Chapter 3
Motion in Two and Three Dimensions
Conceptual Problems
1
[SSM] Can the magnitude of the displacement of a particle be less
than the distance traveled by the particle along its path? Can its
MATH 614
Dynamical Systems and Chaos
Lecture 13:
Dynamics of linear maps.
Hyperbolic toral automorphisms.
Linear transformations
Any linear mapping L : Rn Rn is represented as
multiplication of an n-d
MATH 614
Dynamical Systems and Chaos
Lecture 10:
Bifurcation theory.
Bifurcation theory
The object of bifurcation theory is to study changes that
maps undergo as parameters change.
In the context of o
MATH 614
Dynamical Systems and Chaos
Lecture 15:
Markov partitions.
Solenoid.
General symbolic dynamics
Suppose f : X X is a dynamical system. Given a partition
of the set X into disjoint subsets X ,
MATH 614
Dynamical Systems and Chaos
Lecture 12:
Maps of the circle (continued).
Subshifts of nite type (revisited).
Maps of the circle
T : S 1 S 1,
T an orientation-preserving homeomorphism.
Rotation
MATH 614
Dynamical Systems and Chaos
Lecture 8:
Structural stability.
Sharkovskiis theorem.
Structural stability
Informally, a dynamical system is structurally
stable if its structure is preserved und
MATH 614
Dynamical Systems and Chaos
Lecture 4:
Itineraries.
Cantor sets.
Logistic map
The logistic map is any of the family of quadratic maps
F (x) = x(1 x) depending on the parameter R.
2
If > 1, th
MATH 614
Dynamical Systems and Chaos
Lecture 3:
Classication of xed points.
Logistic map.
Periodic points
Denition. A point x X is called a xed point
of a map f : X X if f (x) = x.
A point x X is call
MATH 614
Dynamical Systems and Chaos
Lecture 7:
Compact sets.
Topological conjugacy (continued).
Denition of chaos (revisited).
Compact sets
Denition. A subset E of a topological space X is compact if
MATH 614
Dynamical Systems and Chaos
Lecture 5:
Cantor sets (continued).
Metric and topological spaces.
Symbolic dynamics.
Cantor Middle-Thirds Set
General Cantor sets
Denition. A subset of the real l
MATH 614
Dynamical Systems and Chaos
Lecture 14:
The horseshoe map.
Invertible symbolic dynamics.
Stable and unstable sets.
The Smale horseshoe map
Stephen Smale, 1960
F
F
The Smale horseshoe map
F
F
Safety Procedures
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Wear Safety glasses
Make all adjustments with the machine at a dead stop.
Examine cutting procedure; plan cuts to avoid backing out of a kerf.
If the m
Which of the following does the full faith and credit clause ensure?
a.
b.
c.
d.
Ensures that official governmental actions of one state is denied by other states.
All 50 states are required to run cr
Cleveland Bailey
ECON 2302
06/27/2016
Assignment 3
1.
1. For each of the following absolute values of price elasticity of demand, indicate whether
demand is elastic, inelastic, perfectly elastic, perf
Cleveland Bailey
ECON 2302
06/27/2016
Assignment 3
1.
1. For each of the following absolute values of price elasticity of demand, indicate whether
demand is elastic, inelastic, perfectly elastic, perf
Cleveland Bailey
Technical Writing 2314
06/16/2016
Outline
My subject is Mutual Fund Investing (Road to Financial Independence). In my article will have
informative Information that will keep readers
Solution of homework assignment 2
Problem 1.
Let Xt and Yt be stationary processes with means X , Y and covariance functions X and Y respectively. We have for Zt = aXt + bYt : EZt = aEXt + bEYt = aX +
MATERIALS 101
INTRODUCTION TO STRUCTURE AND PROPERTIES
WINTER 2012
Problem Set 6
Due: Tuesday, March 6, 11:00 AM
1.
TTT Diagrams
Based on the transformation diagrams for eutectoid steel shown below, w
Quiz 7
1.
The correlation coefficient between the variables EcoFree and GDPpc imples that:
A. As one variable increases, the other decreases
B. As one variable decreases, the other increases
C. There
Amath 301
Homework 4: Due Friday 8/12 at 4pm
Summer 2016
Warning: This homework is long, do not wait until the last minute to
start or you will not finish. Also, given its length, it is probably a goo
CS 3113 / Math 3403, Fall 2011
Assignment 1
Comments and Solutions
1. Write a Octave (or MATLAB) function to compute using the series
=4
X
1 1 1
(1)n
= 4 1 + + . .
2n + 1
3 5 7
n=0
The input to your p
MATH 614
Dynamical Systems and Chaos
Lecture 6:
Symbolic dynamics (continued).
Topological conjugacy.
Denition of chaos.
Interior and boundary
Let X be a topological space. Any open set of the topolog