Unitary Representations and Complex Analysis - MIT Mathematics
Mathematics for Complex Systemsmb1a10/lecture0.pdf · Mathematics for Complex Systems Outline of...
Transcript of Mathematics for Complex Systemsmb1a10/lecture0.pdf · Mathematics for Complex Systems Outline of...
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