Topological degree in analysis of canard-type trajectories in 3-D systems

Abstract

Piecewise linear systems become increasingly important across a wide range of engineering applications spurring an interest in developing new mathematical models and methods of their analysis, or adapting methods of the theory of smooth dynamical systems. One such areas is the design of controllers which support the regimes of operation described by canard trajectories of the model, including applications to engineering chemical processes, flight control, electrical circuits design, and neural networks. In this article, we present a scenario which ensures the existence of a topologically stable periodic (cyclic) canard trajectory in slow-fast systems with a piecewise linear fast component. In order to reveal the geometrical structure responsible for the existence of the canard trajectory, we focus on a simple prototype piecewise linear nonlinearity. The analysis is based on application of the topological degree.

Keywords:

• nonsmooth slow-fast system
• piecewise linear function
• periodic canard solution
• topological degree

AMS Subject Classifications::

• 34E17
• 34C25

Acknowledgements

This research was supported by Science Foundation Ireland grants 05/RFP/ENG062, 06/RFP/MAT048 and by a private bequest to Cork University Foundation. A. Pokrovskii and D. Rachinskii were partially supported by Federal Programme ‘Scientists of Innovative Russia’, grant 2009-1.5-507-007, and Russian Foundation for Basic Research, grant 10-01-93112. V.A. Sobolev was partially supported by Russian Foundation for Basic Research, grant 10-08-00154a, by Programme 22 of Presidium of RAS and Programme 16 of Branch of Physical and Technical Problems of Energetics of Russian Academy of Sciences.