Editing Autonomous system (mathematics) (section)

=== Finite durations ===

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 from its own dynamics, the system will reach the value zero at an ending time and stay there in zero forever after. These finite-duration solutions cannot be [[analytical function]]s on the whole real line, and because they will be non-[[Lipschitz function]]s at the ending time, they don't stand{{clarification needed|date=October 2024}} uniqueness of solutions of Lipschitz differential equations.

As example, the equation:
:<math>y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1</math>
Admits the finite duration solution:
:<math>y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2</math>