two-memcapacitor circuit is selected as a test dynamical system. Symmetry, accuracy and performance analysis of semi- explicit extrapolation ODE solver are ...
______________________________________________________PROCEEDING OF THE 21ST CONFERENCE OF FRUCT ASSOCIATION
6WXG\RI7ZR0HPFDSDFLWRU&LUFXLW0RGHOZLWK 6HPL([SOLFLW2'(6ROYHU 'HQLV1%XWXVRY9DOHULL@7KLVHOHPHQW ZDV FDOOHG D memristor E\ / 2 &KXD ZKR SODFHG LW RQ D SDU ZLWK UHVLVWRUV FDSDFLWRUV DQG LQGXFWRUV /DWHU LQ WKH PDWKHPDWLFDOGHVFULSWLRQRIWKHPHPULVWRUZDVH[WHQGHGWRD ZLGHUFODVVRIPHPULVWLYHV\VWHPV>@ZLWKLQWKHIUDPHZRUN RI ZKLFK WKH PRGHUQ WHUPV RI RWKHU SDVVLYH WZRWHUPLQDO FLUFXLW HOHPHQWV D memcapacitor DQG D meminductor >@ ZHUH LQWURGXFHG $ VOLJKWO\ GLIIHUHQW YLHZ RQ WKH UHODWLRQVKLS EHWZHHQ WKH IRXU EDVLF HOHFWULFDO TXDQWLWLHV LV SUHVHQWHGE\-6KHQHWDOLQSDSHU>@ZKHUH transtorsDQG memtranstors DUH DOVR LQFOXGHG LQWR WKH OLVW RI WKH EDVLF HOHPHQWV 1RZDGD\V LGHDO PHPULVWLYH HOHPHQWV DUH FRQVLGHUHG DV D SRVVLEOH FLUFXLW GHVFULSWLRQ RI UHVLVWLYH VZLWFKLQJ GHYLFHV >@ ±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m DQG Cm QRQOLQHDU HQHUJ\VWRUDJH HOHPHQWV 7KH FLUFXLW DOVR LQFOXGHV DQ LQGXFWRU L DQG D UHVLVWRU ZLWK FRQGXFWDQFH G MXVW DV WKH RULJLQ
)LJ6FKHPDWLFRIWKHFLUFXLWZLWKWZRPHPFDSDFLWRUV
ISSN 2305-7254
______________________________________________________PROCEEDING OF THE 21ST CONFERENCE OF FRUCT ASSOCIATION
7KH PRGHO RI HDFK VPRRWK PHPFDSDFLWRU >@ LV UHSUHVHQWHGE\WKHHTXDWLRQV
°uc m nV q ® °¯Mc mV nV aV bV
LV IRU ³GLDJRQDOO\ LPSOLFLW´ 7KH\ XVXDOO\ QHHG OHVV FRPSXWDWLRQDOHIIRUWVWKDQWKHRWKHUH[WUDSRODWLRQPHWKRGVDQG DUHEHWWHUVXLWHGIRUFKDRWLFSUREOHPV>@7KHVHPHWKRGVH[LVW IRUV\VWHPVRIRUGHUN)RUDJHQHUDOIRUPRI,93
d x dt
ZKHUH ı LV WKH WLPH LQWHJUDO RI WKH PHPFDSDFLWRU FKDUJH q DQG ij LV WKH WLPH LQWHJUDO RI WKH YROWDJH uF DFURVV WKH WHUPLQDOVRIWKHPHPFDSDFLWRU
(DFKGHULYDWLYHRIDVWDWHYDULDEOHxiLVUHODWHGWRDULJKW KDQGVLGHIXQFWLRQDV
d xi dt
)LQDOO\GHILQLQJx qLy ız ıc Ld Ge GDQGLQWURGXFLQJababUHIHUUHGWR WKHV\VWHP GHVFULEHGE\ FDQEHQRUPDOL]HG
xi n
xi n hfi t xn xn xi n xN n
:H WDNH ³(XOHU&URPHU´ LQ TXRWHV VLQFH WKH RULJLQDO (XOHU&URPHUPHWKRGLVXVXDOO\DSSOLHGRQO\WR+DPLOWRQLDQ V\VWHPV 7KH DPHWKRGV DUH VXLWDEOH IRU D JHQHUDO FDVH RI ,93'HQRWHfitnxn fin7KH7D\ORUH[SDQVLRQRI xinRQ RQHLQWHJUDWLRQVWHSRI \LHOGV x n
x n hf n
x n
