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yt YR -RR$BtR Y FsBA$st O Y sARBjZ %sjZ BP Y JZ$j$A$Z8 8st$ hjI4 stI Y RZj8tsk PZt$Bt B s $%t AkU

O # d>n nH N n , wV;D

4 # 4 lwnD wnD wn N nD = L d

R wnD wnD =

M, wVxD

B # d4 w4 N nD ) 4

O4 ^(( L

dR((

3 wVED

yt s js sR s$B B,s8s, b$Y $Zjs 4Z- RZPsR Y PBjjBb$t -RR$BtR YBjIPB Y FsBA$st stI Y 8$ B^$tRU

O # d>n nH N n ) p

wd L 1

p BR W

wVD

dM

n N n ) d L 1 " BR wVMD

n N n ) d

d 1

w p L "

WBR

]wVD

n N n ) R$t w p L w " D "

WwVd(D

n N n ) dp

wd 1

p BR W

wVddD

n N n ) n N n ) ( wVd1DbY$t p $R Y sI$ZR BP Y 8st$ s-$R wD $R Y 7YsPstB% RY$P stI Y $8ItBR s sI$sj I$%s$% yt Y sARt BP jsR8s RRZ " $R R$8jk BB$Btsj B PB $Zjs 4Z- RZPsR y $R YPB 8$RR$Aj B js w " D " Ak 1 " $t eJ wVddDstI $t Y RZARJZt -RR$BtRy RYBZjI A tBI Ys b Ys% B8$I sjj RBtITBI 8R $t $t%R sR s$B $t

eJR wVMDTwVd1D W%YjRR $ ZtR BZ Ys YR -RR$BtR s R$jj sZs tBZYPB Y sjZjs$Bt BP Y j%st T8R $t Y $t8BI JZs$Bt qY Btjk & PB8Y tjI Bt$AZ$BtR B eJR wVMDTwVd1D $R st $tR$t$hst tB8sj$9s$Bt BP Y$t8BI PJZtk

d

VtI$- U *$sBtsj 8s$- j8tR

qY I$sBtsj 8s$- j8tR wU , U , U D s BAs$tI Ak sjk$t Ys%s$t BIZ >R eJ wd1DH B Y B8BttR BP Y $t8BI JZs$Bt IR$AIAk eJR wVdDTwVnD 7$t n N n R$t $R st BII PZt$Bt BP sjj Y 8R b$Y n Nn$t eJR wVdDTwVnD %st$RY sR s RZj BP s%s$t qY 8s$t$t 8R s, Y PBjjBb$tPB8U

U ) ((

rO

p ~ wnD Xz E

R~

p O wnD S (

(

E

rO

p ~ wnD Xz E

R~

p O wnD S 3

w dD

eJZs$Bt w dD $8j$$jk $tjZIR Y JZ$I .w D Bt$AZ$BtR $t Y $t%R sRs$B -stR$Bt qY tjI 8R Bts$t $Y st sII$$Btsj PsB wdbED U d B b$Y R B Y s$tI .w D 8RqB 9BY BI $t Y JZst$$R $t Y hR stI Y RBtI 8R BP eJ w dD s

JZsj B sY BY V %st$RY$t As, $%R Y VjP%?t Bt$tZZ8 I$RR$Bt js$BtqY B8$sj BB$Isj B$BtR B Y s%sI -RR$BtR $t stZjs As,R s.w D R$t Y .wD 8R s BII PZt$BtR BP y RYBZjI A B$tI BZ Ys stkR8sjj B$BtR Ys Bt$AZ JZsjjk B Y bB As,I -RR$BtR IB tB s& YRZZ BP eJ w dD sR YR B$BtR st A sARBAI Ak s $%$sj TIht$$Bt BP G YPB tI B Rj Y B$BtR Ys s s I$&t Abt Y As,I8R V R$8j stsjkR$R RYBbR Ys sjj RZY B$BtR B8 Btjk PB8 Y BIZR BP Yj$ts w$t D Bt$AZ$BtR B Y FsBA$st stI B Y 8$ B^$tR qY$R BAR%s$Bt8stR Ys Y sZsk BP eJR wVMDTwVd(D $R sZsjjk RZ^$t B sjZjs Y JZ$IB$BtR qY RZj$t -RR$Bt PB Y I$sBtsj 8s$- j8t YsR Y PB8UU E ) (

