Ginger Breadman map, pretty chaotic picture

Freshman's dream

• Set of points on the real line \Bbb{R} that is nowhere dense (in particular it contains no intervals), yet has a positive measure - Fat Cantor set.*

• The Trojan asteroids, forming triplets along with Jupiter and the Sun, essentially move according to this scheme. These systems are not always stable.*

• "Poincaré got a big mess, and [the horseshoe] put order in the mess."*

• Thus "... the mathematics created on the beaches of Rio ..." (Hornig)

 was the horseshoe and the higher dimensional Poincare's conjecture*

• When all three hallmarks are present (sensitivity, periodic cycles and mixing), we have a phenomenon called "chaos". Unfortunately, Hollywood films have overused the word, so most people think that it means something altogether more wishy-washy, as opposed to a well-defined and precise mathematical term*

Pg-162 of James Glieck book on Chaos - 

• With Luck your calculations converge toward a solution, Luck has a way of vanishing, however, whenever a problem is especially interesting.*

• After all, self similarity seemed to be the signature of self turbulence, fluctuations up fluctuations, whores upon whorls.*

• What else, when chaos draws all forces inward to shape a single leaf - images of chaos.*

• Why should it be that as things become small they also become incomprehensible. Pg 163.*

• But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. Pg 227

• The boundary between two or more attractors in a dynamical system served as a threshold of a kind that seems to govern so many ordinary processes, from the breaking of materials to the making of decisions. Each attractor in such a systemhas its basin, as a river has a watershed basin that drains to it. Each basin has a boundary. A promising new field of mathematics and physics was the study of fractal basin boundaries. Pg 233

• Fractal basin boundaries - Even when a dynamical system's long term behavior is not chaotic, chaos can appear at the boundary between one kind of steady behavior or the other. Often a dynamical system has more than one equilibrium state, like a pendulum that can come to a halt at either of two magnets placed at tits base. Each equilibrium is an attractor, and the boundary between the attractors can be complicated but not smooth. Pg 235 

Difference between strange attractor and chaotic attractor

• It is a common misconception that the term strange attractor is simply another way of saying chaotic attractor. Though a chaotic attractor is certainly a strange attractor, the reverse is not necessarily true. Instead, the term strange actually refers to the fractal geometry on which chaotic behavior may or may not exist. Indeed, there exist examples of strange attractors which are nonchaotic. 

• To investigate the connection between fractal geometry and chaotic dynamics, we have introduced several mathematical tools. The Lyapunov spectrum is used primarily to identify sensitivity to initial conditions in a dynamical system. If sensitivity exists, then we say that the system is chaotic. If the attractor in the system has a fractal geometry – as indicated by the Kaplan-Yorke dimension or the phase sensitivity measure – then the attractor is strange. Since it is not always easy to identify chaotic behavior and strangeness analytically, we give special attention to how these tools are used computationally.

• The group that came to call itself the Dynamical Systems Collective - others sometimes called it the Chaos Cabal.

• What happens when a quantum butterfly flaps its wings?

Chaos theory tells us that small changes in initial conditions can lead to widely divergent behaviour. But what if your system is a quantum system? The uncertainty principle tells us that you can't know any initial conditions exactly so how can we merge the classical ideas of chaos into the world of quantum uncertainty. 

• Arithmetic dynamics: 

Round-off errors in computer arithmetic are a manifestation of chaos over the p-adic numbers.

Link: http://www.maths.qmul.ac.uk/~fvivaldi/research/ArithmeticDynamics4Layman.pdf 

Books -

Currently I am reading the book -

James Gleick: Chaos making  a new science

"There exists always a textbook which illustrates the difficult part more clearly and in a nice way. Finding that textbook for the topic is the tricky part! and presenting my experience here might help some one in the same situation."

Links to resources:

Stable Manifold Theorem

What's an intuitive explanation for the fact that C is algebraically closed but R isn't?

Course handouts - King

ELEMENTARY INTRODUCTION TO FOLIATION THEORY

Bifurcations other than the common bifurcations! unfolding bifurcations!