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Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following $T$-periodic smooth differential system $$ x'=F_0(t,x)+\\sum_{i=1}^k \\vare"},"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":"1611.04807","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-11-15T12:36:35Z","cross_cats_sorted":[],"title_canon_sha256":"12d2b746569ffa2550bdc99c89b40e365c1d91266dfed11b83c03678ce72a536","abstract_canon_sha256":"82d48e591d4828d685e6180ea3bf042b6ae0cdf31b7055a6544c8455f0e9b6ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:09.206618Z","signature_b64":"aJhkAWUESND41vLrwMRnfS8UVMNt9WUsCbzfAvbnyJT4/5HbT7wy409FhFhD0c6DG4IC/YViz4zRhrWSSV/JDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e1145563f8778e9e8723d752f05f6a5d3069e2b406db08c5962c252f4656f55","last_reissued_at":"2026-05-18T00:38:09.206098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:09.206098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Douglas D. 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