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.

References

^ 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.
^ Norton, Douglas E. (1995). "The fundamental theorem of dynamical systems". Commentationes Mathematicae Universitatis Carolinae. 36 (3): 585–597. ISSN 0010-2628.
^ Razvan, M. R. (2004). "On Conley's fundamental theorem of dynamical systems". International Journal of Mathematics and Mathematical Sciences. 2004 (26): 1397–1401. arXiv:math/0009184. doi:10.1155/S0161171204202125. ISSN 0161-1712.
^ Hurley, Mike (1991). "Chain recurrence and attraction in non-compact spaces". Ergodic Theory and Dynamical Systems. 11 (4): 709–729. doi:10.1017/S014338570000643X. ISSN 0143-3857.
^ Kvalheim, Matthew D.; Gustafson, Paul; Koditschek, Daniel E. (2021). "Conley's Fundamental Theorem for a Class of Hybrid Systems". SIAM Journal on Applied Dynamical Systems. 20 (2): 784–825. arXiv:2005.03217. doi:10.1137/20M1336576. ISSN 1536-0040.
^ Hafstein, Sigurdur; Giesl, Peter (2015). "Review on computational methods for Lyapunov functions". Discrete and Continuous Dynamical Systems - Series B. 20 (8): 2291–2331. doi:10.3934/dcdsb.2015.20.2291. ISSN 1531-3492.