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![Page 1: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/1.jpg)
Some New Frontiers in Mathematical Tipping Point Theory
Christian Kuehn
Vienna University of TechnologyInstitute for Analysis and Scientific Computing
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Overview: Critical Transitions / Tipping Points
Possible paradigm:
◮ Qualitative theory: bifurcations, noise-induced jumps, etc.quite well understood...
![Page 3: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/3.jpg)
Overview: Critical Transitions / Tipping Points
Possible paradigm:
◮ Qualitative theory: bifurcations, noise-induced jumps, etc.quite well understood...
◮ Quantitative theory: scaling laws, early-warning signs, etc.a lot more open problems...
![Page 4: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/4.jpg)
Overview: Critical Transitions / Tipping Points
Possible paradigm:
◮ Qualitative theory: bifurcations, noise-induced jumps, etc.quite well understood...
◮ Quantitative theory: scaling laws, early-warning signs, etc.a lot more open problems...
Example: “Scaling of saddle-node bifurcations: degeneracies and rapid quantitative
changes”, CK, J. Phys. A, 42(4), 045101, 2009
![Page 5: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/5.jpg)
Overview: Critical Transitions / Tipping Points
Possible paradigm:
◮ Qualitative theory: bifurcations, noise-induced jumps, etc.quite well understood...
◮ Quantitative theory: scaling laws, early-warning signs, etc.a lot more open problems...
Example: “Scaling of saddle-node bifurcations: degeneracies and rapid quantitative
changes”, CK, J. Phys. A, 42(4), 045101, 2009
Topics today:
1. Noise-induced transitions for waves in SPDEs.
2. Self-organized criticality in adaptive networks.
3. A different view on large data.
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Topic 1: Spatio-Temporal Stochastic Dynamics
◮ Bounded domain → ’finite-dim.’ bifurcations, warning signs.
◮ Unbounded domain → ???
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Topic 1: Spatio-Temporal Stochastic Dynamics
◮ Bounded domain → ’finite-dim.’ bifurcations, warning signs.
◮ Unbounded domain → ???
Natural class to study (evolution SPDE):
∂u
∂t=
∂2u
∂x2+ f (u) + ’noise’, u = u(x , t).
![Page 8: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/8.jpg)
Topic 1: Spatio-Temporal Stochastic Dynamics
◮ Bounded domain → ’finite-dim.’ bifurcations, warning signs.
◮ Unbounded domain → ???
Natural class to study (evolution SPDE):
∂u
∂t=
∂2u
∂x2+ f (u) + ’noise’, u = u(x , t).
Example: Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP):
∂u
∂t=
∂2u
∂x2+ u(1− u).
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Background - FKPP
∂u
∂t=
∂2u
∂x2+ u(1− u).
◮ Model for waves u = u(x − ct) in biology, physics, etc.
◮ Take x ∈ R and localized initial condition u(x , t = 0).
◮ Many variants e.g. Nagumo PDE.
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Background - FKPP
∂u
∂t=
∂2u
∂x2+ u(1− u).
◮ Model for waves u = u(x − ct) in biology, physics, etc.
◮ Take x ∈ R and localized initial condition u(x , t = 0).
◮ Many variants e.g. Nagumo PDE.
Basic propagating front(s):
◮ u ≡ 0 and u ≡ 1 are stationary.
◮ Wave connecting the two states:
u(η) = u(x − ct), limη→∞
u(η) = 1, limη→∞
u(η) = 0.
◮ Propagation into unstable state u = 0 since
Duf = Du[µu(1− u)] ⇒ Duf (0) = (µ− 2u)|u=0 > 0.
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SPDE version of FKPP
∂u
∂t=
∂2u
∂x2+ u(1− u) + σg(u) ξ(x , t), σ > 0.
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SPDE version of FKPP
∂u
∂t=
∂2u
∂x2+ u(1− u) + σg(u) ξ(x , t), σ > 0.
