A Mathematical Framework for Critical Transitions:
From Bifurcations to Complex Systems
Christian Kuehn
Vienna University of Technology
Institute for Analysis and Scientific Computing
Outline
1. Introduction & Motivation
2. Fast-Slow Systems and Definitions
3. A Theorem on Scaling for Stochastic Systems
4. Examples (predator-prey, epidemics, epileptic seizures)
5. Stability and Instability in Complex Systems
6. More Examples (Games on Networks, Epidemics Data)
Outline
1. Introduction & Motivation
2. Fast-Slow Systems and Definitions
3. A Theorem on Scaling for Stochastic Systems
4. Examples (predator-prey, epidemics, epileptic seizures)
5. Stability and Instability in Complex Systems
6. More Examples (Games on Networks, Epidemics Data)
◮
Work on epileptic seizures joint with:
C. Meisel, MPI-PKS & CGC Hospital, Dresden
◮
Work on networks / complex systems joint with:
G. Zschaler, MPI-PKS, Dresden and T. Gross, U. Bristol
Critical Transitions and Tipping Points in Applications
Geoscience (climate change, climate subsystems, earthquakes)
◮ Lenton et al., Tipping elements in the Earth’s climate system. PNAS, 2008
◮ Wieczorek et al., Excitability in ramped systems. Proc. R. Soc. A, 2010
◮ Thompson and Sieber, Predicting climate tipping as a noisy bifurcation: a
review. IJBC, 2011
Ecology (extinction, desertification, ecosystem control)
◮ Drake and Griffen. Early warning signals of extinction in deteriorating
environments. Nature, 2010
◮ Dakos et al. Slowing down in spatially patterned systems at the brink of
collapse. Am. Nat., 2011
◮ Veraart et al., Recovery rates reflect distances to a tipping point in a living
system. Nature, 2012
Critical Transitions and Tipping Points in Applications
Geoscience (climate change, climate subsystems, earthquakes)
◮ Lenton et al., Tipping elements in the Earth’s climate system. PNAS, 2008
◮ Wieczorek et al., Excitability in ramped systems. Proc. R. Soc. A, 2010
◮ Thompson and Sieber, Predicting climate tipping as a noisy bifurcation: a
review. IJBC, 2011
Ecology (extinction, desertification, ecosystem control)
◮ Drake and Griffen. Early warning signals of extinction in deteriorating
environments. Nature, 2010
◮ Dakos et al. Slowing down in spatially patterned systems at the brink of
collapse. Am. Nat., 2011
◮ Veraart et al., Recovery rates reflect distances to a tipping point in a living
system. Nature, 2012
Human Physiology (epileptic seizures, asthma attacks)
◮ Mormann et al., Seizure prediction: the long and winding road. Brain, 2007
◮ Venegas et al., Self-organized patchiness in asthma as a prelude to catastrophic
shifts. Nature, 2005
Finance (market crashes, risk analysis, asset price bubbles)
◮ D. Sornette, Why Stock Markets Crash. PUP, 2003
◮ Jarrow et al., How to detect an asset bubble. SIAM J. Finan. Math., 2011
Engineering (voltage collapse, fracture)
◮ Chertkov et al., Voltage collapse and ODE approach to power flows. 2011
Goals and Perspectives...
x
x
(a)
(b)
t
t
Goals and Perspectives...
x
x
(a)
(b)
t
t
◮
Prediction of the “critical transition” or “tipping point”.
◮
Dynamical mechanisms and mathematical models.
◮
Time series analysis and data assimilation.
◮
Data availability, experiments, theory of extreme events, . . .
Typical Characteristics of Critical Transitions
Following Scheffer et al, Nature, 2009:
(1) A qualitative change from regular dynamics occurs.
(2) Rapid change in comparison to the regular dynamics.
(3) The system crosses a special threshold near a transition.
(4) The new state is far away from its previous state.
(5) There is often small noise i.e. observed data has a major
deterministic component with “random fluctuations”.
