Abstract
A 14-dimensional generalized Lorenz system of ordinary differential equations is constructed and its bifurcation sequence is then studied numerically. Several fundamental differences are found which serve to distinguish this model from Lorenz's original one, the most unexpected of which is a family of invariant two-tori whose ultimate bifurcation leads to a strange attractor. The strange attractor seems to have many of the gross features observed in Lorenz's model and therefore is an excellent candidate for a higher dimensional analogue.
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Curry, J. H.: Bounded solutions of finite dimensional approximations to the Boussinesq equations. SIAM J. Math. Anal. (to appear)
Guckenheimer, J.: A strange attractor. In: Lecture notes in applied mathematical sciences, Vol. 19, pp. 368–391. Berlin-Heidelberg-New York: Springer 1976
Joseph, D. D.: On the stability of the Boussinesq equations. Arch. Rat. Mech. Anal.20, 59–71 (1965)
Kaplan, J., Yorke, J. A.: Preturbulence: A metastable regime in the system of Lorenz. Preprint (1976)
Lanford, O. E. III: Qualitative and statistical theory of dissipative systems. Preprint (1977)
Lanford, O. E. III: Bifurcation of periodic solutions into invariant tori; The work of Ruelle and Takens. Nonlinear problems in the physical sciences and biology. In: Lecture notes in mathematics, Vol. 322. Berlin-Heidelberg-New York: Springer 1973
Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmos. Sci.20, 130–141 (1963)
Marsden, J., McCracken, M.: The Hopf bifurcation and its application. In: Lecture notes in applied mathematical sciences, Vol. 19. Berlin-Heidelberg-New York: Springer 1976
McLaughlin, J.: Successive bifurcations leading to stochastic behavior. J. Stat. Phys.15, 307–326 (1976)
McLaughlin, J., Martin, P. C.: Phys. Rev. A12, 186 (1975)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. math. Phys.20, 167–192 (1971)
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747–817 (1967)
Williams, R. F.: Structure of Lorenz attractors. Preprint (1976)
Robbin, K. A.: A new approach to subcritical instabilities and turbulent transitions in a simple dynamo. Proc. Cambridge Phil. Soc. (1977)
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Communicated by J. Moser
On leave from Department of Mathematics, Howard University, Washington, DC, USA
The National Center for Atmospheric Research is sponsored by the National Science Foundation
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Curry, J.H. A generalized Lorenz system. Commun.Math. Phys. 60, 193–204 (1978). https://doi.org/10.1007/BF01612888
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DOI: https://doi.org/10.1007/BF01612888