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