Typical Characteristics of Critical Transitions
Following Scheffer et al, Nature, 2009:
(1) A qualitative change from regular dynamics occurs.
(2) Rapid change in comparison to the regular dynamics.
(3) The system crosses a special threshold near a transition.
(4) The new state is far away from its previous state.
(5) There is often small noise i.e. observed data has a major
deterministic component with “random fluctuations”.
Towards a mathematical theory:
◮
Define critical transitions / tipping points using (1)-(5).
◮
Verify the known early-warning signs are indeed generic.
◮
Go beyond basic elements and extend the theory.
Identify Generic Models: Fast-Slow Systems
Fast variables x ∈ Rm , slow variables y ∈ Rn , time scale separation 0 < ǫ ≪ 1.
dx
dt
dy
dt
= x ′ = f (x, y )
= y ′ = ǫg (x, y )
ǫt=s
←→
ǫ dx
ds
dy
ds
↓ ǫ=0
x ′ = f (x, y )
y′ = 0
fast subsystem
= ǫẋ
= ẏ
= f (x, y )
= g (x, y )
↓ ǫ=0
0 = f (x, y )
ẏ = g (x, y )
slow subsystem
Identify Generic Models: Fast-Slow Systems
Fast variables x ∈ Rm , slow variables y ∈ Rn , time scale separation 0 < ǫ ≪ 1.
dx
dt
dy
dt
= x ′ = f (x, y )
= y ′ = ǫg (x, y )
ǫt=s
←→
ǫ dx
ds
dy
ds
↓ ǫ=0
x ′ = f (x, y )
y′ = 0
fast subsystem
= ǫẋ
= ẏ
= f (x, y )
= g (x, y )
↓ ǫ=0
0 = f (x, y )
ẏ = g (x, y )
slow subsystem
◮
C := {f = 0} = critical manifold = equil. of fast subsystem.
◮
C is normally hyperbolic if Dx f has no zero-real-part eigenvalues.
◮
Fenichel’s Theorem: Normal hyperbolicity ⇒ “nice” perturbation.
◮
Loss of Normal hyperbolicity ⇒ complicated dynamics.
Example - A fold bifurcation of the fast subsystem with slow drift:
= −y − x 2 ,
= 1.
ǫẋ
ẏ
0.2
y
0.2
candidate γ0
y
(a)
0
−0.2
γǫ
−0.2
C
−0.7
−1.2
(b)
O(ǫ2/3 )
C
0
x
1.2
−0.7
−1.2
0
Figure: (a) Singular limit ǫ = 0. (b) ǫ = 0.02.
x
1.2
A Natural Definition...
Let p = (xp , yp ) ∈ C0 where C0 is not normally hyperbolic then p
is a critical transition if ∃ a candidate γ0 and times tj−1 , tj s.t.
(C1) γ0 (tj−1 , tj ) ⊂ S0 ⊂ C0 , S0 normally hyperbolic attracting,
(C2) p = γ0 (tj ) is a transition point between subsystems,
(C3) and γ0 (tj−1 , tj ) is oriented from γ0 (tj−1 ) to γ0 (tj ).
A Natural Definition...
Let p = (xp , yp ) ∈ C0 where C0 is not normally hyperbolic then p
is a critical transition if ∃ a candidate γ0 and times tj−1 , tj s.t.
(C1) γ0 (tj−1 , tj ) ⊂ S0 ⊂ C0 , S0 normally hyperbolic attracting,
(C2) p = γ0 (tj ) is a transition point between subsystems,
(C3) and γ0 (tj−1 , tj ) is oriented from γ0 (tj−1 ) to γ0 (tj ).
◮
Covers (1)-(4) of the phenomenological features:
regular dynamics
rapid change
special threshold
new state is far away
↔
↔
↔
↔
slow subsystem flow
slow-to-fast transition
fast subsystem bifurcation (⇒ genericity)
measure on candidate orbit
A Natural Definition...
Let p = (xp , yp ) ∈ C0 where C0 is not normally hyperbolic then p
is a critical transition if ∃ a candidate γ0 and times tj−1 , tj s.t.
