Abstract
The mathematical description of any physical process begins with a repeatable experiment. A mathematical theory is then sought which duplicates the experimental results. Usually physical laws or principles are applied to the experimental system to obtain the form and structure of the mathematical model. In many applications, the parameters in the mathematical model are not precisely known, and must also be obtained from the experiment.
This research was supported by the Alexander von Humboldt Senior Scientist Program and carried out at the Institute for Information Sciences, University of Tübingen.
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References
FritzHugh, R.: Mathematical models of excitation and propagation in nerve. In: Biological Engineering, ed. by H.P. Schwan, pp. 1–84. McGraw-Hill, New York, 1969.
Thorn, R.: Stabilité Structural et Morphogenésé, Benjamin, Reading, Mass., 1972.
Stakgold, I.: Branching of solutions of nonlinear equations, S.I.A.M. Rev. 13, 289–332 (1971) .
Keller, J. and Antman, S.: Bifurcation Theory and Nonlinear Eigenvalue Problems, Benjamin, Reading, Mass., 1969.
Zeeman, E.: Differential equations for the heart beat and nerve impulse. In: Towards a Theoretical Biology, vol. 4, ed. by C.H. Waddington, pp. 8–67. Edinburgh University Press, Edinburgh, 1973.
Sattinger, D.: Topics in Stability and Bifurcation, Lecture Notes in Mathematics No. 309, Springer Verlag, New York, 1973.
Frank-Kamenetzky, D.: Diffusion and Heart Exchange in Chemical Kinetics, Princeton University Press, Princeton, N.J., 1955.
Lemberg, H.L. and Rice, S.A.: A re-examination of the theory of phase transitions in crystalline heavy methane, Physica 63, 48–64 (1973) .
Rubinstein, H.: The Stefan problem, Translations of Mathematical Monographs 27 (1971).
Gurel,O.: Bifurcations in nerve membrane dynamics, Intern. J. Neuro-science 5, 281–286 (1973).
Landahl, H.D. and Licko, V.: On coupling between oscillators which model biological systems, Int. J. of Chronobiology 1 ,245–252 (1973).
Cronin, J.: Biomathematical model of aneurysm of the circle of Willis: a qualitative analysis of the differential equation of Austin, Mathematical Biosciences 16, 209–225 (1973).
Cronin, J.: The Danziger-Elmergreen theory of periodic catatonic schizophrenia, Bull. Math. Biophysics 35., 689–707 (1973).
Pimpley, G.: On Predator-Prey Equations Simulating an Immune Response, Lecture Notes in Mathematics No. 322, Springer Verlag, New York, 1973.
Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems, J. Functional Anal. 7, 487–513 (1971).
Mahler, H. and Cordes, E.: Biological Chemistry, Harper and Row, New York, 1966.
George, J. and Damle, P.S.: On the numerical solution of free boundary problems, preprint.
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George, J. (1974). Mathematical Models and Bifurcation Theory in Biology. In: Conrad, M., Güttinger, W., Dal Cin, M. (eds) Physics and Mathematics of the Nervous System. Lecture Notes in Biomathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80885-2_31
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DOI: https://doi.org/10.1007/978-3-642-80885-2_31
Publisher Name: Springer, Berlin, Heidelberg
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