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The Forced van der Pol Equation II: Canards in the reduced system

Bold, Katherine and Edwards, Chantal and Guckenheimer, John and Guharay, Sabyasachi and Hoffman, Kathleen and Hubbard, Judith and Oliva, Ricardo and Weckesser, Warren (2003) The Forced van der Pol Equation II: Canards in the reduced system. SIAM Journal on Applied Dynamical Systems, 2 (4). pp. 570-608. ISSN 1536-0040. http://resolver.caltech.edu/CaltechAUTHORS:BOLsiamjad03

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Abstract

This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system.


Item Type:Article
Additional Information:© 2003 Society for Industrial and Applied Mathematics. Received by the editors December 3, 2002; accepted for publication (in revised form) by M. Golubitsky July 21, 2003; published electronically December 22, 2003. This research was partially supported by the National Science Foundation and the Department of Energy. The results were obtained during a Research Experiences for Undergraduates Program at Cornell University sponsored by the National Science Foundation.
Record Number:CaltechAUTHORS:BOLsiamjad03
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:BOLsiamjad03
Alternative URL:http://dx.doi.org/10.1137/S1111111102419130
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:557
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:23 Aug 2005
Last Modified:26 Dec 2012 08:40

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