Possible choices for ’noise process’ ξ(x , t)
◮ white in time ξ = B, E[B(t)B(s)] = δ(t − s)
◮ space-time white ξ = W , E[W (x , t)W (y , s)] = δ(t − s)δ(x − y)
◮ Q-trace-class noise
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SPDE version of FKPP
∂u
∂t=
∂2u
∂x2+ u(1− u) + σg(u) ξ(x , t), σ > 0.
Possible choices for ’noise process’ ξ(x , t)
◮ white in time ξ = B, E[B(t)B(s)] = δ(t − s)
◮ space-time white ξ = W , E[W (x , t)W (y , s)] = δ(t − s)δ(x − y)
◮ Q-trace-class noise
Possible choices for ’noise term’ g(u)
◮ g(u) = u, ad-hoc (Elworthy, Zhao, Gaines,...)
◮ g(u) =√2u, contact-process (Bramson, Durrett, Muller, Tribe,... )
◮ g(u) =√
u(1− u), capacity (Muller, Sowers,... )
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Propagation Failure
FKPP SPDE exhibits propagation failure
∂u
∂t=
∂2u
∂x2+ u(1− u) + σg(u) ξ(x , t), g(0) = 0.
i.e. solution may get absorbed into u ≡ 0.
xxx ttt
uuu (a) (b) (c)
Figure : g(u) = u, ξ = B . (a) σ = 0.02, (b) σ = 0.3 and (c) σ = 1.2.
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Scaling near transition: single-point observer statistics:
u =1
T − t0
∫ T
t0
u(0, t) dt, Σ =
[
1
T − t0
∫ T
t0
(u(0, t)− u)2 dt
]1/2
.
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Scaling near transition: single-point observer statistics:
u =1
T − t0
∫ T
t0
u(0, t) dt, Σ =
[
1
T − t0
∫ T
t0
(u(0, t)− u)2 dt
]1/2
.
0 0.5 1 1.5 2
0
0.4
0.8
1.2
0 0.5 1 1.5 20
1
2
u
c
u + Σ
u − Σ
σ
σ
Figure : Average over 200 sample paths; t ∈ [10, 20].
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More on Stochastic FKPP - Further Results
◮ other noise terms g(u)
◮ other noise processes ξ(x , t)
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More on Stochastic FKPP - Further Results
◮ other noise terms g(u)
◮ other noise processes ξ(x , t)
◮ Allee effect / Nagumo nonlinearity
◮ noncompact support for initial condition
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More on Stochastic FKPP - Further Results
◮ other noise terms g(u)
◮ other noise processes ξ(x , t)
◮ Allee effect / Nagumo nonlinearity
◮ noncompact support for initial condition
◮ initial transients as warning signs
xxx ttt
uuu (a) (b) (c)
Figure : Nagumo, g(u) = u, changing nonlinearity.
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Topic 2: Self-organized Criticality in Adaptive Networks
◮ Adaptive networks with simple local rules can self-organize.
◮ Steady state is “critical” (near a ’phase transition’).
◮ Suggestion: optimal information processing.
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Topic 2: Self-organized Criticality in Adaptive Networks
◮ Adaptive networks with simple local rules can self-organize.
◮ Steady state is “critical” (near a ’phase transition’).
◮ Suggestion: optimal information processing.
0 15000 300001
2
3
4
1 2 3 40.0
0.5
1.0
t
K
K
C(a) (b)
Figure : SOC example. K = average connectivity. C = frozen fraction.
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Modified Bornholdt-Rohlf Boolean Network
1. nodes vi (t) ∈ {±1}, directed edges eij(t) ∈ {−1, 0,+1}.2. dynamical update rule (t = 0, random graph), define
fi (t) =∑
j
eij(t)vj(t) + µvi (t) + σri , ~r ∼ N (0, 1)
vi (t + 1) =
{
sgn[fi (t)] if fi (t) 6= 0,vi (t) if fi (t) = 0.
