Devil's staircase route to chaos in a forced relaxation oscillator

Lluis Alsedà; Antonio Falcó

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 109-128
  • ISSN: 0373-0956

Abstract

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We use one-dimensional techniques to characterize the Devil’s staircase route to chaos in a relaxation oscillator of the van der Pol type with periodic forcing term. In particular, by using symbolic dynamics, we give the behaviour for certain range of parameter values of a Cantor set of solutions having a certain rotation set associated to a rational number. Finally, we explain the phenomena observed experimentally in the system by Kennedy, Krieg and Chua (in [10]) related with the appearance of secondary staircases intercalated into the primary staircases which were found by van der Pol and van der Mark (in [17]).

How to cite

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Alsedà, Lluis, and Falcó, Antonio. "Devil's staircase route to chaos in a forced relaxation oscillator." Annales de l'institut Fourier 44.1 (1994): 109-128. <http://eudml.org/doc/75051>.

@article{Alsedà1994,
abstract = {We use one-dimensional techniques to characterize the Devil’s staircase route to chaos in a relaxation oscillator of the van der Pol type with periodic forcing term. In particular, by using symbolic dynamics, we give the behaviour for certain range of parameter values of a Cantor set of solutions having a certain rotation set associated to a rational number. Finally, we explain the phenomena observed experimentally in the system by Kennedy, Krieg and Chua (in [10]) related with the appearance of secondary staircases intercalated into the primary staircases which were found by van der Pol and van der Mark (in [17]).},
author = {Alsedà, Lluis, Falcó, Antonio},
journal = {Annales de l'institut Fourier},
keywords = {Devil's staircase route to chaos; relaxation oscillator of the Van der Pol type; symbolic dynamics},
language = {eng},
number = {1},
pages = {109-128},
publisher = {Association des Annales de l'Institut Fourier},
title = {Devil's staircase route to chaos in a forced relaxation oscillator},
url = {http://eudml.org/doc/75051},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Alsedà, Lluis
AU - Falcó, Antonio
TI - Devil's staircase route to chaos in a forced relaxation oscillator
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 109
EP - 128
AB - We use one-dimensional techniques to characterize the Devil’s staircase route to chaos in a relaxation oscillator of the van der Pol type with periodic forcing term. In particular, by using symbolic dynamics, we give the behaviour for certain range of parameter values of a Cantor set of solutions having a certain rotation set associated to a rational number. Finally, we explain the phenomena observed experimentally in the system by Kennedy, Krieg and Chua (in [10]) related with the appearance of secondary staircases intercalated into the primary staircases which were found by van der Pol and van der Mark (in [17]).
LA - eng
KW - Devil's staircase route to chaos; relaxation oscillator of the Van der Pol type; symbolic dynamics
UR - http://eudml.org/doc/75051
ER -

References

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  1. [1]L. ALSEDÀ, A. FALCÓ, The bifurcations of a piecewise monotone family of circle maps related to the Van der Pol equation, Procedings of European Conference on Iteration Theory, Caldes de Malavella, World Scientific, (1987). 
  2. [2]L. ALSEDÀ, J. LLIBRE, R. SERRA, Bifurcations for a circle map associated with the Van der Pol equation, Sur la théorie de l'itération et ses applications, Colloque internationaux du CNRS Toulouse, 332 (1982). Zbl0523.34043MR87i:58126
  3. [3]L. ALSEDÀ, J. LLIBRE, M. MISIUREWICZ, Combinatorial dynamics and entropy in dimension one, Advanced Series on Nonlinear Dynamics, World Scientific, Singapore, 1993. Zbl0843.58034MR95j:58042
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  7. [7]J. GUCKENHEIMER, P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. Zbl0515.34001
  8. [8]R. ITO, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981). Zbl0484.58027MR82i:58061
  9. [9]A. KATOK, Some remarks on Birkhoff and Mather twist theorems Ergod., Theor. Dynam. Sys., 2 (1982). Zbl0521.58048
  10. [10]M.P. KENNEDY, K.R. KRIEG, L.O. CHUA, The Devil's Staircase : The Electrical Engineer's Fractal, IEEE Trans. on Circuits and Systems., 36 (1989), 1133-1139. 
  11. [11]M. LEVI, Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math., Soc., 244 (1981). Zbl0448.34032MR82g:58052
  12. [12]N. LEVINSON, A second order differential equation with singular solutions, Ann. of Math., 50 (1949). Zbl0045.36501MR10,710b
  13. [13]J. MOSER, Stable and random motions in dynamical systems, Princeton University Press (1973). 
  14. [14]M. MISIUREWICZ, Rotation intervals for a class of maps of the real line into itself, Ergod. Theor. Dynam. Sys., 6 (1986). Zbl0615.54030MR87k:58131
  15. [15]S.E. NEWHOUSE, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, IHES, 1977. Zbl0445.58022
  16. [16]B. VAN DER POL, On relaxation oscillations, Phil. Mag., 2 (1926). JFM52.0450.05
  17. [17]B. VAN DER POL, J. VAN DER MARK, Frequency demultiplication, Nature, 120 (1927). 

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