IMPORTANT
https://www.ocps.net/lc/east/hun/academics/programs/ib/Documents/Extended%20Essay%20Guidelines%20and%20Requirements%202-27-13.pdf
Background
“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”
---Albert Einstein
Feedback often causes chaotic behavior in a system.
e.g. stock market :
prices moving up or down gives incentive for people to buy or sell stocks, causing it to move up or down chaotically.
(example from http://fractalfoundation.org/resources/what-is-chaos-theory/)
x[n+1] = f(x[n])
The next term in sequence is a function of the previous term. This means every term is determined by the previous term by a function.
e.g. Fibonacci's sequence
x[n] = x[n-1] + x[n-2]
Random or Chaotic?
Randomness is also called stochastic. Unlike chaotic, even if the two initial state of sequences are the same, random process would give different sequences.
Chaotic sequences are generated **deterministically** from a **dynamic system.**
http://www.math.tamu.edu/~mpilant/math614/chaos_vs_random.pdf
Deterministic--- causal effect in physics.
"It is the concept that events within a given paradigm are so causally bound that prior states of any object or event completely determine its later states."
"For a given time interval only one future state follows from the current state"
http://en.wikipedia.org/wiki/Determinism
Dynamic--- "a concept where fixed rules describes the time dependence of a point in geometrical space."
"At any given time a dynamical system has a state given by a set of vectors, which can be represented by a point in an appropriate state space."
"Small changes in the state of the system create small changes in the numbers, due to the evolution rule of dynamic system"
http://en.wikipedia.org/wiki/Dynamical_system
A bounded sequence of values {xi} (from i=1 till infinity) is chaotic if
1. {xi} is not asymptotically periodic
2. No Lyapunov exponent vanishes
3. The largest Lyapunov exponent is strictly positive
http://www.math.tamu.edu/~mpilant/math614/chaos_vs_random.pdf
asymptotically periodic--- its terms approaches a periodic sequence
Lyapunov exponent--- 'quantity that characterizes the rate of separation of infinitesimally close trajectories'
wiki
Predictability?
In a complex system, like the weather, we can never know all the initial conditions to enough precision and accuracy. With any small defect or error in the initial conditions, the error is amplified since it increases exponentially, leaving any form of prediction useless.
http://fractalfoundation.org/resources/what-is-chaos-theory/
Calculus (empty)
-= Reading 'From Calculus to Chaos' =-
"Chaos in nonlinear ordinary diﬀerential equations
Explore the connection between the various kinds of homoclinic bifurcations
and the onset of chaos in ordinary diﬀerential equations.
Investigate the occurrence of stochasticity in Hamiltonian systems, as an
integrable system is perturbed more and more strongly.
The period doubling sequence for unimodal (one-humped) maps is well known.
What happens for other maps, e.g. cubics?"
http://www.maths.ox.ac.uk/files/imported/current-students/undergraduates/projects/pdf/project-ideas.pdf
Logistic Map (current focus)
https://www.math.ku.edu/~lerner/m221s11/LogisticLabII.pdf
http://fricke.co.uk/Teaching/cs523/Project1/ExampleSolution2.pdf
http://physics.bc.edu/MSC/430/T1/Application1.html
http://www.emba.uvm.edu/~jxyang/teaching/Math266new/notes_6_1.htm
http://en.wikipedia.org/wiki/Logistic_function
It doesn't focus on looking for roots in a function, but instead, studies the dynamics
http://physics.bc.edu/MSC/430/T1/Application1.html
g(x) = r x (1-x)
linear equations--- the dependent variable and its various derivatives appear in a linear way.
non-linear equations--- if its various derivative has higher powers
Calculus to Chaos
recursion---a process of repeating objects in a self-similar way
http://physics.bc.edu/MSC/430/T1/Application1.html
Fractals
"A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop."
http://fractalfoundation.org/resources/what-is-chaos-theory/
"Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.
Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals.
For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc."
http://fractalfoundation.org/resources/what-is-chaos-theory/