CHAPTER 18 - Theory of Initial Value Problems

Abstract

This chapter presents a brief overview of the theory of initial value problems (IVP). The existence and uniqueness for IVPs is based on a functional. The unknown variable in the differential equation is a function of the form x(t). The proof of the theorem is very much like the computation of inverse functions. The proof uses two maxima. In a sense, the maximum M below gives existence of solutions and the maximum L gives uniqueness. An important detail in the proof is a prior estimate of the time that a solution could last. When considering a microscopic view of a nonlinear equilibrium point, for every screen resolution θ and every bound β on the time of observation and observed scale of initial condition, there is a magnification large enough so that if |a| ≤ β and |b| ≤ β, then the error observed at that magnification is less than θ for 0 ≤ t ≤ β and, in particular, the solution lasts until time β. Also, if there is a lot of magnification, but not by an infinite amount, then it may be seen that a separation between the linear and nonlinear system after a very long time. The solutions of a dynamical system do not necessarily tend to an attracting point or to infinity. There are nonlinear oscillators that have stable oscillations in the sense that every solution (except zero) tends to the same oscillatory solution.