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PreprintCC BY 4.0arXiv

On global Lyapunov functions being Morse

  • Wouter Jongeneel
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Abstract

Lyapunov functions have been a cornerstone of dynamical systems theory ever since the early 1900s. Moreover and akin to Morse theory, Lyapunov functions have been the key in linking topology to dynamical systems theory. In this note we continue along these lines. Suppose that some vector field on Euclidean space has a unique equilibrium point that is globally asymptotically stable (GAS). Under these conditions, it is known that there is always a smooth Lyapunov function to certify stability. One may wonder if this function can always be chosen to be Morse. We show that if the equilibrium point is hyperbolic, this is indeed true. Moreover, we show that if this would be true in general, then the generalized 4-dimensional smooth Poincaré conjecture (SPC4) must be true. Even so, we provide an example that is GAS, but does not admit a Morse Lyapunov function, closing this route of proving SPC4.

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Cite

BibTeXRIS
BibTeX
@misc{jongeneel2026global,
  title = {{On global Lyapunov functions being Morse}},
  author = {Wouter Jongeneel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05626},
  eprint = {2610.05626},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05626},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On global Lyapunov functions being Morse
AU  - Wouter Jongeneel
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05626
UR  - https://arxiv.org/abs/2610.05626
AB  - Lyapunov functions have been a cornerstone of dynamical systems theory ever since the early 1900s. Moreover and akin to Morse theory, Lyapunov functions have been the key in linking topology to dynamical systems theory. In this note we continue along these lines. Suppose that some vector field on Euclidean space has a unique equilibrium point that is globally asymptotically stable (GAS). Under these conditions, it is known that there is always a smooth Lyapunov function to certify stability. One may wonder if this function can always be chosen to be Morse. We show that if the equilibrium point is hyperbolic, this is indeed true. Moreover, we show that if this would be true in general, then the generalized 4-dimensional smooth Poincaré conjecture (SPC4) must be true. Even so, we provide an example that is GAS, but does not admit a Morse Lyapunov function, closing this route of proving SPC4.
ER  -