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In \\cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \\cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi"},"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":"1005.3186","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-18T13:08:48Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"a9c6e18cedbff74ec45a14393991c18b02ad081f5ebb818ace05b74caf80904b","abstract_canon_sha256":"3fbb9d652b52c5b93f1b116b2ac4307a6cc5394513b4a9c0a587a38670cb2bbf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:07:17.598186Z","signature_b64":"7ZaX+EHhtIW+jP0iRp967WyLrLEaKDH0uNgrTFjOsjefBIC6A10pJ2R1SNiAHkThZjjXnGNLk62789chsoXRBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5583823d3375bb683c64c080f94fd60220ed03a5e732e2a6e7f693e7fe770648","last_reissued_at":"2026-05-18T02:07:17.597656Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:07:17.597656Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generic Morse-Smale property for the parabolic equation on the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Genevi\\`eve Raugel (LM-Orsay), Romain Joly (IF)","submitted_at":"2010-05-18T13:08:48Z","abstract_excerpt":"In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. 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