Editing Dynamical system (section)

==History==

Many people regard French mathematician [[Henri Poincaré]] as the founder of dynamical systems.<ref>Holmes, Philip. "Poincaré, celestial mechanics, dynamical-systems theory and "chaos"." ''Physics Reports'' 193.3 (1990): 137–163.</ref> Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the [[Poincaré recurrence theorem]], which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.

[[Aleksandr Lyapunov]] developed many important approximation methods. His methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system.

In 1913, [[George David Birkhoff]] proved Poincaré's "[[Poincaré–Birkhoff theorem|Last Geometric Theorem]]", a special case of the [[three-body problem]], a result that made him world-famous. In 1927, he published his ''[https://archive.org/details/dynamicalsystems00birk/ Dynamical Systems]''. Birkhoff's most durable result has been his 1931 discovery of what is now called the [[ergodic theorem]]. Combining insights from [[physics]] on the [[ergodic hypothesis]] with [[measure theory]], this theorem solved, at least in principle, a fundamental problem of [[statistical mechanics]]. The ergodic theorem has also had repercussions for dynamics.

[[Stephen Smale]] made significant advances as well.  His first contribution was the [[Horseshoe map|Smale horseshoe]] that jumpstarted significant research in dynamical systems.  He also outlined a research program carried out by many others.

[[Oleksandr Mykolaiovych Sharkovsky]] developed [[Sharkovsky's theorem]] on the periods of [[discrete dynamical system]]s in 1964. One of the implications of the theorem is that if a discrete dynamical system on the [[real line]] has a [[periodic point]] of period&nbsp;3, then it must have periodic points of every other period.

In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer [[Ali H. Nayfeh]] applied [[nonlinear dynamics]] in [[mechanics|mechanical]] and [[engineering]] systems.<ref name="Rega">{{cite book |last1=Rega |first1=Giuseppe |chapter=Tribute to Ali H. Nayfeh (1933–2017) |title=IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems |date=2019 |publisher=[[Springer Science+Business Media|Springer]] |isbn=9783030236922 |url=https://books.google.com/books?id=pAilDwAAQBAJ&pg=PA1 |pages=1–2}}</ref> His pioneering work in applied nonlinear dynamics has been influential in the construction and maintenance of [[machines]] and [[structures]] that are common in daily life, such as [[ships]], [[crane (machine)|cranes]], [[bridges]], [[buildings]], [[skyscrapers]], [[jet engines]], [[rocket engines]], [[aircraft]] and [[spacecraft]].<ref name="fi">{{cite web |title=Ali Hasan Nayfeh |url=https://www.fi.edu/laureates/ali-hasan-nayfeh |website=[[Franklin Institute Awards]] |publisher=[[The Franklin Institute]] |access-date=25 August 2019 |date=4 February 2014}}</ref>