(X

~ wd L ; " D wzEbRD

p

~ ((

E

X

~ wd ;wL " DD wzEbRD

p

~3

w 1D

1(

VtI$- ]U !&TI$sBtsj 8s$- j8tR

qY s PBZ B&TI$sBtsj j8tR Ys s j%st B Y $t8BI JZs$Bt qYRs U U U stI U qY$ %sjZs$Bt $t%Bj%R Y Rs$YPBbsI$ts$Bt BP Y -RR$Bt $t eJ wdnD b$Y U stI O IhtI Ak eJR wD stI wVMD !tjkjBbR BI tBtT%st$RY$t 8R $t $t%R sR s$B -stR$Bt tI B A s$tI $tYR sjZjs$BtR Vt sII$$Btsj R$8j$hs$Bt B8R PB8 Y BtI$$Bt E ( d bY$YsjjBbR Bt B tj Y I$&t Abt E,E d stI E L d %kbY - $t YJZst$$R z ! , z ! stI z ! sR YR JZst$$R s k$sjjk BP Y BI BPZt$k wtB BP Y BI BP ED PB Y 8BIR BP $tR qY RZj$t -RR$BtR PB YB&TI$sBtsj j8tR s I8$tI Ak Y PBjjBb$t JZs$BtRU

U ) ((w1L " D

~(( d

p

Xz E

R~ (( " ((

Xz E d

R~

L

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E dp " L

E Xz E dR

~Xz E

R~

~ E wL w " D " D (

( L dp E

Xz E

R~wL w " D " D (

(Xz E d

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w]dD

U ) (( w1L " D

~(( d

p

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R~ (( " ((

Xz EL d

R~

L

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E Xz EL dR

~Xz E

R~

L

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pL wD " Q (( d

p E

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R~wL w " D " D (

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w]1D

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p

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R~ (( " ((

Xz E

R~

L

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E dp " L

E Xz E dR

~Xz E

R~

L

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( dp E

Xz E d

R~wL w " D " D (

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w]nD

1d

U ) ((w1L " D

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p

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L

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E Xz EL dR

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R~

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( L dp E

Xz EL d

R~wL w " D " D (

(Xz E

R~ 3

w];D

yt YR -RR$BtR b ZR s tBs$Bt ~ PB Y 4Z- RZPs s%sI VjP%?t %jB$kbY$Y $R Y VjP%?t %jB$k sjZjsI PB s jBsj 8sRR ItR$k 'wD AZ PB Y BtTs-$R8st$ hjI 4 G s$t Y sI$sj ItIt BP ~ wD sR bjj sR Y sI$sj ItItBP Y $t%R sR s$B # bp Y 7YsPstB% RY$P PsB " wD stI Y RsPk PsBRwD

V,tBbjI8tR

qY$R bB, bsR RZBI Ak =7 * BP etk ]Bts WB *eT3p(nTEeiTx;n1E AkqY =t$I $tIB8 et$t$t stI _YkR$sj 7$tR iRsY ]BZt$j stI Ak eZsB8qY bB, bsR BtIZI sjk ZtI Y eZBst 3ZR$Bt *%jB8t V8t

11

iPtR

>dH N GBt i F 3Bt, 7 3 _sZj * i iBAR e * 3I$,RBt i Ws9$,$st < _s, jj W N 9 i ZItk 7 ]BYt p G <s88 3 ] FBAR * sI 7 7 Ijk * Zjj O Wssks8s * !btR stI e F 7kts,BbR,$_YkR i% N EE dM; wddD

>1H yqei _YkR$R e- pBZ Bt et$ _s$jR <s$t stI ]Zt *$% yqei_YkR$R sR$R eI$BR WZj 3ZR$Bt n 1;d wdD

>nH F _ pBIAjBI < V <Bj$R 7 _BIR p q V <ZkR8stR G t _jsR8s_YkR ]BtBj 3ZR$Bt nx 1 wdnDV 3sRBj$ * qRs 7 7YssB% < N , W $98st V pBtIYsj,s i 3< stR$tt stI Bt$AZBR B Y e3*VTFeq GB,Bs88 _jsR8s_YkR ]BtBj 3ZR$Bt ;; dx w1((1D

>;H ] ]Yt N ]Yt 7 ]Yst Vtt _YkR dEd 1d wdMxD] ]Yt stI 7 ]Yst _YkR 3jZ$IR 1 nEx wdMED

>xH * G iBRR p N ]Yt stI 7 sYs`st _YkR 3jZ$IR 1x Ex1 wdM1D>EH V _jsR8s _YkR ]BtBj 3ZR$Bt 1; dd; wdM1D>H < $8Zs O ZRs8s 7s$ZRs pF s8 qBA$s W8BB q BtIBY

q W$RY$st$ ! *s ]BRs q !9,$ q !$,sbs 7 B$ks8s V B$B,s pO 3Z] ]Yt stI zy VPstsR:% WZj 3ZR$Bt nM dn(n wdMD

