[#P3110] Generic analytic Arnold diffusion in a priori stable systems

Problem. For a real-analytic steep integrable Hamiltonian \(h(I)\) with at least three degrees of freedom, is Arnold diffusion present for a generic sufficiently small real-analytic perturbation \(H_\varepsilon(I,\theta)=h(I)+\varepsilon f(I,\theta)\), in the sense that some orbit changes its action by an amount independent of \(\varepsilon\)?

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1Context

Known frontier: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Open boundary: A general analytic genericity theorem for a priori stable steep systems remains open.

2Problem setup

Definition 1 (a priori stable). The unperturbed system is integrable without a built-in hyperbolic subsystem.

Definition 2 (Arnold diffusion). An orbit has order-one drift in action variables under an arbitrarily small perturbation.

Remark 1. KAM tori occupy large measure and Nekhoroshev theory gives very long stability. In three or more degrees of freedom those tori do not separate the energy surface, leaving thin resonance channels where diffusion is expected.

3What counts as a solution

State a standard Baire or prevalence notion of genericity and prove order-one diffusion for the corresponding analytic perturbations.
Or disprove the generic claim by an open or residual obstruction class under the same topology.

1Status

What counts as a solution

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.^[1][2]

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Generic analytic Arnold diffusion in a priori stable systems. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. The exact unresolved remainder is: A general analytic genericity theorem for a priori stable steep systems remains open. A complete resolution must meet the following acceptance conditions: - State a standard Baire or prevalence notion of genericity and prove order-one diffusion for the corresponding analytic perturbations. - Or disprove the generic claim by an open or residual obstruction class under the same topology. Open full record →
Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.^[1][2] Relevance to this problem Locates the present research frontier immediately below Generic analytic Arnold diffusion in a priori stable systems. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. This leaves the following boundary unresolved: A general analytic genericity theorem for a priori stable steep systems remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.^[1][2] Relevance to this problem Documents why Generic analytic Arnold diffusion in a priori stable systems was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - generic analytic Arnold diffusion a priori stable open problem - Arnold diffusion analytic genericity stable Hamiltonian The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open. Related: Claim 1 Open full record →
A general analytic genericity theorem for a priori stable steep systems remains open.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 A general analytic genericity theorem for a priori stable steep systems remains open. Relevance to this problem Turns the remaining uncertainty in Generic analytic Arnold diffusion in a priori stable systems into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: A general analytic genericity theorem for a priori stable steep systems remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - State a standard Baire or prevalence notion of genericity and prove order-one diffusion for the corresponding analytic perturbations. - Or disprove the generic claim by an open or residual obstruction class under the same topology. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.

Equivalent-formulation queries: generic analytic Arnold diffusion a priori stable open problem; Arnold diffusion analytic genericity stable Hamiltonian
Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.
Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.

How the 4 records connect

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ProblemGeneric analytic Arnold diffusion in a priori stable systems

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2See also

Hilbert's sixteenth problem, second partdynamical systems
Birkhoff conjecture for integrable convex billiardsdynamical systems
Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems

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“Generic analytic Arnold diffusion in a priori stable systems.” TheoremDB. P3110. Problem statement; statement text SHA-256 24bb43c18d7896384870922f50037672b70cdc880429e4c47e8d9c236146aa81. https://theoremdb.org/statement/?ref=P3110

BibTeX

@misc{theoremdb-problem-24bb43c18d7896384870922f50037672b70cdc880429e4c47e8d9c236146aa81,
  title = {{Generic analytic Arnold diffusion in a priori stable systems}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 24bb43c18d7896384870922f50037672b70cdc880429e4c47e8d9c236146aa81},
  url = {https://theoremdb.org/statement/?ref=P3110}
}

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This problem includes 4 records joined by 3 typed links, sourced from arxiv.org^[1], current as of August 1, 2026.

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1References

Packet source. Patrick Bernard, “Arnold's Diffusion: from the a priori unstable to the a priori stable case”. arXiv:1203.2893 (2012). a priori stable discussion. ↗preprint · secondary source · arXiv:1203.2893, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Explains the transition to the hard stable case and outlines prospective mechanisms.Also cited at P. Bernard, Arnold's diffusion: from the a priori unstable to the a priori stable case. a priori stable discussion.Source used to assess the problem's recorded status.For Generic analytic Arnold diffusion in a priori stable systems: This is the dated publication status for the canonical target Generic analytic Arnold diffusion in a priori stable systems.Source named by the research packet.
Qinbo Chen and Rafael de la Llave, “Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems”. Nonlinearity, vol.35, no.4, 2022. DOI 10.1088/1361-6544/ac50bb. arXiv:2103.03847 (2021). main genericity theorem. ↗preprint · primary source · arXiv:2103.03847, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves analytic genericity in a broad a priori unstable class, isolating the stable extension.Source used to assess the problem's recorded status.For Generic analytic Arnold diffusion in a priori stable systems: Proves analytic genericity in a broad a priori unstable class, isolating the stable extension.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3110] Generic analytic Arnold diffusion in a priori stable systems

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