Conley's fundamental theorem of dynamical systems
Conley's fundamental theorem of dynamical systems or Conley's decomposition theorem states that every flow of a dynamical system with compact phase portrait admits a decomposition into a chain-recurrent part and a gradient-like flow part.{{Cite book |last=Conley |first=Charles |title=Isolated invariant sets and the morse index: expository lectures |date=1978 |publisher=American Mathematical Society |others=National Science Foundation |isbn=978-0-8218-1688-2 |series=Regional conference series in mathematics |location=Providence, RI}} Due to the concise yet complete description of many dynamical systems, Conley's theorem is also known as the fundamental theorem of dynamical systems.{{Cite journal |last=Norton |first=Douglas E. |date=1995 |title=The fundamental theorem of dynamical systems |url=https://eudml.org/doc/247765 |journal=Commentationes Mathematicae Universitatis Carolinae |volume=36 |issue=3 |pages=585–597 |issn=0010-2628}}{{Cite journal |last=Razvan |first=M. R. |date=2004 |title=On Conley's fundamental theorem of dynamical systems |journal=International Journal of Mathematics and Mathematical Sciences |language=en |volume=2004 |issue=26 |pages=1397–1401 |arxiv=math/0009184 |doi=10.1155/S0161171204202125 |issn=0161-1712 |doi-access=free }} Conley's fundamental theorem has been extended to systems with non-compact phase portraits{{Cite journal |last=Hurley |first=Mike |date=1991 |title=Chain recurrence and attraction in non-compact spaces |url=https://www.cambridge.org/core/product/identifier/S014338570000643X/type/journal_article |journal=Ergodic Theory and Dynamical Systems |language=en |volume=11 |issue=4 |pages=709–729 |doi=10.1017/S014338570000643X |issn=0143-3857|url-access=subscription }} and also to hybrid dynamical systems.{{Cite journal |last1=Kvalheim |first1=Matthew D. |last2=Gustafson |first2=Paul |last3=Koditschek |first3=Daniel E. |date=2021 |title=Conley's Fundamental Theorem for a Class of Hybrid Systems |url=https://epubs.siam.org/doi/10.1137/20M1336576 |journal=SIAM Journal on Applied Dynamical Systems |language=en |volume=20 |issue=2 |pages=784–825 |doi=10.1137/20M1336576 |issn=1536-0040|arxiv=2005.03217 }}
Complete Lyapunov functions
Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional Lyapunov functions that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.
In the particular case of an autonomous differential equation defined on a compact set X, a complete Lyapunov function V from X to R is a real-valued function on X satisfying:{{Cite journal |last1=Hafstein |first1=Sigurdur |last2=Giesl |first2=Peter |date=2015 |title=Review on computational methods for Lyapunov functions |url=http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11536 |journal=Discrete and Continuous Dynamical Systems - Series B |language=en |volume=20 |issue=8 |pages=2291–2331 |doi=10.3934/dcdsb.2015.20.2291 |issn=1531-3492|doi-access=free }}
- V is non-increasing along all solutions of the differential equation, and
- V is constant on the isolated invariant sets.
Conley's theorem states that a continuous complete Lyapunov function exists for any differential equation on a compact metric space. Similar result hold for discrete-time dynamical systems.