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In mathematics, the Hilbert–Arnold problem is a problem in dynamical systems theory concerning the estimation of limit cycles. It asks whether in a generic finite-parameter family of smooth vector fields on a sphere with a compact parameter base, the number of limit cycles is uniformly bounded across all parameter values. The problem is historically related to Hilbert's sixteenth problem and was first formulated by Russian mathematicians Vladimir Arnold and Yulij Ilyashenko in the 1980s.

Background

The problem arises from considering modern approaches to Hilbert's sixteenth problem. While Hilbert's original question focused on polynomial vector fields, mathematical attention shifted toward properties of generic families within certain classes. Unlike polynomial systems, typical smooth systems on a sphere can have arbitrarily many hyperbolic limit cycles that persist under small perturbations. However, the question of uniform boundedness across parameter families remains meaningful and forms the basis of the Hilbert–Arnold problem.

Due to the compactness of both the parameter base and phase space, the Hilbert–Arnold problem can be reduced to a local problem studying bifurcations of special degenerate vector fields. This leads to the concept of polycycles—cyclically ordered sets of singular points connected by phase curve arcs—and their cyclicity, which measures the number of limit cycles born in bifurcations.

Local Hilbert-Arnold problem

The local version of the Hilbert-Arnold problem asks whether the maximum cyclicity of nontrivial polycycles in generic k-parameter families (known as the bifurcation number ${\displaystyle B(k)}$) is finite, and seeks explicit upper bounds. The local Hilbert–Arnold problem has been solved for ${\displaystyle k=1}$ and ${\displaystyle k=2}$, with ${\displaystyle B(1)=1}$ and ${\displaystyle B(2)=2}$. For ${\displaystyle k=3}$, a solution strategy exists but remains incomplete. A simplified version considering only elementary polycycles (where all vertices are elementary singular points with at least one nonzero eigenvalue) has been more thoroughly studied. Ilyashenko and Yakovenko proved in 1995 that the elementary bifurcation number ${\displaystyle E(k)}$ is finite for all ${\displaystyle k>0}$.

In 2003, mathematician Vadim Kaloshin established the explicit bound ${\displaystyle E(k)<25^{k^{2}}}$.

See also

• Hilbert's sixteenth problem
• Limit cycle
• Dynamical system
• Bifurcation theory

References

1. Ilyashenko, Yu. (1994). "Normal forms for local families and nonlocal bifurcations". Asterisque, Vol. 222, 233-258.
2. Ilyashenko, Yu.; Kaloshin, V. (1999). "Bifurcations of planar and spatial polycycles: Arnold's program and its development". Fields Inst. Commun., 24, 241-271.
3. Kaloshin, V. (2001). "The Hilbert-Arnold problem and estimates for the cyclicity of polycycles on the plane and in space". Functional Analysis and Its Applications, 35(2), 78-81.
4. Ilyashenko, Yu.; Yakovenko, S. (1991). "Finitely-smooth normal forms of local families of diffeomorphisms and vector fields". Russian Mathematical Surveys, 46(1), 3-39.
5. Kaloshin, V. (2003). "The Existential Hilbert 16th Problem and an Estimate for Cyclicity of Elementary Polycycles". Inventiones Mathematicae, 151, 451-512.

• Dynamical systems
• Unsolved problems in mathematics