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In each case the number and amplitude of the limit cycles emerging from the period annulus are calculated following the same strategy: we reduce of all of them to locally equivalent perturbed integrable systems of the form: $dH(x,y)+\\epsilon(f(x,y)dy-g(x,y)dx)=0$, with $H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov function, $M(h)=\\int_{H=h}fdy-gdx$, associated to each particular problem. We obtain the"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"nlin/0210024","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"nlin.PS","submitted_at":"2002-10-14T07:55:41Z","cross_cats_sorted":[],"title_canon_sha256":"2a1d568b32905152cc48d503432d1514bafab2c159ec844054a7a6dbf3555137","abstract_canon_sha256":"5bfc35d1da4efeaecb3862c91b284794dd8aead1a4ac42386f4003448ccbebc1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:19.664607Z","signature_b64":"fF2SiKD8qe40T68omI7/rft2imIdP082ILpJUlnPoARkab4s/dutDlmtTWOzpgndPOOrqDTVUxhq6aKFk9QoCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"789720b7dea9d022c24ce6602a655f2c7d0eec71dacd0f48910abf18762e17a2","last_reissued_at":"2026-05-18T01:38:19.663961Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:19.663961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Number and Amplitude of Limit Cycles emerging from {\\it Topologically Equivalent} Perturbed Centers","license":"","headline":"","cross_cats":[],"primary_cat":"nlin.PS","authors_text":"Jose-Luis Lopez, Ricardo Lopez-Ruiz","submitted_at":"2002-10-14T07:55:41Z","abstract_excerpt":"We consider three examples of weekly perturbed centers which do not have {\\it geometrical equivalence}: a linear center, a degenerate center and a non-hamiltonian center. In each case the number and amplitude of the limit cycles emerging from the period annulus are calculated following the same strategy: we reduce of all of them to locally equivalent perturbed integrable systems of the form: $dH(x,y)+\\epsilon(f(x,y)dy-g(x,y)dx)=0$, with $H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov function, $M(h)=\\int_{H=h}fdy-gdx$, associated to each particular problem. 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