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2265-17
Advanced School on Understanding and Prediction of Earthquakes and other Extreme Events in Complex Systems
Arkady Kryazhimskiy
26 September - 8 October, 2011
International Institute for Applied Systems Analysis Laxenburg
Austria
Recovery Times as Indicators of Stability Loss
Recovery Times as Indicators of Stability Loss
Trieste, October 4, 2011
Arkady Kryazhimskiy
Loss in stability
Loss in stability
Loss in stability
catastrophe
Loss in stability
Loss in stabilityGrowth in recovery time
Shift of steady stateCatastrophe
catastrophe
Mathematical interpretation
Mathematical interpretation
)(xfx Steady state
)()()( 00 xfxfxf c
0)( 0 xf
0xx
)( 0xfc
StateRate of change
Mathematical interpretation
0
Stability
c )( 0xfc
0c
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
Mathematical interpretation c )( 0xfc
time
)exp(0 ct 0
0
0cc grows
catastrophe
Mathematical interpretation
time
)exp(0 ct 0c0
0
0c
c grows
catastrophe
Mathematical interpretation
time
)exp(0 ct 0c0
0
0ccatastrophe
c grows
Mathematical interpretation
time
)exp(0 ct 0c0
0
0c
c grows
catastrophe
Multi-dimensional system
Multi-dimensional system
Tnxxx ),...,( 1)(xfx ii
Tnxx ),...,( 1
0
ikckn
k k
ii x
xf
1
0 )(
)( ikcC C
Multi-dimensional systemyCy C
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
2
1
Multi-dimensional systemyCy C
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
2
11y
2y
Multi-dimensional systemyCy C
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
2
11y
2y
Multi-dimensional systemyCy C
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
2
11y
2y
Multi-dimensional systemyCy C
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
2
11y
2y
Multi-dimensional systemyCy C
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
2
11y
2y
Multi-dimensional systemyCy C
1
2
2y
1y
nyy ,...,1 n ...0 1
222111 )exp()exp( ytyt
Multi-dimensional systemyCy C
1
2
2y
1y
222111 )exp()exp( ytyt 0|| 0
Multi-dimensional systemyCy C
1
2
2y
1y
222111 )exp()exp( ytyt 0
|| 0
|| 0
Multi-dimensional systemyCy C
1
2
2y
1y
222111 )exp()exp( ytyt 0
-recovery time
|| 0
|| 0
),( 0
Multi-dimensional system
-recovery time ),( 0
Expected recovery time
-recovery time ),( 0
Expected recovery time
1
20
),( 0
-recovery time ),( 0
Expected recovery time
1
20
),( 0
-recovery time ),( 0
Expected recovery time
1
20
),( 0
-recovery time ),( 0
Expected recovery time
1
20
),( 0
kE k /)],(...),([),( 001
-recovery time ),( 0
Slow drift to catastrophe
Slow drift to catastrophe
slow time
random shock-recovery tests
),( Esmall
catastrophe
)(xfx p pC 01 p
Most vulnerable coordinate
|| 0i
time
Most vulnerable coordinate
|| 0i
|| 0i
time
Most vulnerable coordinate
|| 0i
|| 0i
time),( 0 i
Most vulnerable coordinate
|| 0i
|| 0i
),( 0 i time
Most vulnerable coordinate
|| 0i
|| 0i
),( 0 i time
Most vulnerable coordinate
|| 0i
|| 0i
),( 0 i
),( iE
nn yy )(...)(0
10
10
time
)(||log)](||log[),(1
100
oyE iiii
Most vulnerable coordinate
nn yy )(...)(0
10
10
)(||log||log
),(),(1
11
oyy
EE ijji
0i identically distributedAssumption:
||max|| 11 iik yy k most vulnerable
Most vulnerable coordinate
)(||log)](||log[),(1
100
0 oy iiii
Slow drift to catastrophe
slow time
random shock-recovery tests
),( iEsmall
)(xfx p pC 01 p
)(||log)](||log[),(1
100
0 oy iiii
Slow drift to catastrophe
slow time
random shock-recovery tests
),( iEsmall
catastrophek most vulnerable
),( kE
)(xfx p pC 01 p
Thank you
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