question about ODEs and processes they describe

From: Alexander Korovyev (korovyev_at_rambler.ru)
Date: 06/13/04 

Date: 13 Jun 2004 06:53:42 -0700

>From the very beginning of my course on ordinary differential
equations I was told that this theory helps to describe and study
finite-dimensional, deterministic and differentiable processes. The
state of such a process is a point of a so-called phase space and at
any given time-point fully describes the process' present, past and
future. So the process' evolution (phase trajectory) is identified
with a smooth curve in a differentiable manifold (phase space). Since
every state must determine some phase trajectory (process' past,
present and future relative to the state's time-point), phase
trajectories clearly must not intersect. Ok, so far so good. But what
about ODE of this kind: dx/dt = x^(2/3)? How many parameters are
needed to characterize a state of a process described by this ODE? It
looks like the phase space is 2-dimensional but it is not! Suppose the
process starts at x = 0, is it possible to tell its evolution? What
information do we need for this? Was I told a lie that all ODEs
describe finite-dimensional, deterministic and differentiable
processes? I'd be happy to hear any clarification on this issue.
Thanks.

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