{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:SJ2LHQT2UB7R4VF5GUI3WHHN6L","short_pith_number":"pith:SJ2LHQT2","canonical_record":{"source":{"id":"2507.01623","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-02T11:52:39Z","cross_cats_sorted":[],"title_canon_sha256":"8f7862cca7125ba675b432681044441d4c8882261d5432c4706a1dfbc6bd4d80","abstract_canon_sha256":"465c5d18ece8acaf486c9d3a2b0b2e4fa0f93c8ed12b2a593b8c3d7237110f46"},"schema_version":"1.0"},"canonical_sha256":"9274b3c27aa07f1e54bd3511bb1cedf2f866c2046db78f2908a79ef4d6034095","source":{"kind":"arxiv","id":"2507.01623","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.01623","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"2507.01623v4","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.01623","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"SJ2LHQT2UB7R","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_16","alias_value":"SJ2LHQT2UB7R4VF5","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_8","alias_value":"SJ2LHQT2","created_at":"2026-06-02T02:04:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:SJ2LHQT2UB7R4VF5GUI3WHHN6L","target":"record","payload":{"canonical_record":{"source":{"id":"2507.01623","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-02T11:52:39Z","cross_cats_sorted":[],"title_canon_sha256":"8f7862cca7125ba675b432681044441d4c8882261d5432c4706a1dfbc6bd4d80","abstract_canon_sha256":"465c5d18ece8acaf486c9d3a2b0b2e4fa0f93c8ed12b2a593b8c3d7237110f46"},"schema_version":"1.0"},"canonical_sha256":"9274b3c27aa07f1e54bd3511bb1cedf2f866c2046db78f2908a79ef4d6034095","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:06.886381Z","signature_b64":"zy0Kt/VDn61LqPvPmeiMbXRBnKxDPyBFN/Z67H66mH7YgKJXo3HlxyJ2pesRfImab/pIQdlreb3vzoZGXQbPBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9274b3c27aa07f1e54bd3511bb1cedf2f866c2046db78f2908a79ef4d6034095","last_reissued_at":"2026-06-02T02:04:06.885869Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:06.885869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.01623","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T02:04:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wJlrSUfFRXiFe7O1dsl8HRucfWpyoRgH2tF3+hoTa9lbgci9h/gRSz3KWZMo+CGnOsSta0XkeQfUcBydx9I1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T06:41:09.411988Z"},"content_sha256":"7a7139a0b6aa21d8094fb98762a1e65cf6c7f99309e68c1d3de4bc6c4873c2d2","schema_version":"1.0","event_id":"sha256:7a7139a0b6aa21d8094fb98762a1e65cf6c7f99309e68c1d3de4bc6c4873c2d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:SJ2LHQT2UB7R4VF5GUI3WHHN6L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"FitzHugh-Nagumo equation: bifurcations, slow-fast system and dynamics near infinity","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alexandre A. P. Rodrigues, Nasrin Sadri","submitted_at":"2025-07-02T11:52:39Z","abstract_excerpt":"In this paper, we present a qualitative and bifurcation analysis of the three-parameter FitzHugh-Nagumo system and its compactified formulation. The study is structured according to three parameter-dependent regimes, for which the associated phase portraits are characterized.\n  In one of these regimes, the system exhibits a double-zero bifurcation with $Z_2$ symmetry, corresponding to a codimension-two degeneracy. We compute explicit bifurcation and transition curves arising in the unfolding of this singularity, including pitchfork, Hopf, Belyakov, and double homoclinic bifurcations, and we co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01623","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.01623/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T02:04:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JUUadhg6SPHrOk+/UXtY5hy3SkH1eI+bCrJcDdQjViUUrSVGlZgSFgwHmBECF4dOSIugTRlNxaTW8Oieq565DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T06:41:09.412357Z"},"content_sha256":"44912a45c9762d7a47ed07db0a85ae742c1e8761d3f9993580b7a096e058a958","schema_version":"1.0","event_id":"sha256:44912a45c9762d7a47ed07db0a85ae742c1e8761d3f9993580b7a096e058a958"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L/bundle.json","state_url":"https://