(C1) γ0 (tj−1 , tj ) ⊂ S0 ⊂ C0 , S0 normally hyperbolic attracting,
(C2) p = γ0 (tj ) is a transition point between subsystems,
(C3) and γ0 (tj−1 , tj ) is oriented from γ0 (tj−1 ) to γ0 (tj ).
◮
Covers (1)-(4) of the phenomenological features:
◮
regular dynamics
rapid change
special threshold
new state is far away
Extension to stochastic
↔ slow subsystem flow
↔ slow-to-fast transition
↔ fast subsystem bifurcation (⇒ genericity)
↔ measure on candidate orbit
systems immediate γǫ → γǫ,σ .
◮
γ0,0 yields B-tipping, γ0,σ N-tipping (→ Ashwin); γǫ,δ perturbations.
Natural: time series correspond to sample paths γǫ,σ .
◮
Extensions to PDEs, DDEs, etc. are possible via paths.
◮
What about Noise? - Stochastic Fast-Slow Systems...
Consider the fast-slow stochastic differential equation
dxs
dys
= 1ǫ f (xs , ys )ds +
= g (xs , ys )ds.
σ
√
F (xs , ys )dWs ,
ǫ
where Ws is standard Brownian motion.
(1)
What about Noise? - Stochastic Fast-Slow Systems...
Consider the fast-slow stochastic differential equation
dxs
dys
= 1ǫ f (xs , ys )ds +
= g (xs , ys )ds.
σ
√
F (xs , ys )dWs ,
ǫ
where Ws is standard Brownian motion.
Suppose (1) has a deterministic attracting slow manifold
Cǫ = {(x, y ) ∈ Rm+n : x = hǫ (y ) = h0 (y ) + O(ǫ)}
Theorem (Berglund and Gentz)
Sample paths of (1) stay near Cǫ with high probability.
(1)
Towards Early-Warning Signs
(W1) The system recovers slowly from perturbations: slowing down.
(W2) The autocorrelation increases before a transition.
(W3) The variance increases near a critical transition.
(W4) . . .
(W1)-(W3) are related.
Towards Early-Warning Signs
(W1) The system recovers slowly from perturbations: slowing down.
(W2) The autocorrelation increases before a transition.
(W3) The variance increases near a critical transition.
(W4) . . .
(W1)-(W3) are related.
Example: Deterministic fold bifurcation x ′ = −y − x 2 , y ′ = ǫ.
√
◮ Slow subsystem ǫ = 0, attracting critical manifold x =
−y.
◮
◮
Fast subsystem ǫ = 0, parameterized family x ′ = −y − x 2 .
Variational equation:
√
X ′ = −2 −yX
Slowing down as y → 0− .
⇒
√
X (t) = X (0)e −2
−yt
Useful Stochasticity - Variance near a Fold
dxt
dyt
1
ǫ (−yt
(xt , yt )
0
y
=
=
− xt2 ) dt +
1 dt.
Xt
C0
0.06
0.03
−0.2
σ
√
dWt ,
ǫ
Xt := xt −
√
(b)
−yt
0
(a)
−0.03
−0.4
−0.6
−0.6
−0.5
−0.4
−0.3
−0.2
y
−0.1
0
0.0003
(c)
Var
0.0002
−0.8
Var ∼ √1
as y → 0−
−0.6
−0.4
−y
0.0001
−1
−0.2
0
0.2
0.4
0.6
0.8
1
x
0
−0.7
−0.5
−0.3
−0.2
y
−0.1
Figure: (x0 , y0 ) = (0.9, −0.92) [red dot], σ = 0.01, ǫ = 0.01.
0
Sketch of Proof: Early-Warning for Folds (→ Berglund)
◮
◮
Focus on attracting slow manifold Cǫ = {x = h0 (y )}.
Variational equation for linearized process:
σ
1
dξsl = (−2h0 (ys )ξsl )ds + √ dWs
ǫ
ǫ
Sketch of Proof: Early-Warning for Folds (→ Berglund)
◮
◮
Focus on attracting slow manifold Cǫ = {x = h0 (y )}.
