# Calculus and Analysis

An Introduction to Dynamical Systems and Chaos
Next: Introduction: So what's a An Introduction to Dynamical Systems and Chaos Marc Spiegelman, LDEO September 22, 1997

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, which originated with the work of the French mathematician RenĂ© Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s, 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[edit]
Catastrophe theory

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

Dynamical System
A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system.

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Vector Calculus

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## Highlights of Calculus | MIT Video Course

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.

Mathematics | 18.03 Differential Equations, Spring 2006

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.

## Weierstrass functions

Table of Integration Formulas to Memorize

Find Derivative of y = x^x
Note that the function defined by y = x x is neither a power function of the form x k nor an exponential function of the form b x and the formulas of Differentiation of these functions cannot be used. We need to find another method to find the first derivative of the above function. If y = x x and x > 0 then ln y = ln (x x) Use properties of logarithmic functions to expand the right side of the above equation as follows.

On-line Tutorials

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