{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:QIEGEHIVDRQHGWQCJJZOC3O3KV","short_pith_number":"pith:QIEGEHIV","canonical_record":{"source":{"id":"1811.05354","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-11-02T02:53:26Z","cross_cats_sorted":["math.PR","nlin.CD"],"title_canon_sha256":"ab0d91d743c8cdbd4755337a47f45180b651e0e753ae986c72d5a92f59e6fd62","abstract_canon_sha256":"85fc1e3b30d18df904f2f5b9b34b49ccd528f237dafb58bba3a0e5859da65b74"},"schema_version":"1.0"},"canonical_sha256":"8208621d151c60735a024a72e16ddb55512491ccdbc88b10c719da28ccb70c34","source":{"kind":"arxiv","id":"1811.05354","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05354","created_at":"2026-05-18T00:00:46Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05354v1","created_at":"2026-05-18T00:00:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05354","created_at":"2026-05-18T00:00:46Z"},{"alias_kind":"pith_short_12","alias_value":"QIEGEHIVDRQH","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QIEGEHIVDRQHGWQC","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QIEGEHIV","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:QIEGEHIVDRQHGWQCJJZOC3O3KV","target":"record","payload":{"canonical_record":{"source":{"id":"1811.05354","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-11-02T02:53:26Z","cross_cats_sorted":["math.PR","nlin.CD"],"title_canon_sha256":"ab0d91d743c8cdbd4755337a47f45180b651e0e753ae986c72d5a92f59e6fd62","abstract_canon_sha256":"85fc1e3b30d18df904f2f5b9b34b49ccd528f237dafb58bba3a0e5859da65b74"},"schema_version":"1.0"},"canonical_sha256":"8208621d151c60735a024a72e16ddb55512491ccdbc88b10c719da28ccb70c34","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:46.313173Z","signature_b64":"3TRxCBbdqeTKJc25UoC+pc/4hf+gzcIsVVgttwDbP3FeULg6gxFcOJ7U6meXejtcWe7i1DiKZ9MzMe0C9EqvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8208621d151c60735a024a72e16ddb55512491ccdbc88b10c719da28ccb70c34","last_reissued_at":"2026-05-18T00:00:46.312814Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:46.312814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.05354","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:00:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zpNYs6MezcIPrOI8qDEOpT/o/e3tdDeVB562klMmXwBceYHD/VPZK8Vz8gxTpqEM63DkCTM/e3k4/poRH2l6BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T11:24:11.967507Z"},"content_sha256":"3bc31e0cad757a6fc489eba53e7a51de97c7cbc1736035f3fda2c54875e15e1c","schema_version":"1.0","event_id":"sha256:3bc31e0cad757a6fc489eba53e7a51de97c7cbc1736035f3fda2c54875e15e1c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:QIEGEHIVDRQHGWQCJJZOC3O3KV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bifurcation in mean phase portraits for stochastic dynamical systems with multiplicative Gaussian noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","nlin.CD"],"primary_cat":"math.DS","authors_text":"Athanasios Tsiairis, Hui Wang, Jinqiao Duan","submitted_at":"2018-11-02T02:53:26Z","abstract_excerpt":"We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker-Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e., saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05354","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:00:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3XuyALs2/TpdlTZwtoJAAQ6su6weOjkhGpa1SxYJ+qznVqnDTB6VpmDtK4yc5iMsME9Sq1V4cVwzVlJLAXztCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T11:24:11.967860Z"},"content_sha256":"f232c0561e8a799e9d1e475b3af83eb944472d48c6520f61751a4c8db8066a98","schema_version":"1.0","event_id":"sha256:f232c0561e8a799e9d1e475b3af83eb944472d48c6520f61751a4c8db8066a98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV/bundle.json","state_url":"https://pith.science/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV/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":"2026-07-04T11:24:11Z","links":{"resolver":"https://pith.science/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV","bundle":"https://pith.science/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV/bundle.json","state":"https://pith.science/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QIEGEHIVDRQHGWQCJJZOC3O3KV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QIEGEHIVDRQHGWQCJJZOC3O3KV","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":"85fc1e3b30d18df904f2f5b9b34b49ccd528f237dafb58bba3a0e5859da65b74","cross_cats_sorted":["math.PR","nlin.CD"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-11-02T02:53:26Z","title_canon_sha256":"ab0d91d743c8cdbd4755337a47f45180b651e0e753ae986c72d5a92f59e6fd62"},"schema_version":"1.0","source":{"id":"1811.05354","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05354","created_at":"2026-05-18T00:00:46Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05354v1","created_at":"2026-05-18T00:00:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05354","created_at":"2026-05-18T00:00:46Z"},{"alias_kind":"pith_short_12","alias_value":"QIEGEHIVDRQH","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QIEGEHIVDRQHGWQC","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QIEGEHIV","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:f232c0561e8a799e9d1e475b3af83eb944472d48c6520f61751a4c8db8066a98","target":"graph","created_at":"2026-05-18T00:00:46Z","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"},"paper":{"abstract_excerpt":"We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker-Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e., saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, ","authors_text":"Athanasios Tsiairis, Hui Wang, Jinqiao Duan","cross_cats":["math.PR","nlin.CD"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-11-02T02:53:26Z","title":"Bifurcation in mean phase portraits for stochastic dynamical systems with multiplicative Gaussian noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05354","kind":"arxiv","version":1},"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":"sha256:3bc31e0cad757a6fc489eba53e7a51de97c7cbc1736035f3fda2c54875e15e1c","target":"record","created_at":"2026-05-18T00:00:46Z","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":"85fc1e3b30d18df904f2f5b9b34b49ccd528f237dafb58bba3a0e5859da65b74","cross_cats_sorted":["math.PR","nlin.CD"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-11-02T02:53:26Z","title_canon_sha256":"ab0d91d743c8cdbd4755337a47f45180b651e0e753ae986c72d5a92f59e6fd62"},"schema_version":"1.0","source":{"id":"1811.05354","kind":"arxiv","version":1}},"canonical_sha256":"8208621d151c60735a024a72e16ddb55512491ccdbc88b10c719da28ccb70c34","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8208621d151c60735a024a72e16ddb55512491ccdbc88b10c719da28ccb70c34","first_computed_at":"2026-05-18T00:00:46.312814Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:46.312814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3TRxCBbdqeTKJc25UoC+pc/4hf+gzcIsVVgttwDbP3FeULg6gxFcOJ7U6meXejtcWe7i1DiKZ9MzMe0C9EqvCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:46.313173Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.05354","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bc31e0cad757a6fc489eba53e7a51de97c7cbc1736035f3fda2c54875e15e1c","sha256:f232c0561e8a799e9d1e475b3af83eb944472d48c6520f61751a4c8db8066a98"],"state_sha256":"60c3c95f49ac91f7111c03f03483e833919129e1a91776efec522f1f9d2dff80"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FyykzlziOBsohZLtY2SpdynKNHaUXIl0G1YIqUS7F0f93buysUC7ey4YBHiTMMPhcA2mHlpwB3cofK4RHMhGCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T11:24:11.969811Z","bundle_sha256":"26a9e8dcf751da08c12a7edf1493acccb96b600d0db22bd73e5155d8d7cd9134"}}