Intro to Complexity

Complexity

Common properties

Core disciplines

Goals

Measures of complexity

Shannon information

Fractal dimension

....

cross-disciplinary insights into complex systems (success)

General theory (controvertial)

Methodologies

Experimental work

Theoretical work

Computer simulation of CS

Problems if interest in Science

Problems of Simplicity

Problems of Disorganized Complexity

Involve just a few variables

Examples

Pressure & Temperature, in Thermodynamics

Current, Resistance, Voltage, in Electricity

Population vs. Time, in population dynamics

problems of the end 19th - begin 20th century,
in biology, chemistry, physics

Problems of Organized Complexity

Simple components or agents

Nonlinear interactions among components

No central control

Emergent behaviors

hierarchical organization

information processing

dynamics

evolution & learning

Dynamics

Information

Computation

Evolution

Study of continually changing structure & behavior of systems

Study of representation, symbols & communication

Study of how systems process information & act on the results

Study of how systems adapt to constantly changing environment

Dynamics & Chaos

Dynamical Systems Theory

Gives a vocabulary & set of toools for describing dynamics

The branch of Math of how systems change over time

Calculus, Differential equations, Iterated maps,
Algebraic topology, etc.

Fractals

Chaos

Difference with Randomness

Deterministic chaos

Seemingly random behavior with sensitive
dependence on initial conditions

Perfect prediction is possible even if
we're looking at a chaotic system

Universality in Chaos

While chaotic systems are not predictable in detail, a wide class of chaotic systems has highly predictable, “universal” properties.

Involve billions or trillions of variables

Temperature example

Understanding laws of temperature and pressure as emerging from trillions of disorganazed air molecules in the room or the athmosphere

Science of Averages

When we look at understanding temperature, we don't look at
the particular position and energy of every individual air molecule.

Systems are understood through taking
averages over a large set of variables

Comes under the rubric of Statistical Mechanics

Statistical Mechanics

key - assumes very little interaction among variables

that allows to take meaningful averages

Involve a moderate to large number of variables

key - due to their strong non-linear interactions, the variables cannot be meaningfully averaged

Problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole

This organic whole refers to the emergent behavior of the system

General

Model

a simplified representation of a real-world phenomena

can be a math equation, a computer program or a drawing

system ~ model

Pouncaret

Other concepts

Dynamics

the manner in which the system changes

Iteration

Logistic function

Let's explore it's dynamics

Logistic Map

the most famous equation in the field of the Chaos theory

example: population growth

Non-linear systems

Systems, whose parts interact in a non-linear way

Periodic-doubling route to chaos

System's Attractor

Periodic attractor

FIxed-point attractor

Maps & Diff. Equations

To describe the evolution of some systems we use maps,
and for others we use differential equations

With a map, the time steps are discrete

With differential equations, they are textcontinuous

prediction becomes impossible

Robert May

even with a simple deterministic model,
long-term prediction is impossible

a simple, completely deterministic equation that,
when iterated, can display chaos (depending on the valu of R)

Together with Quantum Mechanics => impossible to ...

Chaotic (strange) attractor

Bifurications in the Logistic Map

Feigenbaum constant

Sin Map

Significance

Significance of Dynamics & Chaos for Complex Systems

Dynamics gives us vocabulary for describing complex behavior

Complex, unpredictable behavior from simple, Deterministic rules

There are fundamental limits to detailed prediction

At the same time there is universality: "order in chaos"