Mathematics for Complex Systems

● Outline of course● Part I: Nonlinear dynamics

– 1d flows (phase portraits, fixed points, bifurcations)– 2d flows (linear systems, oscillations, index theory)– 3d flows (Lorenz equations – chaos)– 1d-maps (Chaos, Period-doubling, Universality and

Renormalization approaches)– Fractals

● Part II: Networks– Network models– Dynamical processes on networks

Aims

● Equip everybody with a basic mathematical “toolset” to analyse complex systems

● Gain elementary knowledge about :● Nonlinear dynamics (basic knowledge about

differential equations, being able to analyse a given problem and understand the “language” and know some basics about non-linear systems)

● Networks (have a broad familiarity with the discipline and what it is about)

Format

● Lectures and seminars● 1 lecture, 1 seminar per week● Times:

– Monday (lecture) – 2pm, 16-2025 – Wednesday (seminar) – 12am, 06-1083

● Lecture: new material● Seminar: apply and deepen knowledge● “drop in clinics” on request

● Evaluations● 10 minute talks on selected topics● Written exams at the end of the semester

Evaluation

● Written exam ~ 90mins● Test some elementary skills and knowledge

● Talks:● 10mins● If you like two people can combine, talking about

different aspects of the same paper ~ 20 mins● Pick your favourite paper in which a dynamical

systems model of an applied system is built and analysed and present a short summary

● I'll make some suggestions if you don't have one● Talks will be in the seminars in the second half of

the semester

Resources

● Lecture slides:

● Nonlinear Dynamics:● Arrowsmith and Place, “An introduction to

Dynamical Systems”, Cambridge University Press● S. Strogatz, “Nonlinear Dynamics and Chaos” ,

Westview Press

● Networks

M. Newman, “Networks: An Introduction”, Oxford University Press

http://users.ecs.soton.ac.uk/mb8/maths.html

Some toy problems to see why you should know something about (non-)

linear dynamics

● Suppose we have a population of x(0) individuals. Per interval of time each individual reproduces with likelihood r. How large is the population at time t?

● Discrete approach? Continuous approach? Simulation?

Simulation ... Numerical Integration in Python

● See script population.py

More difficult: Zombieworld

● Let's presume we have blue guys (zombies) and red guys (humans)

● Experimental observation tells us:● At a typical instant in time every individual interacts

with one other individual● Blue – blue (zombie meets zombie): one zombie is

killed with probability b● Red-red (human meets human): a new human is

created with probability r● Blue-red (zombie meets human): a human is

converted into a zombie with probability c

Question

● Can zombie and human populations both grow? If so, at which rate?

● Are there scenarios when both humans and zombies will die out?

● For playing around with this have a look at the script zombies.py (see course webpage)

General thoughts (1)

● We want to model a (dynamical) system. What do we need to do?

● Basically, need two “ingredients”● A description of the state of the system by a set of

(real) numbers (the minimal number of independent variables are the degrees of freedom)

● A rule how the system evolves from one state to the next.

● This rule can either be discrete (iterated maps) or continuous (differential eq.'s)

General thoughts (2)

● Examples: How many variables needed?

(=minimum number of coordinates to specify configuration)● A pendulum: (2) angle + angular velocity● A double pendulum: (4) 2 angles+2 angular

velocities● Earth-Moon-Sun: (18) 3 (3 positions + 3 velocities)● A fish population of three species in a lake:

– (3) if we assume food resource to be unlimited– (4) if food resource is renewable

● A point mass on a rotating stick in 2d: (2) (location on stick + velocity)

General thoughts (3)

● What kinds of behaviour are possible and when? Very roughly:● Equilibrium● Exponential growth/decline● Oscillations● Damped oscillations● Self-stabilizing oscillations● “Chaotic” behaviour?

● Any other ideas?● Aim: Want to understand which dynamics

arises from what rules.

Why is this useful for me?

● Calibrate and verify your ABM● Does it show the right behaviour in simple

situations?● Does it reproduce analytical results?

● Do I really need to simulate to solve my problem?● Maybe a mean-field approach can solve it?● Check level of question asked vs. level of detail

provided by approach

● Find out what is interesting about your ABM

Nonlinear Dynamics

History (1)

● Mid 1600: Newton (and Leibniz) invented differential eq.'s and solved 2-body problem

● Subsequent generations tried to solve 3-body problem ... in vain

● Poincare in late 1800's: qualitative rather than quantitative questions

● Focus on nonlinear oscillations (radio, radar, laser), extension of Poincare's results to classical mechanics (Birkhoff, Kolmogorov, Arnol'd, Moser, ...)