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Modified Bornholdt-Rohlf Boolean Network
1. nodes vi (t) ∈ {±1}, directed edges eij(t) ∈ {−1, 0,+1}.2. dynamical update rule (t = 0, random graph), define
fi (t) =∑
j
eij(t)vj(t) + µvi (t) + σri , ~r ∼ N (0, 1)
vi (t + 1) =
{
sgn[fi (t)] if fi (t) 6= 0,vi (t) if fi (t) = 0.
3. Tv node dynamics steps, Ta := ⌊Tv/2⌋, measure activity
Ai :=1
Tv − Ta
Tv−1∑
t=Ta
vi (t)
.
4. topological update rule, choose one node i randomly
|Ai | > 1− δ create an edge eij(t) 6= 0,|Ai | ≤ 1− δ delete an edge eij(t) = 0.
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SOC Ingredients & Observations
Ingredients:
◮ Large time scale separation Tv = 1/ǫ ≫ 1 needed
topology dynamics ↔ slow node dynamics ↔ fast.
◮ SOC is robust to small noise 0 < σ ≪ 1.
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SOC Ingredients & Observations
Ingredients:
◮ Large time scale separation Tv = 1/ǫ ≫ 1 needed
topology dynamics ↔ slow node dynamics ↔ fast.
◮ SOC is robust to small noise 0 < σ ≪ 1.
Observations:
◮ Steady state near fast subsystem bifurcation point?!
◮ Information processing ↔ perturbations ↔ finite-time.
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SOC Ingredients & Observations
Ingredients:
◮ Large time scale separation Tv = 1/ǫ ≫ 1 needed
topology dynamics ↔ slow node dynamics ↔ fast.
◮ SOC is robust to small noise 0 < σ ≪ 1.
Observations:
◮ Steady state near fast subsystem bifurcation point?!
◮ Information processing ↔ perturbations ↔ finite-time.
Question: Are there optimal values of (ǫ, σ)?
◮ Yes for ǫ (’time-scale resonance (TR)’)
◮ Yes for σ (’steady-state stochastic resonance (SSR)’)
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Back to the Bornholdt-Rohlf Model... and Noise
0.1 0.2 0.3 0.4 0.50
1
2
3
0.1 0.2 0.3 0.4 0.50.1
0.2
0.3
0.4
0.5
EK (a) (b)KT
σ σ
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Back to the Bornholdt-Rohlf Model... and Noise
0.1 0.2 0.3 0.4 0.50
1
2
3
0.1 0.2 0.3 0.4 0.50.1
0.2
0.3
0.4
0.5
EK (a) (b)KT
σ σ
◮ Non-monotone error, small noise → noise optimality.
◮ SOC tipping, large noise → noise-induced phase transition.
![Page 29: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/29.jpg)
◮ First thought: It is just stochastic resonance.
◮ Second throught: No, since we have SOC steady state.
![Page 30: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/30.jpg)
◮ First thought: It is just stochastic resonance.
◮ Second throught: No, since we have SOC steady state.
dx = (yx − x3)dt + σdWdy = ǫ(x∗ − |x |)dt
0 0.1 0.2
−0.1
0
0.1
10−6
10−4
10−2
0.02
0.03
0.04
Ex
x
y(a)
(b)
σ
![Page 31: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/31.jpg)
◮ First thought: It is just stochastic resonance.
◮ Second throught: No, since we have SOC steady state.
dx = (yx − x3)dt + σdWdy = ǫ(x∗ − |x |)dt
0 0.1 0.2
−0.1
0
0.1
10−6
10−4
10−2
0.02
0.03
0.04
Ex
x
y(a)
(b)
σ
Important new concept - steady-state stochastic resonance (SSR).
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Topic 3: Remark on Large Data Sets
◮ Approach 1: Data assimilation into large-scale models.
◮ Approach 2: Abstract scaling laws and genericity.
