{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6PPDECWFDYSDVYUYGXGOQAZRP4","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":"c6737bc66c0b401acf4305b442ff0b7a54601694ae57640c4127aa36bcf5679d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-06T03:08:15Z","title_canon_sha256":"869a27fe88d5272de0affe083dd8ef96bafc57aa04bdc685e14100edbb5cd899"},"schema_version":"1.0","source":{"id":"1510.01416","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.01416","created_at":"2026-05-18T01:30:56Z"},{"alias_kind":"arxiv_version","alias_value":"1510.01416v1","created_at":"2026-05-18T01:30:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.01416","created_at":"2026-05-18T01:30:56Z"},{"alias_kind":"pith_short_12","alias_value":"6PPDECWFDYSD","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6PPDECWFDYSDVYUY","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6PPDECWF","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:a054a1fcc1adb1dc2b2e102bc1ac05a69ee48c200f4287124345eb4346b4a926","target":"graph","created_at":"2026-05-18T01:30:56Z","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":"The mode-locking regions of a dynamical system are the subsets of the parameter space of the system within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a curious chain structure with points of zero width called shrinking points. In this paper we perform a local analysis about an arbitrary shrinking point. This is achieved by studying the symbolic itineraries of periodic solutions in nearby mode-locking regions and performing an asymptotic analysis on one-dimensional slow manifolds in order to build a comprehensive theoretical fram","authors_text":"David J.W. Simpson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-06T03:08:15Z","title":"The structure of mode-locking regions of piecewise-linear continuous maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01416","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":"sha256:76ac794f7403f1bc7fdb1cf64675146f6b175836bb91733e406acbc93ddf0976","target":"record","created_at":"2026-05-18T01:30:56Z","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":"c6737bc66c0b401acf4305b442ff0b7a54601694ae57640c4127aa36bcf5679d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-06T03:08:15Z","title_canon_sha256":"869a27fe88d5272de0affe083dd8ef96bafc57aa04bdc685e14100edbb5cd899"},"schema_version":"1.0","source":{"id":"1510.01416","kind":"arxiv","version":1}},"canonical_sha256":"f3de320ac51e243ae29835cce803317f267c16c91f567cafdfaf62907e8f9028","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3de320ac51e243ae29835cce803317f267c16c91f567cafdfaf62907e8f9028","first_computed_at":"2026-05-18T01:30:56.940946Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:56.940946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RCRD4N3GAjZ7rdND7KXL5hUvzgNlH12qduCdBLBEoiWyRsJsi1jgzAxAjt/5Mfjfq6mZ/+vW+q58NIgwiGeuBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:56.941395Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.01416","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:76ac794f7403f1bc7fdb1cf64675146f6b175836bb91733e406acbc93ddf0976","sha256:a054a1fcc1adb1dc2b2e102bc1ac05a69ee48c200f4287124345eb4346b4a926"],"state_sha256":"b9fbb580c300ff194d0c995989145ac0a1574dd29c0b84a52d9afec41352958c"}