Conley's fundamental theorem of dynamical systems explained

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

Notes and References

1. Book: Conley, Charles . Isolated invariant sets and the morse index: expository lectures . 1978 . American Mathematical Society . National Science Foundation . 978-0-8218-1688-2 . Regional conference series in mathematics . Providence, RI.
2. Norton . Douglas E. . 1995 . The fundamental theorem of dynamical systems . Commentationes Mathematicae Universitatis Carolinae . 36 . 3 . 585–597 . 0010-2628.
3. Razvan . M. R. . 2004 . On Conley's fundamental theorem of dynamical systems . International Journal of Mathematics and Mathematical Sciences . en . 2004 . 26 . 1397–1401 . math/0009184 . 10.1155/S0161171204202125 . 0161-1712 . free .
4. Hurley . Mike . 1991 . Chain recurrence and attraction in non-compact spaces . Ergodic Theory and Dynamical Systems . en . 11 . 4 . 709–729 . 10.1017/S014338570000643X . 0143-3857.
5. 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 . en . 20 . 2 . 784–825 . 10.1137/20M1336576 . 1536-0040. 2005.03217 .
6. Hafstein . Sigurdur . Giesl . Peter . 2015 . Review on computational methods for Lyapunov functions . Discrete and Continuous Dynamical Systems - Series B . en . 20 . 8 . 2291–2331 . 10.3934/dcdsb.2015.20.2291 . 1531-3492. free .