Unformatted text preview: we would have
. This contradicts the theorem on
escape from compact sets. So cannot be finite. Similarly, cannot be finite. Therefore the
solution defines a flow by lemma 4.2. □
Remark. If a physical system were exactly described by an initial value problem of form , its
vector field would presumably be bounded. ■
Example. The overdamped pendulum,
Theorem 4.4’ (Reparameterization of Time). If
, , defines a flow by theorem 4.3’. ■
then is equivalent to
Chapter 4A upon reparameterization of time. The vector field of
bounded. defines a flow since it is Proof. The solution of  has a maximum interval of existence
using Since and . Define is increasing and the transformation is one-to-one. By the chain rule . Therefore
. Clearly is bounded. It remains to show that the partial derivatives
. Let are . View , as a function of and We need to show , fixing the other components of
. Then . We are given . Note . and
except at values where
Observe that . Then .
Then . It is sufficient to show that all of except possibly at those values. 5
Chapter 4A Further, if , From the last expression in *3’+, Thus is continuous at remains bounded as . Therefore .□ and Example. The undamped pendulum
defines a complete flow on upon reparameterization of time. ■ Example. The reparameterization of time  also converts some vector fields on domains with
boundaries into flows defined for all times. The Lotka-Volterra system for the competitive
interaction of two species is
The domain is
. The right hand of this system is in the interior of
. Although the domain has boundaries, trajectories do not escape the domain by approaching
the boundaries. Sketch the domain and the flow on the boundaries. However, for certain
initial conditions they may escape to in finite time. (Suppose and are positive,
. The trajectory will escape to in finite negative time.) Upon reparameterization of
time by  this system defines a flow for all time. ■ There are also cases where
where is a proper open subset of . The following
theorem and its proof are found in Perko (3rd edition, theorem 2 in section 3.2).
Theorem (Global Existence on Open Domains with Boundaries). Suppose
. Then there is a function
such that is an open subset
is equivalent to
Chapter 4A upon reparameterization of time. The vector field
Idea of Proof.
and let where defines a complete flow. is given by . Define the closed set .
is a distance from to the boundary of : .
The effect of is to slow trajectories that approach any boundary of . The maximum interval
of existence for the reparameterized system is
Sketch bo ndary and tra e tory approaching boundary Example. The initial value problem
flow. ■ and , which we analyzed in chapter 3. is . Upon reparameterization of t...
View Full Document
- Fall '14
- Stability theory, Lyapunov