Variational equation for linearized process:
σ
1
dξsl = (−2h0 (ys )ξsl )ds + √ dWs
ǫ
ǫ
◮
Define Xs := σ −2 Var(ξsl ) and “observe”
ǫẊ
ẏ
◮
= −4h0 (y )X + 1,
= 1.
Conclusion (up to leading order)
σ2
=O
Var(xs ) = √
4 −y
as y → 0− and σ fixed.
1
√
−y
Main Result - Overview
Theorem (K. 2011)
Classification of generic critical transitions for all fast subsystem
bifurcations up to codimension two:
◮
Fold, Hopf, (transcritical), (pitchfork)
◮
Cusp, Bautin, Bogdanov-Takens
◮
Gavrilov-Guckenheimer, Hopf-Hopf
Main Result - Overview
Theorem (K. 2011)
Classification of generic critical transitions for all fast subsystem
bifurcations up to codimension two:
◮
Fold, Hopf, (transcritical), (pitchfork)
◮
Cusp, Bautin, Bogdanov-Takens
◮
Gavrilov-Guckenheimer, Hopf-Hopf
The main results are:
1. (Existence:) Conditions on slow flow to get a critical transition.
2. (Scaling:) Leading-order covariance scaling Hǫ (y ) for
Cov(xs ) = σ 2 [Hǫ (y )] + O(σ 4 ).
3. ((ǫ, σ)-expansion:) Higher-order calculations for the fold.
4. (Technique:) Covariance estimates without martingales.
Example 1: The Bazykin Predator-Prey System
dx1 =
dx2 =
h
i
x1 x2
2
1+y1 x1 − 0.01x1 i dt +
h
x1 x2
−x2 + 1+y
− y2 x22 dt +
1 x1
x1 −
σ1
√
dW (1) ,
ǫ
σ2
√
dW (2) ,
ǫ
dy1 = ǫg1 (x, y )dt,
dy2 = ǫg2 (x, y )dt,
0.4
y2
0.35
R2
LP
R1
0.3
0.25
H
0.2
0.15
0.1
0.3
BT
R3
LP
0.35
0.4
0.45
y1
Figure: Partial bifurcation diagram; ǫ = 0 = σ1 = σ2 .
0.5
Fits of Vi →
−3
1
x 10
−4
−4
6
x 10
1
Vi = Var(xi ) ↓
x 10
V2
V1 (b)
(c)
(a)
Vi
0.5
3
0.5
0
0.32
0.38
y1
0.44
0
0.32
0.37
y1
0.42
0
0.32
0.37
y1
0.42
Figure: Averaged over 50 sample paths (ǫ, σ) = (3 × 10−5 , 1 × 10−3 );
Vi = Var(xi (y )) for i ∈ 1, 2; V1 (red) V2 (black).
◮
Theory for Bogdanov-Takens point predicts:
O(1/y1 )
K
2
√
Cov(xs (y )) = σ
.
K
O(1/ −y1 )
Fits of Vi →
−3
1
x 10
−4
−4
6
x 10
1
Vi = Var(xi ) ↓
x 10
V2
V1 (b)
(c)
(a)
Vi
0.5
3
0.5
0
0.32
0.38
y1
0.44
0
0.32
0.37
y1
0.42
0
0.32
0.37
y1
0.42
Figure: Averaged over 50 sample paths (ǫ, σ) = (3 × 10−5 , 1 × 10−3 );
Vi = Var(xi (y )) for i ∈ 1, 2; V1 (red) V2 (black).
◮
Theory for Bogdanov-Takens point predicts:
O(1/y1 )
K
2
√
Cov(xs (y )) = σ
.