x n hf n h f n f c n Oh
N
x N n hf N n h ¦ f j n f Nc n Oh
x N n
j
7KHV\VWHP FDQEHXVHGDVDWHVWSUREOHPIRUUHVHDUFK RI QXPHULFDO LQWHJUDWLRQ PHWKRGV VLQFH LW KDV FRPSOH[ QRQOLQHDULWLHV DQG FKDRWLF VROXWLRQ IRU PDQ\ SDUDPHWHU YDOXHV$FFRUGLQJWR>@IRUWKHLQLWLDOFRQGLWLRQV WKH SRVVLEOH SDUDPHWHUV RI WKH V\VWHP WR H[KLELW FKDRWLF EHKDYLRU DUH a b a b c d e ,,, ,17(*5$7,210(7+2'6 $PHWKRGIRULQWHJUDWLRQRIFKDRWLF2'(VVKRXOGVDWLVI\ VHYHUDO UHTXLUHPHQWV 2QH RI WKH PRVW LPSRUWDQW LV WKDW LWV DELOLW\WRGULYHDV\VWHPLQWRDTXDVLFKDRWLFORQJSHULRGLFDO RVFLOODWLRQV PRGHVKRXOGEHDVOHVVDVSRVVLEOH$OVRLWKDV WREHQXPHULFDOO\HIILFLHQWVLQFHFRPSXWDWLRQRIELIXUFDWLRQ GLDJUDPVDQGG\QDPLFDOPDSVLVDFRPSOLFDWHGWDVN )URP >@ D IDPLO\ RI VHPLLPSOLFLW H[WUDSRODWLRQ PHWKRGV LV NQRZQ 7KHVH PHWKRGV DUH GHULYHG IURP WKH (XOHU&URPHUDOJRULWKPDQGDUHFDOOHGWKHDPHWKRGV³D´
----------------------------------------------------------------------------
f i t x x xi xN
,QWHJUDWH FRQVHTXHQWO\ZLWKUHVSHFWWRxx«XVLQJ WKH H[SOLFLW ³(XOHU&URPHU´ LQWHJUDWLRQ ZKHQ SUHYLRXVO\ FDOFXODWHG VWDWH YDULDEOHV DUH WDNHQ LQWR WKH ULJKWKDQG VLGH IXQFWLRQ
$SSO\LQJ LQWHJUDO RSHUDWLRQ FDQ EH WUDQVIRUPHG WR WKHIROORZLQJIRUP
x ca z b z ° ® y d e a y b y ea z b z ° z ea y b y a z b z x ¯
ZKHUH D EROG IRQW GHQRWHV D YHFWRU WKH EDVLF DPHWKRG LV ZULWWHQDVIROORZV
$SSO\LQJ WKH .LUFKKRII V ODZ DQG WKH YROW±DPSHUH FKDUDFWHULVWLFV RI WKH FLUFXLW HOHPHQWV RQH FDQ REWDLQ D V\VWHPRIRUGLQDU\GLIIHUHQWLDOHTXDWLRQV diL ° L dt uc °° dq G uc uc Guc ® dt ° ° dq G u u i c c L °¯ dt
dqL °L dt Mc °° dV GMc Mc GMc ® dt ° ° dV GM M q c c L ¯° dt
Ft x
65
)URP RQHFDQVHHWKDWDVDQRUGLQDOQXPEHURIDVWDWH YDULDEOH LQFUHDVHV WKH TXDQWLW\ RI WHUPV E\ h LQFUHDVHV FRUUHVSRQGLQJO\ 7KH PHWKRG LV FDOOHG WKH H[SOLFLW D PHWKRG DEthx 7R FRPSOHWH WKH 7D\ORU VHULHV DQ DSSOLFDWLRQRIDQadjointPHWKRGLVUHTXLUHG7KLVPHWKRGLV LPSOLFLW DQG ZLOO EH GHQRWHG DV DIthx 7KH LPSOLFLW D PHWKRG LV DOPRVW VLPLODU WR EXW WKH LQWHJUDWLRQ RI HDFK YDULDEOHLVLPSOLFLWDQGWKHLURUGHULVUHYHUVHxNxN« implicit
xi n hfi t x n x n xi n x N n
xi n
7KLVJLYHVDQRWKHU7D\ORUH[SDQVLRQ
x N n
x N n hfn h f N n f Nc n Oh x N n hf N n
x N n
h f N n f Nc n f N n f Nc n Oh
xn
xn hfn h
N
¦f
f c Oh
j n n
j
----------------------------------------------------------------------------
