no. 1
Devil's staircase route to chaos in a forced relaxation oscillator
Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 109-128

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]).

On démontre grâce à l’usage de la dynamique symbolique en dimension un que la transition vers le chaos dans un oscillateur non-linéaire de relaxation avec terme de contrainte périodique se produit à travers un “Devil Staircase” dans le diagramme de bifurcation.

@article{AIF_1994__44_1_109_0,
     author = {Alsed\`a, Lluis and Falc\'o, Antonio},
     title = {Devil's staircase route to chaos in a forced relaxation oscillator},
     journal = {Annales de l'Institut Fourier},
     pages = {109--128},
     year = {1994},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {44},
     number = {1},
     doi = {10.5802/aif.1391},
     zbl = {0793.34028},
     mrnumber = {95b:58098},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1391/}
}
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
SP  - 109
EP  - 128
VL  - 44
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1391/
DO  - 10.5802/aif.1391
LA  - en
ID  - AIF_1994__44_1_109_0
ER  - 
%0 Journal Article
%A Alsedà, Lluis
%A Falcó, Antonio
%T Devil's staircase route to chaos in a forced relaxation oscillator
%J Annales de l'Institut Fourier
%D 1994
%P 109-128
%V 44
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1391/
%R 10.5802/aif.1391
%G en
%F AIF_1994__44_1_109_0
Alsedà, Lluis; Falcó, Antonio. Devil's staircase route to chaos in a forced relaxation oscillator. Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 109-128. doi: 10.5802/aif.1391

[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]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). | Zbl | MR

[3]L. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series on Nonlinear Dynamics, World Scientific, Singapore, 1993. | Zbl | MR

[4]L. Alsedà, F. Mañosas, Kneading theory and rotation interval for a class of circle maps of degree one, Nonlinearity, 3 (1990). | Zbl | MR

[5]M.L. Cartwright, J.E. Littlewood, On nonlinear differential equations of second order II, Ann. of Math., 48 (1947). | Zbl

[6]A. Chenciner, J.M. Gambaudo, Ch. Tresser, Une remarque sur la structure des endomorphismes de degré 1 du cercle, C. R. Acad. Sci., Paris série I, 299 (1984). | Zbl | MR

[7]J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. | Zbl

[8]R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981). | Zbl | MR

[9]A. Katok, Some remarks on Birkhoff and Mather twist theorems Ergod., Theor. Dynam. Sys., 2 (1982). | Zbl

[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]M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math., Soc., 244 (1981). | Zbl | MR

[12]N. Levinson, A second order differential equation with singular solutions, Ann. of Math., 50 (1949). | Zbl | MR

[13]J. Moser, Stable and random motions in dynamical systems, Princeton University Press (1973).

[14]M. Misiurewicz, Rotation intervals for a class of maps of the real line into itself, Ergod. Theor. Dynam. Sys., 6 (1986). | Zbl | MR

[15]S.E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, IHES, 1977. | Zbl | Numdam

[16]B. Van Der Pol, On relaxation oscillations, Phil. Mag., 2 (1926). | JFM

[17]B. Van Der Pol, J. Van Der Mark, Frequency demultiplication, Nature, 120 (1927).

Cité par Sources :