K
O(1/ −y1 )
◮
NOT normal form: hidden scaling law ⇒ “unpredictable”
√
(O(1/y1 ) + O(1/ −y1 ) = O(1/y1 ) as y1 → 0− )
Example 2: Epidemics on Adaptive Networks
SIS dynamics on adaptive networks [Gross et al., 2006]:
p
infected
1−p
infect
susceptible
r
recover
1−r
re-wire
w
1−w
A moment closure pair-approximation (labc =
− ri ,
i ′ = p( µ2 − lII − lSS ) (lII )′ = p( µ2 − lII − lSS )
(lSS
)′
= (r +
w )( µ2
µ
−lII −lSS
2
1−i
− lII − lSS ) −
lab lbc
b )
yields:
+ 1 − 2rlII ,
2p( µ2 −lII −lSS )lSS
.
1−i
#links
where we assume µ = 2 #nodes
= 20, r = 0.002 and w = 0.4.
A moment closure pair-approximation (labc =
− ri ,
i ′ = p( µ2 − lII − lSS ) (lII )′ = p( µ2 − lII − lSS )
(lSS
)′
= (r +
w )( µ2
µ
−lII −lSS
2
1−i
− lII − lSS ) −
lab lbc
b )
yields:
+ 1 − 2rlII ,
2p( µ2 −lII −lSS )lSS
.
1−i
#links
where we assume µ = 2 #nodes
= 20, r = 0.002 and w = 0.4.
Often moment-closure works, and sometimes it doesn’t but
certainly there is finite-size noise:
σ1
dW (1) ,
dx1 = 1ǫ y ( µ2 − x2 − x3 ) − rx1 ds + √
ǫ
i
µ
h
−x2 −x3
σ2
(2) ,
dx2 = 1ǫ y ( µ2 − x2 − x3 ) 2 1−x
+
1
−
2rx
2 ds + √ǫ dW
1
h
i
2y ( µ2 −x2 −x3 )x3
σ3
dx3 = 1ǫ (r + w )( µ2 − x2 − x3 ) −
dW (3) ,
ds + √
1−x1
ǫ
dy
= 1 ds,
Results on Adaptive SIS Epidemics
−5
9
0.5
Var
(a)
x1
4
x 10
12
(b)
x1
x2
x3
x 10
1
Var
(c)
5
6
Var ∼
1
y−ytc
tc
0
Cr
Ca
0
0.025
y
0.05
0
0
0.025
y
0.05
0
0
0.025
y
0.05
Results on Adaptive SIS Epidemics
−5
9
0.5
Var
(a)
x1
4
x 10
12
(b)
x1
x2
x3
x 10
1
Var
(c)
5
6
Var ∼
1
y−ytc
tc
0
Cr
Ca
0
◮
0.025
y
0.05
0
0
0.025
y
0.05
0
0
0.025
y
0.05
Observe theory for scaling law for the transcritical bifurcation
dxs = (xs ys − xs2 )ds + σdW
Var(xs ) = σ 2 O (ys − ytc )−1
as ys → ytc .
◮
Important: Early-warning sign in the link density only!
◮
Also observe a delay (→ way-in way-out function).
Example 3: ECoG Data and Epileptic Seizures
x
x
1
Var
1
Var
(b)
(a)
x
x
1
Var
1
Var
(d)
(c)
Figure: Avg. ECoG data (blue), seizure (black dashed), scaling law (red).
Example 3: ECoG Data and Epileptic Seizures
x
x
1
Var
1
Var
(b)
(a)
x
x
1
Var
1
Var
(d)
(c)
Figure: Avg. ECoG data (blue), seizure (black dashed), scaling law (red).
◮
Scaling law + genericity ⇒ Hopf (→ Terry, U. Sheffield).
Example 3: ECoG Data and Epileptic Seizures
x
x
1
Var
1
Var
(b)
(a)
x
x
1
Var
1
Var
(d)
(c)
Figure: Avg. ECoG data (blue), seizure (black dashed), scaling law (red).
◮
Scaling law + genericity ⇒ Hopf (→ Terry, U. Sheffield).
◮
Early-warning signs for excitable systems (FitzHugh-Nagumo).
Comparison: neuron (micro), cluster (meso), network (macro).