pith.science/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T06:41:09Z","links":{"resolver":"https://pith.science/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L","bundle":"https://pith.science/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L/bundle.json","state":"https://pith.science/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SJ2LHQT2UB7R4VF5GUI3WHHN6L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:SJ2LHQT2UB7R4VF5GUI3WHHN6L","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"465c5d18ece8acaf486c9d3a2b0b2e4fa0f93c8ed12b2a593b8c3d7237110f46","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-02T11:52:39Z","title_canon_sha256":"8f7862cca7125ba675b432681044441d4c8882261d5432c4706a1dfbc6bd4d80"},"schema_version":"1.0","source":{"id":"2507.01623","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.01623","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"arxiv_version","alias_value":"2507.01623v4","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.01623","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_12","alias_value":"SJ2LHQT2UB7R","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_16","alias_value":"SJ2LHQT2UB7R4VF5","created_at":"2026-06-02T02:04:06Z"},{"alias_kind":"pith_short_8","alias_value":"SJ2LHQT2","created_at":"2026-06-02T02:04:06Z"}],"graph_snapshots":[{"event_id":"sha256:44912a45c9762d7a47ed07db0a85ae742c1e8761d3f9993580b7a096e058a958","target":"graph","created_at":"2026-06-02T02:04:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.01623/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we present a qualitative and bifurcation analysis of the three-parameter FitzHugh-Nagumo system and its compactified formulation. The study is structured according to three parameter-dependent regimes, for which the associated phase portraits are characterized.\n  In one of these regimes, the system exhibits a double-zero bifurcation with $Z_2$ symmetry, corresponding to a codimension-two degeneracy. We compute explicit bifurcation and transition curves arising in the unfolding of this singularity, including pitchfork, Hopf, Belyakov, and double homoclinic bifurcations, and we co","authors_text":"Alexandre A. P. Rodrigues, Nasrin Sadri","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-02T11:52:39Z","title":"FitzHugh-Nagumo equation: bifurcations, slow-fast system and dynamics near infinity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.01623","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7a7139a0b6aa21d8094fb98762a1e65cf6c7f99309e68c1d3de4bc6c4873c2d2","target":"record","created_at":"2026-06-02T02:04:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"465c5d18ece8acaf486c9d3a2b0b2e4fa0f93c8ed12b2a593b8c3d7237110f46","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DS","submitted_at":"2025-07-02T11:52:39Z","title_canon_sha256":"8f7862cca7125ba675b432681044441d4c8882261d5432c4706a1dfbc6bd4d80"},"schema_version":"1.0","source":{"id":"2507.01623","kind":"arxiv","version":4}},"canonical_sha256":"9274b3c27aa07f1e54bd3511bb1cedf2f866c2046db78f2908a79ef4d6034095","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9274b3c27aa07f1e54bd3511bb1cedf2f866c2046db78f2908a79ef4d6034095","first_computed_at":"2026-06-02T02:04:06.885869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:06.885869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zy0Kt/VDn61LqPvPmeiMbXRBnKxDPyBFN/Z67H66mH7YgKJXo3HlxyJ2pesRfImab/pIQdlreb3vzoZGXQbPBA==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:06.886381Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.01623","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7a7139a0b6aa21d8094fb98762a1e65cf6c7f99309e68c1d3de4bc6c4873c2d2","sha256:44912a45c9762d7a47ed07db0a85ae742c1e8761d3f9993580b7a096e058a958"],"state_sha256":"4d1aea1623484048ac424e0575033e01a42d3f15db8a810d89e3606fd09421e2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8H5fxpjkISdo+enduLBzDEeWbMzu1XqBUWlVvZoL26/ra3pBoNtqF49PTejilz5IYSxpHiLR/elN6r4VkbD0AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T06:41:09.414265Z","bundle_sha256":"b9d2a8c8ee97ae8a8d65f072038e327a52f34202aefe8c2ffd4a660fb8a4f961"}}