{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:KCTX7UQJQSFKU2FJOKL5PGXMSI","short_pith_number":"pith:KCTX7UQJ","schema_version":"1.0","canonical_sha256":"50a77fd209848aaa68a97297d79aec9217969be18f987fae8774e437b68f810e","source":{"kind":"arxiv","id":"1901.06463","version":1},"attestation_state":"computed","paper":{"title":"Equilibrium Index of Invariant Sets and Global Static Bifurcation for Nonlinear Evolution Equations","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Desheng Li, Zhi-qiang Wang","submitted_at":"2019-01-19T03:56:47Z","abstract_excerpt":"We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_t+A u=f_\\lambda(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\\mathfrak{m}$ may be even. In particular, in case $\\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also gi"},"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":"1901.06463","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-19T03:56:47Z","cross_cats_sorted":[],"title_canon_sha256":"a4a4ea0e733aa45dfe4261249783e50fce4150282e3439ac8d229a6b5318e153","abstract_canon_sha256":"cfed538659e532848ce1380e3244c0d67d39bcc29691ed9c145e88c331583c49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:50.225162Z","signature_b64":"eTtXFhws0pNd6l+IMzVSDuEZSjKaJom7GdyVSajT9tUJn/isbLA+23SJT6xlZvh9upj2ViRbtSjK1vCyrzJQAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50a77fd209848aaa68a97297d79aec9217969be18f987fae8774e437b68f810e","last_reissued_at":"2026-05-17T23:55:50.224596Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:50.224596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equilibrium Index of Invariant Sets and Global Static Bifurcation for Nonlinear Evolution Equations","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Desheng Li, Zhi-qiang Wang","submitted_at":"2019-01-19T03:56:47Z","abstract_excerpt":"We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_t+A u=f_\\lambda(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\\mathfrak{m}$ may be even. In particular, in case $\\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also gi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.06463","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.06463","created_at":"2026-05-17T23:55:50.224688+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.06463v1","created_at":"2026-05-17T23:55:50.224688+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.06463","created_at":"2026-05-17T23:55:50.224688+00:00"},{"alias_kind":"pith_short_12","alias_value":"KCTX7UQJQSFK","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"KCTX7UQJQSFKU2FJ","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"KCTX7UQJ","created_at":"2026-05-18T12:33:21.387695+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI","json":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI.json","graph_json":"https://pith.science/api/pith-number/KCTX7UQJQSFKU2FJOKL5PGXMSI/graph.json","events_json":"https://pith.science/api/pith-number/KCTX7UQJQSFKU2FJOKL5PGXMSI/events.json","paper":"https://pith.science/paper/KCTX7UQJ"},"agent_actions":{"view_html":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI","download_json":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI.json","view_paper":"https://pith.science/paper/KCTX7UQJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.06463&json=true","fetch_graph":"https://pith.science/api/pith-number/KCTX7UQJQSFKU2FJOKL5PGXMSI/graph.json","fetch_events":"https://pith.science/api/pith-number/KCTX7UQJQSFKU2FJOKL5PGXMSI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI/action/storage_attestation","attest_author":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI/action/author_attestation","sign_citation":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI/action/citation_signature","submit_replication":"https://pith.science/pith/KCTX7UQJQSFKU2FJOKL5PGXMSI/action/replication_record"}},"created_at":"2026-05-17T23:55:50.224688+00:00","updated_at":"2026-05-17T23:55:50.224688+00:00"}