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.[1] Due to the concise yet complete description of many dynamical systems, Conley's theorem is also known as the fundamental theorem of dynamical systems.[2][3] Conley's fundamental theorem has been extended to systems with non-compact phase portraits[4] and also to hybrid dynamical systems.[5]

## 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:[6]

* 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.

## See also

* Conley index theory

## References

1. ↑ Conley, Charles (1978). Isolated invariant sets and the morse index: expository lectures. Regional conference series in mathematics. National Science Foundation. Providence, RI: American Mathematical Society. ISBN 978-0-8218-1688-2.
2. ↑ Norton, Douglas E. (1995). "The fundamental theorem of dynamical systems". Commentationes Mathematicae Universitatis Carolinae 36 (3): 585–597. ISSN 0010-2628. https://eudml.org/doc/247765.
3. ↑ Razvan, M. R. (2004). "On Conley's fundamental theorem of dynamical systems" (in en). International Journal of Mathematics and Mathematical Sciences 2004 (26): 1397–1401. doi:10.1155/S0161171204202125. ISSN 0161-1712.
4. ↑ Hurley, Mike (1991). "Chain recurrence and attraction in non-compact spaces" (in en). Ergodic Theory and Dynamical Systems 11 (4): 709–729. doi:10.1017/S014338570000643X. ISSN 0143-3857. https://www.cambridge.org/core/product/identifier/S014338570000643X/type/journal_article.
5. ↑ Kvalheim, Matthew D.; Gustafson, Paul; Koditschek, Daniel E. (2021). "Conley's Fundamental Theorem for a Class of Hybrid Systems" (in en). SIAM Journal on Applied Dynamical Systems 20 (2): 784–825. doi:10.1137/20M1336576. ISSN 1536-0040. https://epubs.siam.org/doi/10.1137/20M1336576.
6. ↑ Hafstein, Sigurdur; Giesl, Peter (2015). "Review on computational methods for Lyapunov functions" (in en). Discrete and Continuous Dynamical Systems - Series B 20 (8): 2291–2331. doi:10.3934/dcdsb.2015.20.2291. ISSN 1531-3492. http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11536.

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