Abstract
The analysis of piecewise smooth bifurcations reveals an alarming proliferation of cases as the dimension of phase space increases. This suggests that a different approach needs to be taken when trying to describe bifurcations. In particular, it may not be helpful to analyze particular bifurcations at the level of detail that is standard for smooth systems.
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References
J. Buzzi, Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps. Int. J. Bifurc. Chaos 9, 1743–1750 (1999)
L. Carroll, Through the Looking-Glass, and What Alice Found There (Macmillan, London, 1871)
M. di Bernardo, Normal forms of border collision in high dimensional non-smooth maps. Proc. IEEE ISCAS 2003(3), 76–79 (2003)
M. di Bernardo, C. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical Sciences, vol. 163 (Springer, London, 2008)
A.F. Filippov, Differential Equations with Discontinuous Right Hand Sides (Kluwer, Netherlands, 1988)
J.M. Gambaudo, I. Procaccia, S. Thomae, C. Tresser, New universal scenarios for the onset of chaos in Lorenz type flows. Phys. Rev. Lett. 57, 925–928 (1986)
P. Glendinning, The anharmonic route to chaos: kneading theory. Nonlinearity 6, 349–367 (1993)
P. Glendinning, Renormalization for the boundary of chaos in piecewise monotonic maps with a single discontinuity. Nonlinearity 27, R143–R162 (2014)
P. Glendinning, Bifurcation from stable fixed point to \(N\)-dimensional attractor in the border collision normal form. Nonlinearity 28, 3457–3464 (2015)
P. Glendinning, Bifurcation from stable fixed point to two-dimensional attractor in the border collision normal form, IMA J. Appl. Math. (2016). doi:10.1093/imamat/hxw001
P. Glendinning, Less is More II: An Optimistic View of Piecewise Smooth Bifurcation Theory, this volume (2017)
P. Glendinning, Shilnikov Chaos, Filippov Sliding and Boundary Equilibrium Bifurcations (in preparation)
J. Milnor, W. Thurston, On iterated maps of the interval, Dynamical Systems LNM, vol. 1342 (Springer, Berlin, 1988), pp. 465–563
A.B. Nordmark, Universal limit mapping in grazing bifurcations. Phys. Rev. E. 55, 266–270 (1997)
H.E. Nusse, J.A. Yorke, Border collision bifurcations including period two to period three bifurcation for piecewise smooth systems. Phys. D 57, 39–57 (1992)
M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143, 349–373 (2001)
Acknowledgements
I am grateful to Mike Jeffrey and Rachel Kuske for conversations that helped crystallize these ideas, and to the Simons Foundation for support at the CRM, Barcelona.
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Glendinning, P. (2017). Less Is More I: A Pessimistic View of Piecewise Smooth Bifurcation Theory. In: Colombo, A., Jeffrey, M., Lázaro, J., Olm, J. (eds) Extended Abstracts Spring 2016. Trends in Mathematics(), vol 8. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55642-0_13
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DOI: https://doi.org/10.1007/978-3-319-55642-0_13
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