Abstract
In this and succeeding papers we consider some foundational elements of a theory of Hopf bifurcation from non-periodic solutions of ordinary differential equations.
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References
Coddington, E., and Levinson, N. (1955).Theory of Ordinary Differential Equations, McGraw-Hill, New York.
Coppel, W. (1978).Dichotomies in Stability Theory (Lecture Notes in Mathematics, vol. 629), Springer-Verlag, Berlin.
De Concini, C., and Johnson, R. (1987). The algebraic-geometric AKNS potentials.Ergodic Theory and Dyn. Sys. 7, 1–24.
Ellis, R. (1967).Lectures on Topological Dynamics, Benjamin, New York.
Fal'ko, N. (1978). Almost-periodic systems with quasiperiodic coefficients.Differential Equations 14, 326–331.
Giachetti, R., and Johnson, R. (1986). The Floquet exponent for two-dimensional linear systems with bounded coefficients.J. Math. Pures Appl. 65, 93–117.
Golubitsky, M., and Schaefer, D. (1985).Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York.
Hopf, E. (1942). Abzweigung einer periodischen Lösung von einer stationÄren Lösung eines Differentialsystems.SÄchs. Akad. Wiss. Leipzig, Math-Phys. Klasse 94.
Hopf, E. (1955). Repeated branching through loss of stability: an example. InProceedings of the Conference on Differential Equations, Maryland.
Johnson, R. (1986). Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients.J. Differential Equations 61, 54–78.
Johnson, R. (1987). Remarks on linear differential equations with measurable coefficients.Proc. Am. Math. Soc. 100, 491–504.
Johnson, R., and Moser, J. (1982). The rotation number for almost-periodic potentials.Commun. Math. Phys. 84, 403–438.
Johnson, R., and Yi, Y-F. (1989). Hopf bifurcation from nonperiodic solutions of differential equations. II.J. Dyn. Differential Equations 11 (in preparation).
Johnson, R., Palmer, K., and Sell, G. (1987). Ergodic properties of linear dynamical systems.SIAM J. Math. Anal. 18, 1–33.
Katok, A. (1972). Minimal diffeomorphisms on a smooth S1 principal bundle. InAll-Union Conference in Topology, Tblisi [in Russian].
Kotani, S. (1982). Lyapounov indices determine absolutely continuous spectrum of stationary random Schrödinger operators. InStochastic Analysis, ed. by K. Ito, North-Holland, 1984, pp. 225–247.
Millionscikov, V. (1978). Typicality of almost-periodic systems with almost-periodic coefficients.Differential Equations 14, 448–449.
Nemytskii, V., and Stepanov, V. (1960).Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NJ.
Newhouse, H., Ruelle, D., and Takens, F. (1978). Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, m⩾3.Commun. Math. Phys. 64, 35–40.
Novikov, V. (1974). On almost reducible systems with almost periodic coefficients.Math. Notes 16, 1065–1071.
Oseledec, V. (1968). A multiplicative ergodic theorem: Lyapounov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc. 19, 197–231.
Palmer, K. (1987). Exponential dichotomies for almost periodic equations. University of Miami, Miami, Florida, preprint.
Ruelle, D., and Takens, F. (1971). On the nature of turbulence.Commun. Math. Phys. 20, 167–192.
Sacker, R. (1964). On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, PhD thesis, Courant Institute, New York University, New York.
Sacker, R., and Sell, G. (1974). Existence of dichotomies and invariant splittings for linear differential equations. I.J. Differential Equations 15, 429–458.
Sacker, R., and Sell, G. (1976). Existence of dichotomies and invariant splittings for linear differential system. III.J. Differential Equations 22, 497–522.
Sacker, R., and Sell, G. (1978). A spectral theory for linear differential systems.J. Differential Equations 27, 320–358.
Sacker, R., and Sell, G. (1980). The spectrum of an invariant submanifold.J. Differential Equations 38, 135–160.
Schwarzmann, S. (1957). Asymptotic cycles.Ann. Math. 66, 270–284.
Sell, G. (1979). Bifurcation of higher-dimensional tori.Arch. Rat. Mech. Anal. 69, 199–230.
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Johnson, R.A. Hopf bifurcation from nonperiodic solutions of differential equations. I. Linear theory. J Dyn Diff Equat 1, 179–198 (1989). https://doi.org/10.1007/BF01047830
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DOI: https://doi.org/10.1007/BF01047830