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 s k , evolving  k ϵ + − p − e − γ Δ t s k + p + k ϵ − if | s k | < 1   s k, new = 0 reset if s k ≥ 1 ⇐ buy  0 reset if s k ≤ − 1 ⇐ sell  ◮ p ± k random number of news ∼ Pois ( n ± ( 1 + gR ± )) Δ t ◮ 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 0.14 time profile Equilibrium Rate 0.12 R + ( t ) − R − ( t ) 0.1 R = R + − R − 0.08 0.06 0.08 buying − selling rate 0.04 0.02 0.06 0 −0.02 0 20 40 60 80 100 0.04 0.02 0 −0.02 0 0.5 1 1.5 2 2.5 amplification g

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 R e q = R + eq − R + eq : in the long run men R = R eq : R eq = lim t → ∞ men s ∈ [ 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 ) − R ref ] R ref is equilibrium if R ref = lim t → ∞ men s ∈ [ t,t + T ] R ( s ) ◮ The long-time mean of feedback loop input is zero. ◮ For large numbers N of traders: ◮ E [ R ( t ) − R ref ] 2 → N → ∞ ,t → ∞ 0 ◮ Resulting equilibrium R ref 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] Parameter sweep Equilibrium fraction of infected  e q 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.002 0.004 0.006 0.008 0.01 infection rate p

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 mean control input & regression curve mean ”cured” − artificially infected N = 400, infection rate p = 0 . 001 1.5 1 0.5 0 −0.5 −1 −1.5 −2 measurements regression −2.5 uncertainty equilibria −3 0 0.5 1 reference fraction of infected  ref

Newton iteration & continuation with uncertainty Linear regression with Gaussian process root  r of regression curve y r with error bar (easy to find with Newton iteration) y  r measurement with error bar linear regression curve y r (  ) with error bar 

Procedure for continuation with uncertainty 1. find roots (bifurcations) of regression curve y r (  ) 2. determine where to measure next: ◮  for which measurement y r (  ) ± σ r (  ) would minimize error bar of root for updated y r , or ◮  where measurement y r (  ) + σ r (  ) changes root the most Both are nonlinear optimization problems on current regression curve y r (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: R ref , g . bias = [ ϵ + − ϵ − ]( t ) = k [ R ( t ) − R ref ] Run with feedback: after transients, read off ◮ men [ ϵ + − ϵ − ] , ◮ R eq = men R ⇒ equilibrium surface in space ( g, ϵ + − ϵ − , R eq ) ∂R eq fold condition : ( R ref , g ) = 1 ∂R ref

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 N = 400 0.9 fraction of infected  men  > 0 0.8 0.7 men  < 0 0.6 0.5 0.4 +5e-3 0.3 -5e-3 0.2 close to tanscritical: 0.1 long transients 0 0 1 2 3 4 5 −3 infection rate p x 10

Oscillations Ring of nonlocally coupled phase oscillators

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

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

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

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

Abrams Chimeras 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