Observing emerging bifurcations in complex systems Jan Sieber - - PowerPoint PPT Presentation

Observing emerging bifurcations in complex systems

Jan Sieber

University of Exeter (UK)

Outline

◮ two didactic examples

◮ agent-based simulation — traders ◮ disease spreading on network

◮ definition of macroscopic unstable equilibria and

bifurcations

◮ folds & Hopf bifurcations

Agent-based simulation — traders [Siettos et al, EPL 2012]

◮ N traders, buying & selling ◮ each trader k has internal state sk, evolving

sk,new =      e−γΔtsk + p+

k ϵ+ − p− k ϵ−

if |sk| < 1 reset if sk ≥ 1 ⇐buy reset if sk ≤ −1 ⇐sell

◮ p± k random number of news ∼ Pois(n±(1 + gR±))Δt ◮ R± avg rate of buys/sells over past period T ◮ g gain ◮ ϵ± jump size ◮ n± rate of good/bad news other than buys/sells

Agent-based simulation — traders

⇒Matlab animation

Example: collective behaviour of agents

0.5 1 1.5 2 2.5 −0.02 0.02 0.04 0.06 0.08

20 40 60 80 100 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14

time profile R+(t) − R−(t)

amplification g R = R+ − R− Equilibrium Rate buying−selling rate

Definition of equilibrium?

◮ positive feedback

◮ others buying ⇒good news ⇒buy ◮ others selling ⇒bad news ⇒sell

◮ stochastic system has stationary density with (in

projection) 3 well-separated local maxima stable equilibria:

◮ everyone buys as fast as possible ◮ everyone sells as fast as possible ◮ balance

◮ Balance Req = R+ eq − R+ eq: in the long run

men R = Req: Req = lim

t→∞ mens∈[t,t+T] R(s)

(mean of conditional stationary density)

Proposed definition of (unstable) equilibrium Include feedback loop: bias p = [ϵ+ − ϵ−](t) = k[R(t) − Rref] Rref is equilibrium if Rref = lim

t→∞ mens∈[t,t+T] R(s) ◮ The long-time mean of feedback loop input is zero. ◮ For large numbers N of traders:

◮ E[R(t) − Rref]2 →N→∞,t→∞ 0 ◮ Resulting equilibrium Rref independent of

choice of feedback loop for N → ∞.

Bifurcation diagram in bias parameter

2nd example — disease spreading on network [Gross et al, PRL 2006]

◮ network with N nodes (individuals) with state either

S (susceptible) or  (infected)

◮ kN links (initially random, k ∼ 10) ◮ at every step:

◮  individual recovers with probability r ◮ infection travels along S link (infects S node)

with probability p

◮ S link is rewired (keep S node, replace  node

by random other S node) with probability 

◮ system has parameter range where disease-free

and endemic equilibrium coexist

2nd example — disease spreading on network [Gross et al, PRL 2006]

0.002 0.004 0.006 0.008 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

infection rate p Equilibrium fraction of infected eq Parameter sweep

Proposed definition of (unstable) equilibrium

◮ Choose reference fraction of infected ref ◮ at every step:

if  < ref, infect ref −  individuals along S links if  > ref, “cure”  − ref individuals

◮ ref is equilibrium value if

men artificically cured = men artificially infected after transients have settled

Proposed definition of (unstable) equilibrium

0.5 1 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5

measurements regression uncertainty equilibria

reference fraction of infected ref mean ”cured”− artificially infected N = 400, infection rate p = 0.001 mean control input & regression curve

Newton iteration & continuation with uncertainty



measurement with error bar Linear regression with Gaussian process

y

linear regression curve yr() with error bar root r of regression curve yr with error bar (easy to find with Newton iteration)

r

Procedure for continuation with uncertainty

• 1. find roots (bifurcations) of regression curve yr()
• 2. determine where to measure next:

◮  for which measurement yr() ± σr() would

minimize error bar of root for updated yr, or

◮  where measurement yr() + σr() changes

root the most Both are nonlinear optimization problems on current regression curve yr (cheap in principle).

• 3. optimal new  not necessary, only sensible 
• 4. stop if expected effect on  is not worth additional

measurement.

Example – traders fold continuation 2 system parameters:

◮ bias ϵ+ − ϵ− (also control input) ◮ self-referentialness g

⇒ 2 base variables: Rref, g. Run with feedback: bias = [ϵ+ − ϵ−](t) = k[R(t) − Rref] after transients, read off

◮ men[ϵ+ − ϵ−], ◮ Req = men R

⇒equilibrium surface in space (g, ϵ+ − ϵ−, Req) fold condition: ∂Req ∂Rref (Rref, g) = 1

Matlab demo

Example – traders fold continuation Comments

◮ during continuation regression surface always

evaluated near boundary ⇒ results less accurate (larger uncertainty) ⇒ large correlation parameter

◮ at end standard continuation for entire regression

surface ⇒ more accurate (interpolation) ⇒ small correlation parameter

Disease on network equilibrium continuation

1 2 3 4 5 x 10

−3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fraction of infected  N = 400 infection rate p men  < 0 +5e-3 men  > 0

• 5e-3

close to tanscritical: long transients

Oscillations

Ring of nonlocally coupled phase oscillators

Chimeras

Abrams Strogatz Laing I Omelchenko O Omelchenko Zakharova Wolfrum Schoell ...

Chimeras

Abrams Strogatz Laing I Omelchenko O Omelchenko Zakharova Wolfrum Schoell ...

Chimeras

Abrams Strogatz Laing I Omelchenko O Omelchenko Zakharova Wolfrum Schoell ...

Chimeras

Abrams Strogatz Laing I Omelchenko O Omelchenko Zakharova Wolfrum Schoell ... for example

Chimeras

Abrams Strogatz Laing I Omelchenko O Omelchenko Zakharova Wolfrum Schoell ... for example global order parameter

Continuum limit (Ott-Antonsen)

OE Omel'chenko, Nonlinearity 2013

Continuum limit (Ott-Antonsen)

OE Omel'chenko, Nonlinearity 2013

Fold "Hopf"

Conclusion

◮ Stabilizing feedback loop makes it possible to

define macroscopic

◮ equilibria (stable/unstable) ◮ periodic orbits (stable/unstable) ◮ fold, Hopf, pitchfork, period-doubling

bifurcations in a computable manner.

◮ Examples studied until now

◮ interaction of traders ◮ disease spread on network ◮ chimeras on rings of phase oscillators