Nonlinear System Theory by John L. Casti (Eds.)

By John L. Casti (Eds.)

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V. Kourganoff, “Basic Methods in Transfer Problems,” Oxford Univ. Press, Oxford, 1952. Predator-prey dynamics are extensively considered in numerous papers and volumes. A good starting point is N. Goel, S. Maitra, and E. Montroll, On the Volterra and other nonlinear models of interacting populations, Rev. Mod. Phys. 43 (1971), 231-276. and in Chapter 12 of M. Hirsch and S. Smale, “Differential Equations, Dynamical Systems and Linear Algebra,” Academic Press, New York, 1974. Mathematical models of international conflict, including missile warfare, are considered in the works J.

Saaty, “Modern Nonlinear Equations,” McGraw-Hill, New York, 1967. H. Davis, “Introduction to Nonlinear Differential and Integral Equations,” Dover, New York, 1962. Mathematical Tools of Nonlinear System Theory I. INTRODUCTION The theory of linear dynamical processes is, mathematically speaking, an advanced exercise in linear algebra and matrix theory. As shown in Chapter 3, the essential ingredients of linear theory involve various conditions concerning the ranks of certain matrices and the properties of certain polynomials.

Check that algebraic dim V ( 9 ) = 1. 4. Yl be the ideal generated by the polynomials f i ( s l r . , s n ) - yi, 0 i = 1,2,. , r, where yi = f i ( x0l , x 02 , .. , x,,). 5. Define 9 equal to the smallest ideal containing and the polynomial S j - g j ( s l ,.. s,,), where g i ( x l , . , x,,) is the ith entry in A X ( $ ‘ ] . 6. Use the equation for the first integrals, together with si - g i ( x , , . , ds/dr = ,ti[s] . This yields a single algebraic relation between iiand x i . To illustrate use of the procedure, we give some examples.

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