{"paper":{"title":"On the independence of the slow and fast scales in multiple-scale expansions, with application to Van der Pol's equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["nlin.PS"],"primary_cat":"math.DS","authors_text":"Gregory Kozyreff, John R. King","submitted_at":"2026-06-25T13:44:45Z","abstract_excerpt":"When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27038","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.27038/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"}