Transcription of Nonlinear OrdinaryDifferentialEquations
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Nonlinear OrdinaryDifferentialEquations
Nonlinear Ordinary Differential Equationsby Peter J. OlverUniversity of Minnesota1. notes are concerned with initial value problems for systems of ordinary dif-ferential equations. Here our emphasis will be on nonlinearphenomena and properties,particularly those with physical relevance. Finding a solution to a differential equationmay not be so important if that solution never appears in the physical model representedby the system, or is only realized in exceptional circumstances. Thus, equilibrium solu-tions, which correspond to configurations in which the physical system does not move,only occur in everyday situations if they are stable.
insufficient resources to survive, and so its growth rate will be negative. In other words, the growth rate ρ(N) > 0 when N < N⋆, while ρ(N) < 0 when N > N⋆, where the carrying capacity N⋆ > 0 depends upon the resource availability. The simplest class of functions that satifies these two inequalities are of the form ρ(N) = µ(N⋆ − ...
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