Pith Number

pith:TPPWEFTA

pith:2025:TPPWEFTAUQY7NMTG5YRYI2RUVV

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Feedback control of the Kuramoto model defined on uniform graphs I: Deterministic natural frequencies

Kazuyuki Yagasaki

In the controlled Kuramoto model on uniform graphs with uniform natural frequencies, exactly 2^n synchronized solutions exist for n nodes, and only one is stable while converging to the target rotation as feedback gain grows.

arxiv:2505.02196 v4 · 2025-05-04 · math.DS

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Claims

C1strongest claim

For the case of node number n≥3, we establish the existence of exactly 2^n synchronized solutions in the controlled Kuramoto model (CKM) and their saddle-node and pitchfork bifurcations, and determine their stability. In particular, we show that only a solution converging to the desired motion in the limit of infinite feedback gain is stable and the others are unstable.

C2weakest assumption

The natural frequencies are uniformly spaced and the underlying graphs are uniform (complete, random dense, or random sparse); this regularity is used to count the synchronized solutions and classify their bifurcations and stability.

C3one line summary

Proves exactly 2^n synchronized solutions exist in the controlled Kuramoto model for n≥3, with only one stable solution that converges to the target motion at high gain, and establishes its asymptotic stability in the continuous limit and for large finite n on random graphs.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-06-03T01:05:04.361324Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9bdf621660a431f6b266ee23846a34ad6e7ead672bf64eda2d772e8932d4b565

Aliases

arxiv: 2505.02196 · arxiv_version: 2505.02196v4 · doi: 10.48550/arxiv.2505.02196 · pith_short_12: TPPWEFTAUQY7 · pith_short_16: TPPWEFTAUQY7NMTG · pith_short_8: TPPWEFTA

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/TPPWEFTAUQY7NMTG5YRYI2RUVV \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 9bdf621660a431f6b266ee23846a34ad6e7ead672bf64eda2d772e8932d4b565

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "ebb7d1553972f9f4da3dbc9c5c17c65b3566baf4778d7b98e16b80edb54889eb",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2025-05-04T17:32:41Z",
    "title_canon_sha256": "933c97cff552111487391aca1e9e2b1d3a3998831101ad4c0bb4fd1b36dcf73b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2505.02196",
    "kind": "arxiv",
    "version": 4
  }
}