______________________________________________________PROCEEDING OF THE 21ST CONFERENCE OF FRUCT ASSOCIATION
7KH VHFRQG DQG WKH IROORZLQJ FROXPQV DUH REWDLQHG E\ WKH$LWNHQ1HYLOOHH[WUDSRODWLRQIRUPXOD
&RPSRVLWLRQRI DQG ZLWKDVWHSVL]Hh
D t h x
DII $ DI t h x
T j k t N T j k
FRPSOHWHV 7D\ORU VHULHV IRU HDFK VWDWH YDULDEOH 0RUHRYHU WKXV REWDLQHG DPHWKRG LV V\PPHWULF ZKLFK IROORZV IURP WKHIDFWWKDWWKHLQWHJUDWLRQ ZLWKQHJDWLYHVWHSVL]H\LHOGV WKH LQWHJUDWLRQ DQG YLFH YHUVD $OVR WKLV PHWKRG LV UHYHUVLEOHIRUȡUHYHUVLEOHV\VWHPV$VLWVDFFXUDF\RUGHULV WKHDEEUHYLDWLRQDLVXVHGIRULW
Tkk LV D VROXWLRQ 7KH FKRVHQ )URP WKH WDEOH xN VHTXHQFH ^nj` DIIHFWV WKH HIILFLHQF\ RI WKH PHWKRG EXW XVXDOO\DVLPSOHURZRIGRXEOHGQDWXUDOQXPEHUVLVHQRXJK &RQVLGHU VLPSOH QG RUGHU V\VWHP LQWHJUDWLRQ DV DQ LOOXVWUDWLYHH[DPSOHIRUWKHDPHWKRG/HWWKHV\VWHPEHDV
)RUVLPSOHUQRWDWLRQGHQRWHfitxnxn«xin« xNn Jxin 5HFDOOWKDWWKHH[SUHVVLRQ xi n hgxin
x y
f x y g x y
)RUWKLVFDVHWKHDPHWKRGZLOOEHZULWWHQDV
FDQEHH[SDQGHGDV xi n
T j k T j k nj p n j k
ZKHUHpHTXDOV IRUV\PPHWULFPHWKRGVOLNHWKHDPHWKRG
,PSOLFLWLQWHJUDWLRQLQ FDQEHLQFRQYHQLHQWLQFDVHWKH IXQFWLRQ fitnxn LV QRQOLQHDU 7KLV PRWLYDWHV WR UHSODFH DQ LPSOLFLWFDOFXODWLRQE\DQH[SOLFLWRQHLQDIROORZLQJZD\
xi n
xi n hgxin hgxin hgxin hg
xn
xn hf xn yn
yn
yn hg xn yn
yn
yn hg xn yn
xn
xn hf xn yn
7KLV LV DQ LQILQLWH VHULHV RI VXEVWLWXWLRQV ZKLFK LV HTXLYDOHQW WR WKH LPSOLFLW IRUPXOD %XW WR DFKLHYH D VHFRQGRUGHURQO\RQHVXEVWLWXWLRQLVUHTXLUHGZKLFKDOORZV XVLQJDQDSSUR[LPDWLRQ
ZKLOHWKHSEDDOJRULWKPZLOOORRNDV
xi n
xi n hg xi n
xn
xi n
xi n hg xi n
yn
LQVWHDGRI ([WUDSRODWLRQPHWKRGVRIKLJKHURUGHUEDVHG RQ WKH D LQWHJUDWRU UHTXLUH D TXDQWLW\ RI H[SDQVLRQV SURSRUWLRQDOWRWKHLURUGHURIDFFXUDF\6XFKDIXOO\H[SOLFLW PHWKRGUHWDLQVV\PPHWU\ $QDEEUHYLDWLRQ ³6('´³6(´ LV IRU VHPLH[SOLFLW LV XVHG IRU DPHWKRG ZLWK WKH H[SOLFLW VXEVWLWXWLRQ IXUWKHU
xn hf xn yn yn hg xn yn
yn
yn hg xn yn
yn
yn hg xn yn
xn
xn hf xn yn
xn
xn hf xn yn
&RQVLGHU DQ LQWHJUDWLRQ RI WKH PHPFDSDFLWRU FKDRWLF SUREOHP E\ D PHWKRG :ULWWHQ LQ &OLNH FRGH LW LV DV VKRZQLQ/LVWLQJ
7KH DPHWKRG LV WKHQ XVHG DV D EDVLF PHWKRG IRU H[WUDSRODWLRQ 7KH LGHD RI H[WUDSRODWLRQ LV DV IROORZV 6XSSRVH D VROXWLRQ RQ RQH ³PDFUR´ VWHS h VKRXOG EH REWDLQHG ZLWK DFFXUDF\ Ohk 7DNH D VHTXHQFH ^nj`RI QDWXUDO QXPEHUV ZKLFK ZLOO EH GLYLVRUV RI WKH VWHSVL]H IRU LQVWDQFH
