Uncertainty transformation via Hopf bifurcation in fast-slow systems
Abstract
Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (i) to certain classes of symmetric copies, (ii) to an almost deterministic output, (iii) to a mixture distribution with differing moments and (iv) to a very restricted class of general distributions. We prove under which conditions the cases (i)-(iv) occur in certain classes vector fields.
- Publication:
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Proceedings of the Royal Society of London Series A
- Pub Date:
- April 2017
- DOI:
- 10.1098/rspa.2016.0346
- arXiv:
- arXiv:1512.03002
- Bibcode:
- 2017RSPSA.47360346K
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Probability;
- Nonlinear Sciences - Pattern Formation and Solitons
- E-Print:
- 19 pages, 5 figure, preprint version