>MH 7 e 7YssB% * qRs Vj * W BAs V 3sRBj$ W ] <sb,R i 3< stR$tt p zBt <jj8stt stI Bt$AZBR B Y e3*VTFeqGB,Bs88 _YkR N V 1M d1 w1((dD

>H 7 e 7YssB% Vj < N , * W BAs W $98st ] * ]Ysjj$RV 3sRBj$ W ] <sb,R q ] <tI F s$jjBZ- 7 * _$tYR * qRs stIBt$AZBR B Y e3*VTFeq GB,Bs88 _YkR _jsR8sR 1(1 w1((1D

>d(H < N , * W BAs W $98st 7 * _$tYR stI 7 e 7YssB% _YkRi% N M dMx((1 w1((dD

1n

>ddH < N , W $98st stI 7 _,, _jsR8s _YkR i 1n M wdD

>d1H F ]stIk W $98st F G zst *s8 stI q !9,$ _YkR N V 1dx 1wdED

>dnH p q V <ZkR8stR 7 e 7YssB% V $,Ys$jB%R,$$ stI G t _YkR_jsR8sR M ;11 w1((dD

>d;H G t F _ pBIAjBI p q V <ZkR8stR 7 _BIR stI e 7Ybs9 F ]B8_YkR d;1 1d wdMD

>dxH N y iZIs,B% stI i 7sI% 7B% _YkR *B,j E ;dx wdEdD

>dEH 3 Bts 7 $Zj$B N ]Yt 7 *$, p 3Bs$s * qRs stI p zjsI _YkR_jsR8sR ;n w1((1D

>dH i Ws9$,$st p F s8 < N , W $98st stI 7 7YssB% Zjj V8_YkR 7B ; n1 w1((1D

>dMH e FB&$t p pB$t$ ] * ]Ysjj$R W ] <sb,R q ] <tI * 3 <Bbjj _s _ st$s * s9Bt 7 e 7YssB% p qRR stI Bt$AZBR B Ye3*VTFeq GB,Bs88 _jsR8s _YkR ]BtBj 3ZR$Bt ;; dn w1((1D

>dH e FB&$t ] * ]Ysjj$R p * ]Btbsk psA V pZI 7 pZtY W ]<sb,R q 7 <tI * <Bbjj p q V <ZkR8stR e Ns99sB _ s ssY, V p _R 7 * _$tYR 7 e 7YssB% stI Bt$AZBR B Y e3*VTFeq GB,Bs88 yttsj stRB As$ $$t Ak s$Btsj 8st$ 4Z-RZPsR $t B,s8s,R 3ZR$Bt etk 1((1 w_B dY yVeV ]BtPt NkBt 1((1DyVeV z$tts e\_dTdn RZA8$I B WZjs 3ZR$Bt w1((nD

1;

3yp=ie ]V_qy!W7

3yp d VjP%?t ]sRsIR BP Rb$tTZ stI BP Rb$tTIBbt kR BAR%I $t s FeqI$RYs R$8ZjstBZRjk b$Y qVeR qY $Y s BP Y hZ $R s 9BB8 BP YjP s y RYBbR qVeR RB8 BP bY$Y s$R PB8 Y V]

3yp 1 7Y8s$ jB BP Y V] 8BI PJZt$R PB bB RjI BjB$Isj 8BI tZ8AR- stI-d qY I$&t JZsIstR BRBtI B I$&t R$tR BP < $'&;( stI < (7&28 7Bj$I j$tR $tI$s BRR$Aj V] 8BIR IsRYI j$tR $tI$s Y VjP%?t Bt$tZZ8Y $jR $tI$s Y $BtR bY Y ]sRsI 8BI sB-$8s$Bt Ps$jR qYj%st %sjZR BP E s RYBbt $t sY JZsIst 3$Z 1s RtR BR$$%PJZt$R stI 3$ 1A $R PB ts$% PJZt$R

3yp n qstRPB8s$Bt BP Y &$% Bt$sj PB sI$sjjk -tII 8BIR PB8 s R$tjbjj wzR EL db1 k (D B s IBZAj bjj wzR EL db1 c (D

