Qualitative theory of differential equations
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The problem of stability is of primary concern in the qualitative theory of differential equations and has occupied mathematicians for the past century. The problem appears when considering solutions to the differential equation x-f(t,x) where x=( x1(t),…,xn(t) )T and f(t,x) is a nonlinear function of x1,…,xn. While no known method of solving this equation explicitly exists even for the case n=2, it is possible to discuss the qualitative properties of x1(t) and x2(t) where x1(t) and x2(t) denote, for example, the populations, at time t, of two competing species. The qualitative attributes under consideration include points of equilibrium and the question of the stability of solutions in neighborhoods of these points.