Calculus and Analysis

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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.

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

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[edit] 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.

Dynamical System

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. Vector Calculus. Disclaimer: CosmoLearning is promoting these educational resources as a courtesy of Gaussian Department of Mathematics (GMath).

Vector Calculus

We make no representations regarding the courses and videos of this content provider or any information related to them. CosmoLearning is not endorsed by GMath and we are not affiliated with them, unless otherwise specified. Any questions, claims or complaints regarding this content should be directed to the original creator. Highlights of Calculus. Since 2008, Academic Earth has worked diligently to compile an ever-growing collection of online college courses, made available free of charge, from some of the most respected universities.

Highlights of Calculus

We proudly build on the precedent set forth by MIT OpenCourseware, believing that everyone deserves access to a world-class education. To facilitate this goal, we have curated links to over 750 online courses and 8,500 individual online lectures, giving students of all ages unparalleled access to college courses they may otherwise never experience. Home - Math 106. Visualizing a function can give a mathematician enormous insight into the function's algebraic and geometrical properties.

Home - Math 106

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".

Weierstrass functions

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. 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.

Find Derivative of y = x^x

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. Calculus Online Book. On-line Tutorials.