Abstract
When studying a nonlinear dynamical system, one is foremost interested in finding all the attractors which represent the long-term stable motions. The attractors could be equilibrium states, periodic motions, quasi-periodic motions, or strange attractors. All the states which are not associated with attractors are transient states. For each of the transient states one wishes to know which attractor the system will evolve to if it starts from that transient state, and how long it takes to go from that transient state to its final attractor. In addition, one is interested in locating the domains of attraction for each attractor and the boundary sets between the domains of attraction. This is the information one wishes to have, preferably all at once, when one is interested in the global behavior of a dynamical system.
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References
Bondy, J. A. and Murty, U. S. R., Graphs Theory with Applications. North-Holland, New York 1985.
Chartrand, G. and Lesniak, L., Graphs and Digraphs, 2nd Edition. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA 1986.
Dushnik, B. and Miller, E. W., Partially ordered sets. American J. Math., 63, 600–610 (1941).
Haray, F., Graph Theory. Addison-Wesley Publishing Co., Reading, MA 1969.
Hirsch, M. W., The dynamical systems approach to differential equations. Bulletin (New Series) of the American Math. Soc., 11, 1–64 (1984).
Hsu, C. S., Cell-to-Cell Mapping, a Method of Global Analysis for Nonlinear Systems. Springer-Verlag, New York 1987.
Hsu, C. S., Global analysis by cell mapping. Int. J. Bifurcation & Chaos, 2, 727–771 (1992).
Rival, I., Editor, Graphs and Order. D. Reidel Publishing Co., Dordrecht, The Netherlands 1985.
Rival, I., Editor, Algorithms and Order. Kluwer Academic Publishers, Dordrecht, The Netherlands 1989.
Robinson, D. F. and Foulds, L. R., Digraphs: Theory and Techniques. Gordon and Breach Science Publishers, New York 1980.
Trotter, W. T. Combinatorics and Partially Ordered Sets—Dimension Theory. The John Hopkins University Press, Baltimore and London 1992.
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Dedicated to Paul M. Naghdi on the occasion of his 70th birthday
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© 1995 Birkhäuser Verlag Basel/Switzerland
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Hsu, C.S. (1995). Dynamical systems considered as ordering machines. In: Casey, J., Crochet, M.J. (eds) Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9229-2_40
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DOI: https://doi.org/10.1007/978-3-0348-9229-2_40
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