3yp ; qstR$$Bt BP VjP%?t ]sRsI 8BIR B qVeR PB BR$$% PJZtk 8BIR 7Bj$IZ%R Rt $tPJZt$R PB jBbR BI sI$sj 8BIR b$Y z ) ; E )wd1 ddD stI b$Y Y PBjjBb$t %sjZR BP $tZ ss8RU R " " bR ) (31x < $'&;( )1 ) (3dn " ) (3(n qY$t IsRYI Z%R 8s, VjP%?t Bt$tZZ8 PJZt$RqY$, IsRYI Z% RtR Y 8BI RYBbt $t JZsIst wD BP 3$ 1s

3yp x qstR$$Bt BP VjP%?t ]sRsI 8BIR B qVeR PB ts$% PJZtk 8BIR 7Bj$IZ%R Rt $tPJZt$R PB jBbR BI sI$sj 8BIR b$Y z ) ; E )wd1 ddD stI b$Y Y Rs8 $tZ ss8R sR $t 3$ ; qY$t IsRYI Z%R 8s,VjP%?t Bt$tZZ8 PJZt$R qY$, IsRYI Z% RtR Y 8BI RYBbt $tJZsIst wID BP 3$ 1A

3yp E 7tsRYBR BP Y 8BI RZZ PB s BR$$% PJZtk 8BI b$Y z ) ; stIE ) wd1 ddD IZ$t Y stR$$Bt PB8 s ]sRsI 8BI B qVe qY jP BjZ8tRYBbR Y sI$sj BhjR BP Y E ) d1 stI E ) dd BjB$Isj B8BttR PBY jBb >q ) w( dDH 8BI BP 3$ ; qY $Y BjZ8t RYBbR Y Z 8BI>q ) wd (DH qY %sjZR BP R PB Y RtsRYBR sU wsD R ) 13dx wAD R ) 13MxwD R ) 13ME wID R ) 13M;

1x

3yp 7tsRYBR BP Y 8BI RZZ PB s ts$% PJZtk 8BI b$Y z ) ; stIE ) wd1 ddD IZ$t Y stR$$Bt PB8 s ]sRsI 8BI B qVe qY jP BjZ8tRYBbR Y sI$sj BhjR BP Y E ) d1 stI E ) dd BjB$Isj B8BttR PBY jBb >q ) w( dDH 8BI BP 3$ ; qY $Y BjZ8t RYBbR Y Z 8BI>q ) wd (DH qY %sjZR BP R PB Y RtsRYBR sU wsD R ) 13dx wAD R ) 13MxwD R ) 13ME wID R ) 13M;

1E

200

190

TAEs

TAEs

Cascade

180

170

160

150

140

6.2 6.4 6.6 6.86.0F

requency

(kH

z)

JG0

2.4

31

-3c

Time (s)

200

160

120

80

40

TAEs

Cascade

Cascades

6.54.0 4.5 5.0 5.5 6.0

Fre

quency

(kH

z)

Time (s)

3yp dU

1

(b) m=M-1

(d) m=M-1

(c) m=M

(a) m=M

Q <0

Q >0tor Q <0tor

Q >0tor Q <0tor

hot

Q <0hotQ >0hot

Q >0hot

Mn

M-1n

M-1/2n

q0

Ω

1

2

ω>0

no mode

3yp 1sU

1M

(b) m=M-1

(d) m=M-1

(c) m=M

(a) m=M

Q >0

Q >0tor Q <0tor

Q >0tor Q <0tor

hot

Q >0hotQ <0hot

Q <0hot

q0

|Ω|

1

2

ω<0

Mn

M-1n

M-1/2n

no mode

3yp 1AU

1

0

0

nq0-m+1/2<0

nq0-m+1/2>0

U

x

3yp nU

n(

2

1.5

1

0.5

03 2.95 2.9 2.85 2.8 2.75

Safety factor q0(t)

l=(1,0)

l=(0,1)

Ω

ω>0

3yp ;U

nd

|Ω|

2

1.5

1

0.5

03 2.95 2.9 2.85 2.8 2.75

Safety factor q0(t)

l=(1,0)

l=(0,1)

ω<0

3yp xU

n1

)(b

(c)

(a)

(d)

0

0

0

0

0

0

0

No mode

Strong continuum damping

-8 -4 0 4 8

x-8 -4 0 4 8

x

Φ11 Φ12

Φ11

Φ12

3yp EU

nn

0

)(b

0

(c)

0

(a)

-8 -4 0 4 8

x

(d)

0

Φ11

0

-8 -4 0 4 8

x

Φ12

Φ11

Φ12

No mode

Strong continuum damping

0

0

3yp U