Network measures based upon wavelet decomposition.
◮
◮
Critical Transitions in (unstructured) Complex Systems
Consider x ∈ RN , N ≫ 1
dx
= f (x).
dt
At a steady state x ∗ , we have
f (x ∗ ) = 0,
A = Df (x ∗ ) determines stability.
Critical Transitions in (unstructured) Complex Systems
Consider x ∈ RN , N ≫ 1
dx
= f (x).
dt
At a steady state x ∗ , we have
f (x ∗ ) = 0,
◮
A = Df (x ∗ ) determines stability.
Large system (→ May) take A a random real symmetric
matrix, use semi-circle law
N−1
N
1
1
∗
∗
,
P(x saddle) = 1−
.
P(x (un-)stable) =
2
2
Critical Transitions in (unstructured) Complex Systems
Consider x ∈ RN , N ≫ 1
dx
= f (x).
dt
At a steady state x ∗ , we have
f (x ∗ ) = 0,
A = Df (x ∗ ) determines stability.
◮
Large system (→ May) take A a random real symmetric
matrix, use semi-circle law
N−1
N
1
1
∗
∗
,
P(x saddle) = 1−
.
P(x (un-)stable) =
2
2
◮
Needs strong assumptions on structure of the matrix A!
Proof of full circular law (→ Tao and Van Vu 2008).
◮
Expected abundance of saddle points in complex systems.
Metastability and Critical Transitions near Saddle Points
Trivial case: planar saddle in R2 at x = (0, 0) locally
⇒ x(t) = y1 (0)e λs t v1 + y2 (0)e λu t v2 .
x ′ = Ax
3
3
(a)
x2
−2
(b)
x1,2
(c)
d
−4
2
2
−6
1
1
−8
0
−0.5
−0.5
B
0
1
2
x1
3
0
0
2
4
6
8
10
t
12
−10
0
2
4
6
8
10
12
t
Metastability and Critical Transitions near Saddle Points
Trivial case: planar saddle in R2 at x = (0, 0) locally
⇒ x(t) = y1 (0)e λs t v1 + y2 (0)e λu t v2 .
x ′ = Ax
3
3
(a)
x2
−2
(b)
x1,2
(c)
d
−4
2
2
−6
1
1
−8
0
−0.5
−0.5
B
0
1
2
x1
3
0
0
2
4
6
8
10
t
12
−10
0
2
4
6
If x(0) close to W s (0) then logarithmic distance reduction
ln kx(t2 ) − x(t1 )k ≈ λs t1 + k1 .
8
10
12
t
Example 4: Evolutionary Game on a Network
Agents/nodes play snowdrift game each time step
b − c/2 b − c
M=
c=cost, b=benefit.
b
0
Re-wire (p), adopt (1 − p) based upon performance.
Example 4: Evolutionary Game on a Network
Agents/nodes play snowdrift game each time step
b − c/2 b − c
M=
c=cost, b=benefit.
b
0
Re-wire (p), adopt (1 − p) based upon performance.
1
(a2)
x1,2
0.5
0.2
0.3
p
0
0.4
0.5
0.5
0.6
V1
0
5
(b1)
0.7
p
d1
(b2)
0.1
V2
d2
0
0.15
0.2
0.25
0
0.93
0.8
p
0.3
0.6
0.94
0.95
p
0.96
−3
1
T 0.2
2
(a3)
x1,2
0.5
0
0.1
0
0.1
1
1
(a1)
x1,2
0.7
0.8
0.9
p
1
−6
(b3)
−3
−6
0.93
0.94
0.95
p
0.96
Example 4: Evolutionary Game on a Network
Agents/nodes play snowdrift game each time step
b − c/2 b − c
M=
c=cost, b=benefit.
b
0
Re-wire (p), adopt (1 − p) based upon performance.
