f x Ctwim m c w s W d th d h s id ti wth th d m s o d m A c for the NES. By 1( ) ( ) x t y t c Fig. 3 Corres Comparing the wo expansion mportant resu mathematical f complexify the where [·] de such that We note that by dynamics (e.g., he (until now decomposed in harmonic comp significant freq dentical theore ime series. Hen where the seco heoretical foun he important ( development o made regarding slow-fast partit of dynamical s dynamics by l methodology ca An example of column of Fig. y comparing w (1) (1) 1 2 ( ) ( ), c t c t pondence betw respo e EMD result ( s thus providi lt was establis fact that a com m-th IMF (k mc enotes Hilbert t ( ) ˆ ( ) k mA t y complexifyin , the complex ad hoc) domi terms of domi ponents in the quency-time sca etical (i.e., (m kx nce, we make t nd half addres ndation for EM (multi-scale) d f the NSI meth g the type or d tions discussed ystems. In fac ocalizing all n an be extended f this correspo 3 we depict a with the exact n 2 () () x t v t ween theory ( onse of LO (fi (8) with the slo ing a theoretic shed in Lee e mplex function w ) ( )t of the tim ( ) ( ) ˆ ( ) ( m k k m t c t transformation ( ) 2 ( ) k mc t ng the identifie amplitudes k inant IMFs in inant IMFs, it f e slow-flow d ales in the dyn ) ( )t ) and num the association ( ) ( ) m k t sses the equiva MD, whereby th dynamics. This hodology of b dimensionality d above and the ct, EMD can pr non-smooth ef d also to non-sm ondence is now comparison be numerical time (2) (2) 1 2 ( ) ( c t c t slow flow dyn rst row) and L ow/fast decom cal basis for E et al. [23] and whose imagina me series ( ) kx t ( ) ) [ ( )] k m t j c t , and ( ) ˆ ( ) k mA t a ( ) 2 (ˆ [ ( )] , k k m m c t ed IMFs in (9) ( ) ( ) m t in the ex terms of the follows that decomposition namics). It follo merical (i.e., c n ( ) ( ) ˆ ( ) m m k k t alence of `slow he dominant IM important res road applicabi of system and e physical inte rovide us with ffects (e.g., du mooth dynamic w given in term etween the mag series we estab )t . namics) and nu LF response o mposition (3) w EMD in term d will form th ary part is the in (8) by defin ( ) [ ˆ ( ) ] ˆ ( ) k m j k mA t e and ( ) ˆ ( ) k m t are ) 1 ( ) tan t we have a dire xpression (3)). slow-flow dyn ( ) k m m for m of the dynam ows that the ab ( ) ( ) k mc t ) multi-s ( [ ˆ ( ) ˆ ( ) m j k mA t e w' complex amp MFs represent sult derived by ility (cf. Sectio d the type and erpretation of t h the added ben ue to clearance cal systems. ms of the prev gnitudes of the blish the comp umerics (EMD of the NES (sec we clearly estab ms of the unde he basis of th Hilbert transfo ning the analyt ( ) ( ) ( ) ] k k m m t t j t e e the instantane ( ) ( ) [ ( )] / ( k k m m c t c ect way to relat . This will pro namics. Given 1, , m N , wh mics (i.e., the bove partitions scale slow-fast ) ( ) ( ) ] k k m t t plitudes. These the underlying y Lee et al. [2 on 3.2). We no d strength of th the results of E nefit of separa es) in the lead vious strongly e slowly varyin pleteness of the D results) for t cond row) blish the possi erlying slow f he NSI method orm of the rea tic complex fun eous envelope ( )t te them to the ovide a way to n, however, th here N is the dimensionality can be related t decompositio e relations prov g slow flow an 3] provides th ote that no ass he nonlinearity EMD should ho ating the smoo ding order IM nonlinear exa ng amplitudes o e decompositio the system (5) ibility of relati flow dynamics d. Motivated b l part is analyt nction and phase of c governing slow o physically int at the time se number of dom y, or the numb d since they rep ons of the mea vide a physics nd, hence, capt he foundation f sumptions have y. It follows th old for a broad oth from non-s MFs. Hence, th mple (5). In th of the low- and on; i.e., ): HF ing the s. This by the tic, we (9) ( ) ( ) k mc t (10) w flow terpret eries is minant mber of present asured (11) -based ture all for the e been hat the d class mooth he NSI he left d high517
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