Editing Dynamical system (section)

== Nonlinear dynamical systems and chaos ==
{{Main|Chaos theory}}
Simple nonlinear dynamical systems, including [[Piecewise linear function|piecewise linear]] systems, can exhibit strongly unpredictable behavior, which might seem to be random, despite the fact that they are fundamentally deterministic. This unpredictable behavior has been called ''[[chaos theory|chaos]]''. [[Anosov diffeomorphism|Hyperbolic systems]] are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the [[tangent space]]s perpendicular to an orbit can be decomposed into a combination of two parts: one with the points that converge towards the orbit (the ''stable manifold'') and another of the points that diverge from the orbit (the ''unstable manifold'').

This branch of [[mathematics]] deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a [[steady state]] in the long term, and if so, what are the possible [[attractor]]s?" or "Does the long-term behavior of the system depend on its initial condition?"

The chaotic behavior of complex systems is not the issue. [[Meteorology]] has been known for years to involve complex—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost trivial systems. The [[Pomeau–Manneville scenario]] of the [[logistic map]] and the [[Fermi–Pasta–Ulam–Tsingou problem]] arose with just second-degree polynomials; the [[horseshoe map]] is piecewise linear.

=== 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>