Editing Dynamical system (section)

=== Solutions of finite duration ===

For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,<ref>{{cite book |author = Vardia T. Haimo |title = 1985 24th IEEE Conference on Decision and Control |chapter = Finite Time Differential Equations |year = 1985 |pages = 1729–1733 |doi = 10.1109/CDC.1985.268832 |s2cid = 45426376 |chapter-url=https://ieeexplore.ieee.org/document/4048613}}</ref> meaning here that in these solutions the system will reach the value zero at some time, called an ending time, and then stay there forever after. This can occur only when system trajectories are not uniquely determined forwards and backwards in time by the dynamics, thus solutions of finite duration imply a form of "backwards-in-time unpredictability" closely related to the forwards-in-time unpredictability of chaos. This behavior cannot happen for [[Lipschitz continuity|Lipschitz continuous]] differential equations according to the proof of the [[Picard–Lindelöf theorem|Picard-Lindelof theorem]]. These solutions are non-Lipschitz functions at their ending times and cannot be analytical functions on the whole real line.

As example, the equation:
:<math>y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1</math>
Admits the finite duration solution:
:<math>y(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2</math>
that is zero for <math>t \geq 2</math> and is not Lipschitz continuous at its ending time <math>t = 2.</math>