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"