F 4 T d a c c c s H m a 4 I o s wlo th th m e o r d Tt r a k Fig. 5 Global 4.2 Local Aspe To each transit discussed previ account for this co-existing loc can help us cla correspondence slow componen Hence, we defi moreover, each again the nonlin 4 (case of 1:3 t IMFs in the co oscillator (LO) 1 1/ 3 (HF N should reprodu where we keep ow-frequency hat during 1:3 he fast frequen modified accor , 1,2 m j t e m m otherwise it wo represent the tr dynamics. The constructio erms of slow represent the i averaging in (1 2, 1,2 k m ) identification the IM ect of NSI: Slo tion in the FEP iously nonline s eventuality; i al models in th arify this issue. e between the t nts of the iden ine the local m h IMO should a near system of transient reson orresponding d ) at frequency NES) and 2 ce the response HF LF N HF p the notations IMF (1) 1 ( ) c t of TRC the fast f ncies vary slow dingly (see dis modulated by ould be off-res ransient nonlin on of the local and fast comp mportant unde 13) and introdu ). Then, we ob on the FEP o MFs (left), and ow-Flow Mode P there corresp ear systems ma i.e., the possibi he same energ The construct theoretical and ntified IMFs of odel as a set of approximately f coupled oscill nance capture). damped dynam y 2 1 (HF 1 (LF NES), e of an identifi (1) 2 (2) 1 (2) 2 LO: ()2 NES: () NES: () c t c t c t for the IMFs f the LO as in frequencies of wly with time, scussion in Lee slowly-varyin onance and its near modal inte l model descri ponents, and f erlying dynam ucing the new tain the follow f system (5): r direct wavelet els ponds a local m ay possess mu ility that depen gy range. The g tion of the loca d empirical slo f the nonlinear f intrinsic mod reproduce one lators (5) and d . The dimensio mics. In this c LO), and two respectively. I ied dominant IM (2) (1) 1 2 2 (1) (2) 2 1 1 (2) (2) 2 2 2 ( ) 2 ( ) 2 ( ) c t c t c t introduced in nsignificant. Th the IMFs lock or we have clo e et al. [24]). T g complex am s effect on the eractions betw ibing the damp focusing exclu mics of the tran complex varia wing slow flow reconstruction t transform sp model of the dy ltiple co-existi nding on the in global transitio al model for a ow-flows discu systems as the dal oscillators e of the identif develop a local onality of the l case there exis o IMFs for the It follows that MF. Hence, we 2 (1) 2 2 2 (2) 1 1 2 (2) 2 2 ( ) Re[ ( ) R ( ) R c t c t c t Section 2 (cf. he rationale fo in 1:3 ratio an osely spaced fa The forcing term mplitudes. This dynamics negl een the LO and ped transition usively on the nsition). This ables, ( ) ( ) k mc t model for the ns based on fr pectra of the I ynamics which ing solutions, nitial condition ons in the FEP given transitio ussed in Section e underlying s (IMOs) that re fied dominant I l model for the local model de st three domin e nonlinear os the local mode e express the lo 2 1 2 (2) 1 (1) 2 (2) 2 [ ( ) ] e[ ( ) ] Re[ ( ) ] j t j t j t t e t e t e . Fig. 3), and w or expressing th nd remain appr fast frequencies ms in (13) are i s particular fo ligible [52]. In d the NES at th is facilitated b e slow dynami can be achiev ( ) ˆ ( ) k m m j c t IMOs, equency and e IMFs (right) h we now proc so when perfo ns the system m P discussed in on in the FEP n 2. This allow slow-flow resp eproduce the m IMFs. As an ex e damped transi epends on the n nant IMFs: an scillator (the N el should be 3ocal model in t we omitted the he local mode roximately con s the slow mod in the form of f orm of excitat n essence, the f he dominant fa by decomposin ics (as mentio ved by applyin ( ) ˆ ( ) m j t m k t e (LO energy estima ceed to constru orming NSI we may possess dif the previous s will be based ws us to interp ponses of the sy measured time s xample, we co ition depicted i number of dom n IMF for the NES) at frequ -DOF, and each the following f e contribution l in the form ( nstant; in cases del would need fast oscillating tion is chosen forcing terms i ast frequencies ng the IMOs ( ned previously ng complexific O- 1, 1 k m ; ates of uct. As e must fferent section on the ret the ystem. series; onsider in Fig. minant linear uencies h IMO form, (13) of the (13) is where d to be g terms n since in (13) s of the (13) in y they cation- NES521
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