Abstract
We now resume the study of piecewise-smooth systems with general dependence on the variable x and the switching multiplier \(\boldsymbol{\lambda }\).
We begin by exploring some of the more exotic behaviours created by dynamics hidden inside the switching layer .
To see how much of our knowledge of linear systems—the content of the last chapter—applies in general, we also establish which features of linear switching are robust to small perturbations that are nonlinear in the switching multipliers and which features are not.
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References
R. Edwards, A. Machina, G. McGregor, and P. van den Driessche. A modelling framework for gene regulatory networks including transcription and translation. Bull. Math. Biol., 77:953–983, 2015.
J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences 42. Springer, 2002.
N. Guglielmi and E. Hairer. Classification of hidden dynamics in discontinuous dynamical systems. SIADS, 14(3):1454–1477, 2015.
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.
M. R. Jeffrey. Hidden degeneracies in piecewise smooth dynamical systems. Int. J. Bif. Chaos, 26(5):1650087(1–18), 2016.
M. R. Jeffrey. The ghosts of departed quantities in switches and transitions. SIAM Review, 60(1):116–36, 2017.
A. Machina, R. Edwards, and P. van den Dreissche. Singular dynamics in gene network models. SIADS, 12(1):95–125, 2013.
E. Plahte and S Kjøglum. Analysis and generic properties of gene regulatory networks with graded response functions. Physica D, 201:150–176, 2005.
Y. Ueda. Strange attractors and the origin of chaos. Nonlinear Science Today, 2:1–16, 1992.
B. Van der Pol. On relaxation-oscillations. The London, Edinburgh and Dublin Phil. Mag. & J. of Sci., 2(7):978–992, 1926.
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Jeffrey, M.R. (2018). Nonlinear Switching (Local Theory): The Phenomena of Hidden Dynamics. In: Hidden Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-02107-8_9
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DOI: https://doi.org/10.1007/978-3-030-02107-8_9
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