested Citation

  • Mike R. Jeffrey, 2018. "One Switch in the Plane: A Primer," Springer Books, in: Hidden Dynamics, chapter 0, pages 31-60, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-02107-8_2
    DOI: 10.1007/978-3-030-02107-8_2
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