Chaos and the Butterfly Effect Presented by S. Yuan

● Chaos Theory is a developing scientific area of study focused on the study of nonlinear systems ● It is related to other fields such as quantum mechanics, yet is able to cover a wide range of scientific areas – math, physics, biology, finance, etc. ● To understand chaos theory, we must define what it deals with – (dynamical) Systems – “Nonlinear”

Definition: System ● Understanding of the relationship between the way a group of objects/object interact ● Is part of a grander subject: dynamics ● A system's dynamics / movements / progress are examined ● Ex. Pendulums, electrical circuits, oscillators, population growth, oscillators, etc.

Systems ● Explanations of dynamical systems through: – Analysis of a chaotic mathematical model (mathematically) – Analytical techniques, plotting – (such as recurrence plots & Poincaré maps)

Ex.s of systems.. ● Belusov-Zhabotinsky reaction – Spiral waves of chemical activity in a shallow disk

● Periodically forced double-well oscillator

“Nonlinear” ● Describes the mathematical model that is not “linear” Until the growth of chaos theory, many math models had been analyzed as if they were linear (with methods such as Calculus) although in nature, they were nonlinear and much more complex ● Nonlinear systems are more unpredictable than linear systems, which can be broken down into parts with each part able to be solved separately and recombined to produce the answer.

How to tell the Difference.. ● Linear System: ● ẋ 2 = -(b/m) ẋ – (k/m)x ● = -(b/m)x2 – (k/m)x1 – linear: all x on right hand side are to the first power, just like in a linear equation – ẋ 1 = x2 – Ẋ 2 = -(b/m)x2 – (k/m)x1

● Nonlinear System: - ẍ + g/L sin x = 0 (pendulum) – ẋ 1 = x2 – ẋ 2 = -g/L sin x1

Topics ● To understand linear and nonlinear systems, we need to learn about how their graphical representation can tell us info about them ● Concepts: fixed points, stability, phase portraits, trajectories, parameters ● Examples of systems: 1-D / 1 st order systems, 2-D / 2 nd order systems

1-D example form of a 1-D system: ẋ = f(x) ● ẋ = -x^3 ● ẋ = x^3 ● ẋ = x^2 ● ẋ = 0

Bifurcations: qualitative changes in dynamics ● Interesting aspect of 1-D systems: their dependence on parameters (value of the constants in front of variables) ● Fixed points are created or destroyed, and stability of these changes ● ẋ = r + x^2 ● Can you determine which phase portrait is for r >0, r<0, r=0 ?

2-D example Form of a 2-D system: ẋ = ax + by ẋ = cx + dy ● ẋ = ax ● ẏ = -y -uncoupled; each is solved separately – The solution: ● x(t) = x0 e^(at) ● y(t) = y0 e^(-t) -Here, a is a parameter, so we must find the phase portraits of all values of a...

● a<-1 – x(t) decays more rapidly than y(t) ● a=-1 – Star, all trajectories are straight lines ● -1<a<0 – Node, trajectories approach x* along x-direction ●

● a=0 – Entire line of fixed points along x-axis ● a>0 – x* = 0 is saddle point – y-axis is stable manifold of saddle point (t → infinity) – x-axis is unstable manifold of saddle point (t → - infinity)

Chaos Theory: a brief summary ● In common usage, the word “chaos” means simply “a state of disorder” ; a “mess” ● For a system to be in chaos it must have the following necessary properties: ● 1. Must be sensitive to initial conditions ● 2. Must be 'topologically mixing' (a system that will evolve over time so that any given region of it will eventually overlap with any other region in it) ● 3. Its periodicity must be dense (Its periodic orbits must be closely packed together)

Chaos Theory: a brief summary, cont. ● A non-linear dynamical system can show one or more of the following types of behavior: – forever at rest – forever expanding (only unbounded systems) – periodic motion – quasi-periodic motion – chaotic motion ● The type of behavior that a system may show depends on the initial state of the system as well as the value of its parameters ● The most difficult type of behavior to characterize/predict is chaotic motion (a non-periodic, complex motion)

Lorenz Equations: a 3-D system modeling convection rolls in the atmosphere ● Where σ, r, b >0 are parameters, the Lorenz equations are: ẋ = σ(y – x) ẏ = rx – y – xz = xy – bz – This simple system can have very chaotic dynamics when examined over a wide range of parameters – Lorenz plotted trajectories in 3 dimensions, to find that they create a “strange attractor”

● ẋ = σ(y – x) ẏ = rx – y – xz = xy – bz ● The system has only 2 nonlinearities: the quadratic terms xy & xz ● The Lorenz system has 2 types of fixed points: – At the origin (x*, y*, z*) = (0, 0, 0) [for all r] – Symmetric pair of fixed pts x* = y* = – +(sqrt(b(r-1)) [for r>1] ● Represent left- or right- turning convection rolls

Partial Bifurcation Diagram for this system.. ● System has – Fixed point at origin circled by a saddle cycle, a type of unstable limit cycle possible only in phase spaces of 3 + dimensions – Not all trajectories are repelled out to infinity, they instead enter a large ellipsoid, confined to this bounded area and moving without intersecting themselves or others forever

● Resembles butterfly wings ● This trajectory appears to cross itself repeatedly but it does not ● Trajectory starts near the origin, swings to the right, then to the center of the spiral on the left, then shoots back to right, then left, etc.

Attractors ● A way of visualizing chaotic motion: make a phase diagram of motion, plotting a dimension on each axis ● Geometrically, an attractor can be – Points (' a point attractor') – Lines – Surfaces/Volumes/manifolds (any n-dimensional space) – or even a complicated set with a fractal structure named a strange attractor

The Butterfly Effect ● Is an example of sensitivity to initial conditions, which suggests that the flapping of a butterfly's wings might create tiny changes in the atmosphere, which can dramatically magnify over time and cause something like a tornado/whirlwind to occur ● The central idea of it: small variations in initial conditions can have dramatically different results after several cycles of the system, therefore as a consequence complex systems such as the weather are difficult to predict after a certain time range ● Found in the weather: the unpredictability of weather patterns up to a certain amount of time (2 days) ● Some time ranges can be much longer (such as the orbit of pluto being the same for 2 million years)

The Butterfly Effect, cont. ● Under the two following conditions, the butterfly effect occurs: ● 1. The system is nonlinear ● 2. Each state of the system is determined precisely by the previous state. (The result of each moment in time is repeatedly cycled through math functions that may determine the system)

Animation of Butterfly Effect ● lorenz.html lorenz.html

Sources ● Wikipedia ● ● “Nonlinear Dynamics and Chaos”