1
1
1
(a1)
x1,2
(a2)
x1,2
0.5
0.5
0.5
0
0.1
0.2
0.3
p
0
0.4
0.5
0.6
V1
0
5
(b1)
0.7
p
d1
(b2)
0.1
V2
d2
0
0
0.1
0.15
0.2
0.25
0
0.93
0.8
p
0.3
0.6
0.94
0.95
p
0.96
−3
1
T 0.2
2
(a3)
x1,2
0.7
0.8
0.9
p
1
−6
(b3)
−3
−6
0.93
0.94
0.95
p
◮
For near-full cooperation with high re-wiring: saddle point.
◮
Early-warning signs: period blow-up and log-distance.
0.96
Example 5: Back to Epidemics
Measles epidemics in cities in the UK between 1944 and 1966.
1
I
(a)
0.8
rc
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
1944
1949
1954
1959
t
1964
0
(b)
0
0.2
0.4
0.6
0.8
rf
Figure: (a) Example time series. (b) ROC(d)-curve (dots, crosses =
different forecast lengths); blue=true instability, red=weak stability.
1
Example 5: Back to Epidemics
Measles epidemics in cities in the UK between 1944 and 1966.
1
I
(a)
0.8
rc
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
1944
1949
1954
1959
t
1964
0
(b)
0
0.2
0.4
0.6
0.8
rf
Figure: (a) Example time series. (b) ROC(d)-curve (dots, crosses =
different forecast lengths); blue=true instability, red=weak stability.
Test logarithmic distance indicator to estimate λu , threshold d.
Receiver-operator-characteristic curve
rc =
#correct predictions
#events/outbreaks
and rf =
#false positives
.
#non-events
1
Overview & Conclusions
Mathematical Theory:
◮ Characterization and definition of critical transitions.
◮ Useful noise: proof of scaling laws up to codimension two.
◮ Hidden laws, coarse-grained networks.
◮ Saddle points as critical transitions.
◮ Logarithmic distance and period blow-up as warning signs.
◮ Data analysis, excitable systems, multiplicative noise, . . .
Overview & Conclusions
Mathematical Theory:
◮ Characterization and definition of critical transitions.
◮ Useful noise: proof of scaling laws up to codimension two.
◮ Hidden laws, coarse-grained networks.
◮ Saddle points as critical transitions.
◮ Logarithmic distance and period blow-up as warning signs.
◮ Data analysis, excitable systems, multiplicative noise, . . .
Applications:
◮ Social connections could be crucial for epidemic prediction.
◮ Epileptic seizures and evidence for Hopf bifurcation.
◮ Economic networks: critical transitions via bursts of defectors.
◮ (systems biology, biomechanics and controllability, . . .)
Overview & Conclusions
Mathematical Theory:
◮ Characterization and definition of critical transitions.
◮ Useful noise: proof of scaling laws up to codimension two.
◮ Hidden laws, coarse-grained networks.
◮ Saddle points as critical transitions.
◮ Logarithmic distance and period blow-up as warning signs.
◮ Data analysis, excitable systems, multiplicative noise, . . .
Applications:
◮ Social connections could be crucial for epidemic prediction.
◮ Epileptic seizures and evidence for Hopf bifurcation.
◮ Economic networks: critical transitions via bursts of defectors.
◮ (systems biology, biomechanics and controllability, . . .)
For references see my webpage (and the arXiv):
◮ http://www.asc.tuwien.ac.at/∼ckuehn/
Overview & Conclusions
Mathematical Theory:
◮ Characterization and definition of critical transitions.
◮ Useful noise: proof of scaling laws up to codimension two.
◮ Hidden laws, coarse-grained networks.
◮ Saddle points as critical transitions.
◮ Logarithmic distance and period blow-up as warning signs.
◮ Data analysis, excitable systems, multiplicative noise, . . .
Applications:
◮ Social connections could be crucial for epidemic prediction.
◮ Epileptic seizures and evidence for Hopf bifurcation.
◮ Economic networks: critical transitions via bursts of defectors.
◮ (systems biology, biomechanics and controllability, . . .)
For references see my webpage (and the arXiv):
◮ http://www.asc.tuwien.ac.at/∼ckuehn/
Thank you for your attention.