Skip to main content

Three Experiments

  • Chapter
  • First Online:
Modeling with Nonsmooth Dynamics

Part of the book series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials ((FIADS,volume 7))

  • 591 Accesses

Abstract

To set up the remainder of this article let us pose three problems. They incorporate various issues arising from basic analysis to simulation to applied modelling. Their seemingly ambiguous dynamics expose remarkably well our current state of knowledge concerning the robustness of nonsmooth models. The problems are: a classic example of ambiguity from seminal texts; a two-gene regulatory system with seemingly ambiguous activation of genes; and an investment game where players’ seemingly steady behaviour destabilizes a company’s trading.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. V. Acary, H. de Jong, B. Brogliato, Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems. Physica D 269, 103–119 (2014)

    Article  MathSciNet  Google Scholar 

  2. J.C. Alexander, T.I. Seidman, Sliding modes in intersecting switching surfaces, I: blending. Houst. J. Math. 24(3), 545–569 (1998)

    MathSciNet  MATH  Google Scholar 

  3. J.C. Alexander, T.I. Seidman, Sliding modes in intersecting switching surfaces, II: hysteresis. Houst. J. Math. 25(1), 185–211 (1999)

    MathSciNet  MATH  Google Scholar 

  4. C. Bonet, T.M. Seara, E. Fossas, M.R. Jeffrey, A unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems. Commun. Nonlinear Sci. Numer. Simul. 50, 142–168 (2017)

    Article  MathSciNet  Google Scholar 

  5. A.F. Filippov, Differential Equations with Discontinuous Righthand Sides (Kluwer, Dordrecht, 1988) (original in Russian 1985)

    Book  Google Scholar 

  6. A.V. Hill, The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. Proc. Physiol. Soc. 40, iv–vii (1910)

    Google Scholar 

  7. M.R. Jeffrey, Dynamics at a switching intersection: hierarchy, isonomy, and multiple-sliding. SIADS 13(3), 1082–1105 (2014)

    Article  MathSciNet  Google Scholar 

  8. M.R. Jeffrey, Hidden dynamics in models of discontinuity and switching. Physica D 273–274, 34–45 (2014)

    Article  MathSciNet  Google Scholar 

  9. M.R. Jeffrey, Hidden Dynamics: The Mathematics of Switches, Decisions, and Other Discontinuous Behaviour (Springer, Berlin, 2019)

    Google Scholar 

  10. M.R. Jeffrey, D.J.W. Simpson, Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise. Nonlinear Dyn. 76(2), 1395–1410 (2014)

    Article  MathSciNet  Google Scholar 

  11. M.R. Jeffrey, G. Kafanas, D.J.W. Simpson, Jitter in dynamical systems with intersecting discontinuity surfaces. IJBC 28(6), 1–22 (2018)

    MathSciNet  MATH  Google Scholar 

  12. D.N. Novaes, M.R. Jeffrey, Regularization of hidden dynamics in piecewise smooth flow. J. Differ. Equ. 259, 4615–4633 (2015)

    Article  MathSciNet  Google Scholar 

  13. E. Plahte, S. Kjøglum, Analysis and generic properties of gene regulatory networks with graded response functions. Physica D 201, 150–176 (2005)

    Article  MathSciNet  Google Scholar 

  14. D.J.W. Simpson, On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations. Discrete Contin. Dyn. Syst. 34(9), 3803–3830 (2014)

    Article  MathSciNet  Google Scholar 

  15. D.J.W. Simpson, R. Kuske, The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise. Stochastics Dyn. 14(4), 1450010 (2014)

    Article  MathSciNet  Google Scholar 

  16. D.J.W. Simpson, R. Kuske, Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete Contin. Dyn. Syst. Ser. B 19(9), 2889–2913 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Jeffrey, M.R. (2020). Three Experiments. In: Modeling with Nonsmooth Dynamics. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-35987-4_4

Download citation

Publish with us

Policies and ethics