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.
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]).
@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}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {44}, number = {1}, year = {1994}, doi = {10.5802/aif.1391}, mrnumber = {95b:58098}, zbl = {0793.34028}, language = {en}, url = {http://www.numdam.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 - http://www.numdam.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 http://www.numdam.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. http://www.numdam.org/articles/10.5802/aif.1391/
[1]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]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). | MR | Zbl
, , ,[3]Combinatorial dynamics and entropy in dimension one, Advanced Series on Nonlinear Dynamics, World Scientific, Singapore, 1993. | MR | Zbl
, , ,[4]Kneading theory and rotation interval for a class of circle maps of degree one, Nonlinearity, 3 (1990). | MR | Zbl
, ,[5]On nonlinear differential equations of second order II, Ann. of Math., 48 (1947). | Zbl
, ,[6]Une remarque sur la structure des endomorphismes de degré 1 du cercle, C. R. Acad. Sci., Paris série I, 299 (1984). | MR | Zbl
, , ,[7]Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. | Zbl
, ,[8]Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981). | MR | Zbl
,[9]Some remarks on Birkhoff and Mather twist theorems Ergod., Theor. Dynam. Sys., 2 (1982). | Zbl
,[10]The Devil's Staircase : The Electrical Engineer's Fractal, IEEE Trans. on Circuits and Systems., 36 (1989), 1133-1139.
, , ,[11]Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math., Soc., 244 (1981). | MR | Zbl
,[12]A second order differential equation with singular solutions, Ann. of Math., 50 (1949). | MR | Zbl
,[13]Stable and random motions in dynamical systems, Princeton University Press (1973).
,[14]Rotation intervals for a class of maps of the real line into itself, Ergod. Theor. Dynam. Sys., 6 (1986). | MR | Zbl
,[15]The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, IHES, 1977. | Numdam | Zbl
,[16]On relaxation oscillations, Phil. Mag., 2 (1926). | JFM
,[17]Frequency demultiplication, Nature, 120 (1927).
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