History (2)

● First high speed computer in 1950's● Lorenz (1963) Chaos and strange attractors

● 1970's boom years of chaos● Ruelle and Takkens: onset of turbulence● May: Chaos and iterated maps in pop. Biology● Feigenbaum: Universal laws governing transition to

chaos

● Also in 1970's● Mandelbrot's fractals● Winfree applied geometric methods to biol. osci's

Some Basics ... (1)

● Two types of systems:● Differential eq's: continuous time● Iterated maps: discrete (“clocked”) time

● Ordinary vs. partial diff. eq's

m x+b x+kx=0∂u∂ t

=∂2u

∂ x2

(one independent variable) (>1 independent variables)

Some Basics ... (2)

● General framework for ODE's

● Is this general enough? (higher order deriv's?)

● Linear vs. nonlinear systems● Linear: f's are linear in x● Nonlinear: much more difficult

d x1

dt=f 1(x1 ,… , xn)

d xn

dt=f n(x1 ,…, xn)⋮ n-th order system of ODE's

Nonlinear Dynamics and Complex Systems

● Why are we mainly interested in nonlinear dynamics in a Complex Systems course?● Linear systems can easily be broken down into

“simple” parts● Solutions for the “simple parts” can be found and

then be recombined to find the general solution● This is the essence of the Laplace method or

Fourier analysis which allows us to find analytical solutions for linear systems

● However: Generally none of this works for nonlinear systems!

Some Basics ... (3)

● “Geometrical approach”

● Given system, want to draw (qualitative behaviour) of all trajectories without solving the system of differential eq's

trajectory

“phase space” (spaceof all dof's)

Some Basics ... (4)

● Autonomous vs. non-autonomous systems● Autonomous: f's have no explicit time dependence● How to deal with explicit time dependence?

– Introduce time as additional variable– Example: forced pendulum:

– Nth order time dep. system becomes (n+1) order autonomous system

– Forced osci: 2nd order linear -> 3rd order non-linear

m x+b x+kx=F cos t x1=x , x2= x , x3=tx1=x2

x3=1x2=1 /m(F cos x3−bx2−kx1)

A classification scheme ...

1d Systems

1d Flows (1)

● First order system:● Interpret differential eq's as a vector field and

dynamics as flows along the field

● An example:● Analytically: separation of variables

● After a bit of manipulation:

● Not so easy to interpret even if we have an analytical solution!– What is asymptotical behaviour of x(t) for arbitrary ICs?

x=f ( x)

x=sin(x )

dt=dx

sin (x)t=∫

dxsin (x)

t=ln∣tan x /2∣+C

1d Flows (2)

● Graphically: x=sin(x )

● dx/dt=0 -> no flow -> fixed points (FP)● Two types: stable and unstable

Fixed points and stability (1)

● General System

dx/dt=f(x)● Imagine fluid flowing

along real line with local velocity dx/dt

● Fixed points are equilibrium solutions with

dx/dt=0=f(x*) such that if x0=x* -> x(t)=x* all t

● Stable: small perturbations damp out● Unstable: small perturbations grow

Fixed points and stability (2)

● Consider● Classify the dynamics of (1) by analyzing fixed

points and their local and global stability!● Fixed points:

● Stability: x1 unstable, x

2 locally stable, but not

globally

● What kind of perturbation could destabilize x2?

x=x2−1x=x2−1=f ( x)

f (x )=0 x1/2=±1

Linear Stability Analysis (1)

● Consider a FP x* (i.e. f(x*)=0) and the fate of a small perturbation ε=x(t)-x* from it:

● Expand f into a Taylor series around x*:

● Perturbation ε:● Grows exponentially if df/dx>0● Declines exponentially if df/dx<0● If df/dx=0 more analysis is needed

ϵ= x=f (x+ϵ)

ϵ=f (x )+ϵ df /dx+O(ϵ2)

1/∣df /dx∣charact. timescale

Linear Stability Analysis (2)

● A simple example:● That is● FP:● Stability?● Stable for odd k and unstable for even k

x=sin(x )

f (x )=sin (x )

f (x )=0 x=k π

df /dx=cos (x )=cos(k π)

What if df/dx=0?

unstable FPstable FP

half-stable FP non-isolated FP

Existence and Uniqueness

● So far: assumed existence of unique solutions● THEOREM:

Suppose f(x) and df/dx are continuous in an open interval of R around x

0.Then the initial value

problem dx/dt=f(x), x(0)=x0 has a solution on some

time interval (-τ,τ) around t=0 and this solution is unique.

● In the following: always assume f(x) is “smooth enough” such that solutions exist and are unique

Impossibility of Oscillations in 1d

● So far: all trajectories tend to or are FP● These are the only possible dynamics for a vector

field on the real line

● Why?● Topological reason: 1d system corresponds to a

flow on the real line. If you flow monotonically on a line you never come back to starting position

±∞

Summary

● Terminology (dof's, ODE vs. PDE, order of a system, linearity, autonomous vs. non-autonomous)

● Separation of variables● Fixed points

● Types● Graphical methods● Linear stability analysis

● Classification of dynamics in 1d