Abstract
We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations.
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Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. 133, 73–169 (1991)
Cartwright, M.L., Littlewood, J.E.: On nonlinear differential equations of the second order. J. Lond. Math. Soc. 20, 180–189 (1945)
Guckenheimer, J., Holmes, P.: Nonlinear oscillators, dynamical systems and bifurcations of vector fields. Appl. Math. Sciences 42, Berlin-Heidelberg-New York: Springer-Verlag, 1983
Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Lecture Notes in Math., 583 Berlin-Heidelberg-New York: Springer Verlag, 1977
Hopf, E.: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Verh. Sächs, Acad. Wiss. Leipzig Math. Phys. 94, 1–22 (1942)
Jakobson, M.: Absolutely continue invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981)
Katok, A.: Lyapunov exponents, entropy and periodic points for diffeomorphisms. Publ. IHES 51, 137–173 (1980)
Levi, M.: Qualitative analysis of periodically forced relaxation oscillations. Mem. AMS 214, 1–147 (1981)
Levinson, N.: A second order differential equation with singular solutions. Ann. Math. 50(1), 127–153 (1949)
Lyubich, M.: Almost every real quadratic map is either regular or stochastic. Ann. Math., (2001)
Marsden, J., McCracken, M.: The Hopf bifurcation and its applications. Appl. Math. Sci. 19 Berlin-Heildelberg-New York: Springer-Verlag, 1976
Mora, L., Viana, M.: Abundance of strange attractors. Acta. Math. 171, 1–71 (1993)
Newhouse, S.: ``Lectures in Dynamical Systems''. In: CIME Lectures, Bressanone, Italy, June 1978, Bascal-Britian: Birkhäuser, 1980, pp. 1–114
Newhouse, S., Ruelle, D., Takens, F.: Occurrence of strange Axiom A attractors near quasi-periodic flows on T m, n ≥ 3. Commun. Math. Phys. 64, 35–40 (1978)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)
Smale, S.: Differentiable dynamical systems. Bull. AMS 73, 717–817 (1967)
Wang, Q., Young, L.-S.: Strange attractors with one direction of instability. Commun. Math. Phys. 218, 1–97 (2001)
Wang, Q., Young, L.-S.: From invariant curves to strange attractors. Commun. Math. Phys. 225, 275–304 (2002)
Wang Q., Young, L.-S.: Strange attractors with one direction of instability in n-dimensional spaces. 2002 preprint
Young, L.-S.: What are SRB measures, and which dynamical systems have them? To appear In a volume in honor of D. Ruelle and Ya. Sinai on their 65th birthdays, J Stat. Phys. (2002)
Zaslavsky, G.: The simplest case of a strange attractor. Phys. Lett. 69A(3), 145–147 (1978)
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Communicated by G. Gallavotti
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Wang, Q., Young, LS. Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations. Commun. Math. Phys. 240, 509–529 (2003). https://doi.org/10.1007/s00220-003-0902-9
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DOI: https://doi.org/10.1007/s00220-003-0902-9