Listing 1&RGHIRUWKHPHPULVWLYHSUREOHP h=h/2; float tmp; int i; int k=4; x=x+h*c*(a2*z+b2*z**3); y=y+h*((d-e)*(a1*y+b1*y**3)+... +e*(a2*z+b2*z**3)); z=z+h*(e*(a1*y+b1*y**3-a2*z-b2*z**3)-x); tmp=z; for (i=0; i@ SUHVHQWV WKH SURRI RI VWDELOLW\ V\PPHWU\ DQG WLPHUHYHUVLELOLW\ RI WKH DPHKRGV GXH WR PHQWLRQHG FRQYHUJHQFHWKH6('PHWKRGVDOVRVKDUHWKHVHSURSHUWLHV
9 &21&/86,21
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a DUH SUHVHQWHG 'DUN FRORU FRUUHVSRQGV WR FKDRWLF UHJLRQV $ VSHFLDO SURSHUW\RIWKHVHPLH[SOLFLWPHWKRGZKLFKFDQQRWEHIRXQG LQWKHH[SOLFLWRULPSOLFLWPHWKRGVFDQEHVHHQLQ)LJD :KHQ WKH VWHSVL]H LV VPDOO D EHKDYLRU RI WKH V\VWHP LV FKDRWLF GXULQJ WKH LQWHUYDO RI a אሾǦͲǤʹʹǡǦͲǤͳͶሿ ZKLFK LV DSSUR[LPDWHO\ HTXDO WR WKDW RI WKH FRQWLQXRXV V\VWHP %XW ZKHQ WKH VWHSVL]H H[FHHGV WKH UHJLRQ RI WKH FKDRWLF EHKDYLRUH[SDQGVRQWKHODUJHULQWHUYDORISDUDPHWHUYDOXHV
----------------------------------------------------------------------------
68
,Q WKLV SDSHU WKH VHPLH[SOLFLW PRGLILFDWLRQ RI WKH D PHWKRGLVSURSRVHG7KHVWXG\RI6('PHWKRGVKRZVWKDWLW FDQ EH PRUH HIILFLHQW WKDQ WKH RWKHU H[WUDSRODWLRQ PHWKRGV RQ H[WHQGHG GDWD W\SHV HVSHFLDOO\ IRU VHSDUDEOH G\QDPLFDO V\VWHPV7KHVHILQLWHGLIIHUHQFHVFKHPHVDUHV\PPHWULFDQG WLPHUHYHUVLEOHIRUȡ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
----------------------------------------------------------------------------
______________________________________________________PROCEEDING OF THE 21ST CONFERENCE OF FRUCT ASSOCIATION
)LJ'\QDPLFDOPDSVRIWKHV\VWHP DDEFGDUHWKHSDUDPHWHUVRIV\VWHP
$&.12:/('*0(17 7KH UHSRUWHG VWXG\ ZDV SDUWLDOO\ VXSSRUWHG E\ 5)%5 UHVHDUFKSURMHFW1R 5()(5(1&(6 >@ / 2 &KXD ³0HPULVWRUWKH PLVVLQJ FLUFXLW HOHPHQW´ in Proc. IEEE Transactions on Circuit Theory YRO &7 QR 6HSW SS± >@ /2&KXDDQG60.DQJ³0HPULVWLYHGHYLFHVDQGV\VWHPV´in Proc. IEEEYROQR)HESS± >@ 0 'L 9HQWUD @ - 6KHQ ' 6KDQJ @ '1 %XWXVRY $, .DULPRY DQG $9 7XWXHYD ³6\PPHWULF H[WUDSRODWLRQVROYHUVIRURUGLQDU\GLIIHUHQWLDOHTXDWLRQV´in Proc. 2016 IEEE NW Russia Young Researchers in Electrical and Electronic Engineering Conference (2016 ElConRusNW) )HE SS±
>@ -0RX.6XQ-5XDQDQG6+H³$QRQOLQHDUFLUFXLWZLWKWZR PHPFDSDFLWRUV´ Nonlinear Dynamics YRO QR 1RY SS± >@ ' 1 %XWXVRY $ 9 7XWXHYD DQG ( 6 +RPLWVND\D ³([WUDSRODWLRQ 6HPLLPSOLFLW 2'( VROYHUV ZLWK DGDSWLYH WLPHVWHS´ in Proc. XIX IEEE Int.Conf. on Soft Computing and Measurements (SCM - 2016)0D\SS± >@ '1 %XWXVRY 9