Nonsmooth Bifurcations in Families of One-Dimensional Piecewise-Linear Quasiperiodically Forced Maps
-
Rafael Martinez-Vergara [1] ; Joan Carles Tatjer [1]
-
[1] Universitat de Barcelona
Universitat de Barcelona
Barcelona, España
-
- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 1, 2026
- Idioma: inglés
- Enlaces
-
Resumen
We study nonsmooth bifurcations of four types of families of one-dimensional quasiperiodically forced maps of the form Fi(x,θ) = ( fi(x, θ ), θ + ω) for i = 1,..., 4, where x is real, θ ∈ T is an angle, ω is an irrational frequency, and fi(x,θ) is a real piecewise linear map with respect to x. The first two types of families fi have a symmetry with respect to x, and the other two could be viewed as quasiperiodically forced piecewise-linear versions of saddle-node and period-doubling bifurcations. The four types of families depend on two real parameters, a ∈ R and b ∈ R. Under certain assumptions for a, we prove the existence of a continuous map b∗(a) where for b = b∗(a) there exists a nonsmooth bifurcation for these types of systems. In particular we prove that for b = b∗(a) we have a strange nonchaotic attractor. It is worth to mention that the four families are piecewise-linear versions of smooth families which seem to have nonsmooth bifurcations. Moreover, as far as we know, we give the first example of a family with a nonsmooth period-doubling bifurcation.
-
Referencias bibliográficas
- 1. Alsedà, L., Costa, S.: On the definition of strange nonchaotic attractor. Fund. Math. 206, 23–39 (2009). https://doi.org/10.4064/fm206-0-2
- 2. Bjerklöv, K.: SNA’s in the quasi-periodic quadratic family. Comm. Math. Phys. 286(1), 137–161 (2009). https://doi.org/10.1007/s00220-008-0626-y
- 3. Cobza¸s, ¸S., Miculescu, R., Nicolae, A.: Lipschitz functions. Vol. 2241. Lecture Notes in Mathematics. Springer, Cham, pp. xiv+591...
- 4. Conway, J.B.: Functions of one complex variable. Second. Vol. 11. Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin (1978)....
- 5. Dueñas, J., Núñez, C., Obaya, R.: Bifurcation theory of attractors and minimal sets in d-concave nonautonomous scalar ordinary differential...
- 6. Dueñas, J., Núñez, C., Obaya, R.: Generalized pitchfork bifurcations in D-concave nonautonomous scalar ordinary differential equations....
- 7. Dueñas, J., Núñez, C., Obaya, R.: Saddle-node bifurcations for concave in measure and d-concave in measure skewproduct flows with applications...
- 8. Fuhrmann, G.: Non-smooth saddle-node bifurcations I: existence of an SNA. Ergodic Theory Dynam. Syst. 36(4), 1130–1155 (2016). https://doi.org/10.1017/etds.2014.92
- 9. Jäger, T.H.: The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. Mem. Amer. Math. Soc. 201(945), vi+106...
- 10. Jorba, A., Tatjer, J.C., Zhang, Y.: Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map. Int. J. Bifurc....
- 11. Jorba, À., Muñoz-Almaraz, F.J., Tatjer, J.C.: On non-smooth pitchfork bifurcations in invertible quasiperiodically forced 1-D maps. J....
- 12. Jorba, À., Tatjer, J.C.: A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps. Discrete Contin....
- 13. Jorba, À., et al.: Old and new results on strange nonchaotic attractors. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17(11), 3895–3928...
- 14. Katznelson, Y.: An introduction to harmonic analysis. Third. Cambridge Mathematical Library. Cambridge University Press, Cambridge, pp....
- 15. Keller, G.: A note on strange nonchaotic attractors. Fund. Math. 151(2), 139–148 (1996)
- 16. Núñez, C., Obaya, R.: A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete Contin. Dyn. Syst....
- 17. van Rooij, A.C.M., Schikhof, W.H.: A second course on real functions. Cambridge University Press, Cambridge-New York, pp. xiii+200...