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Topic 3: Remark on Large Data Sets
◮ Approach 1: Data assimilation into large-scale models.
◮ Approach 2: Abstract scaling laws and genericity.
Examples of Approach II:
1. Metastability, saddle points, epidemics (CK, Zschaler, Gross)
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Topic 3: Remark on Large Data Sets
◮ Approach 1: Data assimilation into large-scale models.
◮ Approach 2: Abstract scaling laws and genericity.
Examples of Approach II:
1. Metastability, saddle points, epidemics (CK, Zschaler, Gross)
2. Epileptic seizures and Hopf bifurcation (Meisel, CK)
![Page 35: Some New Frontiers in Mathematical Tipping Point Theory · Some New Frontiers in Mathematical Tipping Point Theory ChristianKuehn Vienna University of Technology Institute for Analysis](https://reader034.vdocument.in/reader034/viewer/2022042313/5ede1be4ad6a402d6669655a/html5/thumbnails/35.jpg)
Topic 3: Remark on Large Data Sets
◮ Approach 1: Data assimilation into large-scale models.
◮ Approach 2: Abstract scaling laws and genericity.
Examples of Approach II:
1. Metastability, saddle points, epidemics (CK, Zschaler, Gross)
2. Epileptic seizures and Hopf bifurcation (Meisel, CK)
3. Social networks, known events (CK, Martens, Romero)
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References(1) CK, A mathematical framework for critical transitions: bifurcations, fast-slow
systems and stochastic dynamics, Physica D: Nonlinear Phenomena, Vol. 240,No. 12, pp. 1020-1035, 2011
(2) CK, A mathematical framework for critical transitions: normal forms, variance
and applications, Journal of Nonlinear Science, Vol. 23, No. 3, pp. 457-510,2013
(3) CK, Warning signs for wave speed transitions of noisy Fisher-KPP invasion
fronts, Theoretical Ecology, Vol. 6, No. 3, pp. 295-308, 2013
(4) CK, Time-scale and noise optimality in self-organized critical adaptive networks,Physical Review E, Vol. 85, No. 2, 026103, 2012
(5) C. Meisel and CK, Scaling effects and spatio-temporal multilevel dynamics in
epileptic seizures, PLoS ONE, Vol. 7, No. 2, e30371, 2012
(6) CK, E.A. Martens and D. Romero, Critical transitions in social network activity,arXiv:1307.8250, 2013
(7) CK, G. Zschaler and T. Gross, Early warning signs for critical saddle-escape in
complex systems, preprint, 2013
For more references see also:
◮ http://www.asc.tuwien.ac.at/∼ckuehn/
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References(1) CK, A mathematical framework for critical transitions: bifurcations, fast-slow
systems and stochastic dynamics, Physica D: Nonlinear Phenomena, Vol. 240,No. 12, pp. 1020-1035, 2011
(2) CK, A mathematical framework for critical transitions: normal forms, variance
and applications, Journal of Nonlinear Science, Vol. 23, No. 3, pp. 457-510,2013
(3) CK, Warning signs for wave speed transitions of noisy Fisher-KPP invasion
fronts, Theoretical Ecology, Vol. 6, No. 3, pp. 295-308, 2013
(4) CK, Time-scale and noise optimality in self-organized critical adaptive networks,Physical Review E, Vol. 85, No. 2, 026103, 2012
(5) C. Meisel and CK, Scaling effects and spatio-temporal multilevel dynamics in
epileptic seizures, PLoS ONE, Vol. 7, No. 2, e30371, 2012
(6) CK, E.A. Martens and D. Romero, Critical transitions in social network activity,arXiv:1307.8250, 2013
(7) CK, G. Zschaler and T. Gross, Early warning signs for critical saddle-escape in
complex systems, preprint, 2013
For more references see also:
◮ http://www.asc.tuwien.ac.at/∼ckuehn/
Thank you for your attention.