Abstract
In this paper we start with some historical remarks about nonlinear network theory and motivate the geometrical approach of nonlinear dynamical networks by means of a simple example. Then we outline the theoretical foundations of this theory and discuss some results. Finally, we refer to geometric approaches in physics and in other areas of engineerings and explain the common features and the differences between electrical network theory and classical mechanics.
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References
Arnol'd, V.I.: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York-Berlin 1988 (Original in Russian, 1979).
Mathis, W.: Theorie nichtlinearer Netzwerke. Springer-Verlag, Berlin 1987.
Sanders, J.A.; F. Verhulst: Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York-Berlin 1985
Andronov, A.A.; A.A. Vitt; S.E. Khaikin: Theory of Oscillators. Dover Publ., Inc., New York 1987 (first English translation is publ. 1966)
Mathis, W.: Der Netzwerktheoretiker Wilhelm Cauer-1900 bis 1945. Manuskript eines Vortrages of dem Physikertag der Deutschen Physikalischen Gesellschaft, Berlin 2.4.1987.
Moser, J.K.: Bistable Systems of Differential Equations with Applications to Tunnel Diode Circuits. IBM Journ. Res. Dev. 5(1961)226–240
Brayton, R.K.; J. Moser: A theory of nonlinear networks: I+II. Quart. Appl. Math. 22(1964)1–33, 81–104.
Brayton, R.K.: Nonlinear Reciprocal Networks. In: Mathem. Aspects of Electrical Network Analysis. American Math. Soc., Providence 1971.
Smale, S.: On the mathematical foundations of electrical circuit theory. Journ. Diff. Geometry 7(1972)193–210.
Belevitch, V.: Classical Network Theory. Holden-Day, San Francisco 1968
Belevitch, V.: Summary of the History of Circuit Theory. Proc. IRE 50(1962)848–855
Mathis, W.; W. Marten: Unified Theory of Nonlinear Electrical Networks. Proc. 29th Midwest Symposium on Circuits and Systems, August 10–12, 1986, Lincoln (Ne), Publ.: North-Holland, New York, 1987
Mathis, W.; W. Marten: On the Structure of Networks and Duality Theory. Proc. 31st Midwest Symposium on Circuits and Systems, August 9–12, 1988, St.Louis (Missouri).
Guillemin, V.; A. Pollack: Differential Topology. Prentice-Hall, Inc., Englewoods Cliffs (NJ), 1974
Mathis, W.: Differentialgeometrische Beschreibung nichtlinearer Netzwerke. Tagungsberichte der ITG-Diskussionssitzung ‘Neue Anwendungen theoretischer Konzepte in der Elektrotechnik', 20.–21. Februar 1990, Universitäts-Verlag Ulm, 1991.
Marten, W.; W.Mathis; L.O. Chua: Gradient Systems on Pseudo-Riemannian Manifolds as a Tool for Non-Linear Network Dynamics. Intern. Seminar on Non-linear Circuits and Systems, June 16–18, 1992, Moscow.
Crouch, P.E.: Geometric Structures in System Theory. Proc. IEE 128(1981) 242–252.
Oster, G.; A.Perelson; A.Katchalsky: Network Thermodynamics. Nature 234(1971) 393–399.
Meetz, K.; W.L.Engl: Elektromagnetische Felder. Springer-Verlag, Berlin 1980.
Wheeler, J.A.: Einsteins Vision. Springer-Verlag, Berlin 1968.
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Mathis, W. (1992). Geometric theory of nonlinear dynamical networks. In: Pichler, F., Díaz, R.M. (eds) Computer Aided Systems Theory — EUROCAST '91. EUROCAST 1991. Lecture Notes in Computer Science, vol 585. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021004
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DOI: https://doi.org/10.1007/BFb0021004
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