Enclosure Methods as Applied to Linear Periodic ODEs and Matrices

Abstract

This paper is dedicated to the memory of Prof. Dr.-Ing. H. RETTIG, one of the early and continuously leading contributors to numerical and experimental work on gear drive vibration, e.g., [271, [28]. Fur Systeme linearer gewohnlicher DGL mit periodischen Koeffizienten werden Einschlie$ungsmethoden fur die folgenden Verifikationsaufgaben behandelt : (i) der asymptotischen Stabilitat der homogenen DGL und (ii) der Existenz periodischer Liisungen der inhomogenen DGL. Wenn im Freiglied ein stochastischer ,,lntervall-Punktprozess" auftritt, werden zusatzlich (iii) die Mengen der (klassischen) Losungen und Randverteilungen iiber diesen Mengen eingeschlossen. Ein fur li) entwickelter hinreichender Matrix-Test ist direkt anwendbar, um lLil c 1 oder Re (,Ii) < 0 fur die Eigenwerte I i oder li zugeordneter Matrizen zu verijkieren. Numerische Beispiele zeigen die Leistungsfiihigkeit der behandelten Einschlie$ungsmethoden, u. a. in Anwendungen auf Getriebeschwingungen.Concerning linear ODEs with periodic coefficients, enclosure methods are presented for the verification of (i) asymptotic stability of the homogeneous ODEs and (ii) the existence of periodic solutions of the nonhomogeneous ODEs. In the case of forcing stochastic "interval point processes", (ui) the set of (classical) solutions and marginal distribution functions concerning this set are enclosed. A suffient matrix test with respect to (i) is directly applicable for the verification of IAil < 1 or Re (Ai) c 0 for the eigenvalues li or rZi of related matrices. Numerical examples show the power of the presented enclosure methods, among others in applications concerning gear drive vibrations. , &A cucmeM nunefinwx 0 6 w~n o~e n n~x du&$epenyuaflbnbix ypasnenuir (OAY) c nepuoduvecuunru ~os@q!kyuenma~u nped-cmasmmmcA Memobbi sumuenus dm sepughxayuu (i) acuMnmomuvecKoli ycmoavusocmu odnopodnbix OAY u (ii) cy yecmso-saw3 nepuoduvecxuxpezuenz& neodnopodnwx OAY. B cflyvae swnym3amupx cmoxacmuvecuux CtunmepeaJrbnwx movewmx npoyeccosw (iii) eumuaromc~ Mnowecmso (maccuvecwx) pewend u @ynKyui( MapzunaJzbnozo pacnpedeflentur omnocu-Merlbno 3moz0 Mnoxecmsa. AocmamovHyio Maypuqnyto nposeprcy omnocumemio (i) Momto* npuMenumb nenocpedcmsenno d m sepu&uKayuu mozo vmo llil < 1 uflu Re (Ai) < 0 dm co6cmsennbix 3navenuzC ii uflu kt coomsemcsymyux ~a m p u y .LiucflennMe npuMepw no~a3wsamm npou3sodume~anocmb npedcmasmeMwx Memodos srcmovenuscpedu dpyzux d m npuvenenuu Kacamyuxcrr su6payuu nepedamono ZI ~e x a n u 3~a .The periodic continuation of [y,](t^) is obvious. By construction, [y](t) is the smallest interval containingthe set of solutions of (2.1), (7.1), (7.2). As m + co, the sequence of enclosures ([y](t + mpT,)) approaches asymptotically the interval [ym](t). The periodic continuations of -jm and j , are (pTB)-periodic functions of time t. This asymptotic enclosure-periodic state (of the bounds) rests on the assumptions Q(C) < 1, (Al), and (A9).Remarks: 1.) The analysis employed here is related to the one in [9].2.) Unless, p , q E I N in (A 1) are sufiiciently small, j m poss...

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