Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study. (English) Zbl 0499.70034


70K50 Bifurcations and instability for nonlinear problems in mechanics
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54C05 Continuous maps
54C25 Embedding
58C25 Differentiable maps on manifolds
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[1] Arnol’d, V.I.: Loss of stability of self-oscillations close to resonance and versal deformations of equivarient vector fields. Funct. Anal. Appl.11, 1-10 (1977) · Zbl 0411.58013
[2] Aronson, D.G., Chory, M.A., Hall, G.R., McGehee, R.P.: A discrete dynamical system with subtly wild behavior. New approaches to nonlinear problems in dynamics, Holmes, P. (ed.). SIAM 1980
[3] Bowen, R.: On axiomA diffeomorphisms. CBMS Regional Conference Series in Mathematics, No. 35. Providence, Rhode Island: Am. Math. Soc. 1978 · Zbl 0383.58010
[4] Conley, C.: Isolated invariant sets and the Morse index. CBMS Regional Conference Series in Mathematics, No. 38. Providence, Rhode Island: Am. Math. Soc. 1978 · Zbl 0397.34056
[5] Conley, C.: Hyperbolic sets and shift automorphisms. Dynamical systems: theory and applications. In: Lecture Notes in Physics, Vol. 38, Moser, J. (ed.), pp. 539-549. Berlin, Heidelberg, New York: Springer 1975
[6] Curry, J., Yorke, J.: A transition from Hopf bifurcation to chaos: computer experiments on maps inR 2. The structure of attractors in dynamical systems. In: Lecture Notes in Mathematics, Vol. 668, pp. 48-68. Berlin, Heidelberg, New York: Springer 1978
[7] Fenichel, N.: Persistence and smoothness of invariant manifolds for flows, Ind. Univ. Math. J.21, 193-226 (1971) · Zbl 0246.58015
[8] Flaherty, J., Hoppensteadt, F.: Frequency entrainment of a forced van der Pol oscillator. Stud. Appl. Math.58, 5-15 (1978) · Zbl 0384.34023
[9] Guckenheimer, J.: A strange, strange attractor. The Hopf bifurcation theorem and its applications, Marsden, J., McCracken, M. (eds.), pp. 368-381. Berlin, Heidelberg, New York: Springer 1976
[10] Guckenheimer, J.: On the bifurcation of maps of the interval. Invent. Math.39, 165-178 (1977) · Zbl 0354.58013
[11] Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Publ. Math. I.H.E.S.50, 307-320 (1979) · Zbl 0436.58018
[12] Hartman, P.: Ordinary differential equations. New York: Wiley 1964 · Zbl 0125.32102
[13] Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys.50, 67-77 (1976) · Zbl 0576.58018
[14] Iooss, G.: Topics in bifurcation of maps and applications. Math. Stud. 36. Amsterdam: North-Holland 1979 · Zbl 0408.58019
[15] Levi, M.: Qualitative analysis of the periodically forced relaxation oscillators. Mem. AMS244, 1981 · Zbl 0448.34032
[16] Levinson, N.: A second order differential equation with singular solutions. Ann. Math.50, 127-153 (1949) · Zbl 0045.36501
[17] Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci.20, 130-141 (1963) · Zbl 1417.37129
[18] Maynard Smith, J.: Mathematical ideas in biology. Cambridge: Cambridge University Press 1971
[19] Milnor, J., Thurston, W.: On iterated maps of the interval, I, II (preprint) · Zbl 0664.58015
[20] Newhouse, S.: Diffeomorphisms with infinitely many sinks. Topology13, 9-18 (1974) · Zbl 0275.58016
[21] Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Vol. 3. Paris: Gauthiers-Villars 1899
[22] Pounder, J.R., Rogers, T.D.: The geometry of chaos: dynamics of a nonlinear second-order difference equation. Bull. Math. Biol.42, 551-597 (1980) · Zbl 0439.39001
[23] Rademacher, H.: Lectures on elementary number theory. New York: Blaisdell 1964 · Zbl 0119.27803
[24] Ruelle, D.: A measure associated with axiomA attractors. Am. J. Math.98, 19-64 (1976) · Zbl 0355.58010
[25] Ruelle, D.: Strange attractors. The Mathematical Intelligencer2, 126-137 (1980) · Zbl 0487.58014
[26] Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys.20, 167-192 (1971) · Zbl 0223.76041
[27] Smale, S.: Diffeomorphisms with many periodic points. Differential and combinatorial topology, pp. 63-80. Princeton, N.J.: Princeton University Press 1965 · Zbl 0142.41103
[28] Stein, P.R., Ulam, S.M.: Nonlinear transformation studies on electronic computers. Rozprawy Matem.39, 3-65 (1964) · Zbl 0143.18801
[29] Takens, F.: Forced oscillations and bifurcations. Applications of global analysis. Commun. Math. Inst. Rikjsuniversitat Utrecht · Zbl 1156.37315
[30] Williams, R.F.: Expanding attractors. Publ. Math. I.H.E.S.43, 169-203 (1974); see also Proceedings of the Mount Aigual Conference on Differential Topology, Univ. of Montepellier, 1969 · Zbl 0279.58013
[31] Zharkovsky, A.N.: Coexistence of cycles of a continuous map of a line into itself. Ukrain. Mat. Z.16, 61-71 (1974)
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