# Calculus and Analysis

An Introduction to Dynamical Systems and Chaos. Catastrophe theory. In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.

Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s. It considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.

Elementary catastrophes Fold catastrophe Bifurcation. In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos.

It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation (Rasband 1990).

More generally, a bifurcation is a separation of a structure into two branches or parts. . , where denotes the real part, exhibits a bifurcation along the negative real axis and. Dynamical System. Vector Calculus. Highlights of Calculus. Home - Math 106. Visualizing a function can give a mathematician enormous insight into the function's algebraic and geometrical properties.

The easiest way to see what a function looks like is to use a computer as a graphing tool. At times, this technique is the most useful, but drawing the function yourself is always the best way to get a feeling for why the function looks the way it does when graphed. All of you are by now familiar with graphing one-variable functions in the plane and hopefully can easily predict the shape of any fairly simple 2D function just by analyzing its equation. This knowledge came from graphing similar functions over and over again to get a feel for their general shape and critical points. The purpose of this tutorial is to quickly revisit these 2D examples and then to move on to drawing functions situated in 3D and even 4D space.

18.03 Differential Equations, Spring 2006. Weierstrass functions. Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere".

Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr.

Table of Integration Formulas to Memorize. Find Derivative of y = x^x. Calculus Online Book. On-line Tutorials.