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We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system $\\dot{x}=-y$, $\\dot{y}=x$, $\\dot{z}=0$, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree $n$, and second inside the class of all discontinuous piecewise polynomial differential systems of degree $n$ with two pi"},"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":"1501.01987","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-01-08T21:23:22Z","cross_cats_sorted":[],"title_canon_sha256":"286c5c516189c662bf2720566f8c82a09035e070d2fad1b84850593787ab4d0e","abstract_canon_sha256":"e5d358df95d946daf5e9221a92ae248c37664dfc71a3a3445b0668120e227770"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:17.055252Z","signature_b64":"37txQgWULDOhdlWFzNDqpfLtAdnSd2bTo1jVbpBuLMPoT2uXDJtCQ5dSE6gRJ1xIt7tfAACibqmt3ySnt8dWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d657b9ddbaa192366a3e7d837cfb643f24f4b9a510834a904a443a55c93bd307","last_reissued_at":"2026-05-18T02:29:17.054862Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:17.054862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+2)-dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"I.O. Zeli, J. Llibre, M.A. Teixeira","submitted_at":"2015-01-08T21:23:22Z","abstract_excerpt":"The orbits of the reversible differential system $\\dot{x}=-y$, $\\dot{y}=x$, $\\dot{z}=0$, with $x,y \\in R$ and $z\\in R^d$, are periodic with the exception of the equilibrium points $(0,0, z)$. 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