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Chapter 4A
4.1 Definitions
A dynamical system is an evolution rule that defines trajectories in phase space:
.
is a differentiable map that is parameterized by time.
is the solution of the
initial value problem discussed in chapter 3. However, this chap
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Course Administration
Syllabus
Text and errata. For errata see links on website.
My electronic notes are available online.
Introduction
Models of systems undergoing change are dynamical systems.
Dynamical systems are often modeled by differential equati
Statistics 512
Review Homework
Not to be turned in for grade
For each of problems 1 through 4 you are to provide the following:
a)
The experimental design structure (CRD, RCBD).
b)
The null and research hypothesis, test statistic and all important (necess
Homework 3
Due in Laboratory on February 21, 2014
1)
An experiment was designed to explore the growth of the plant (Bienertia Sinuspersici), a species with C4 photosynthesis and is
salt tolerant. Four levels of salt concentration were of interest to the r
Homework 2
Due in Laboratory on February 14, 2014
In homework 1 we analyzed the experimental data assessing the ability of a plant species to be grown in different
concentrations of saline. The experimental description is repeated here for your benefit.
A
Homework 2
Due in Laboratory on February 14, 2014
In homework 1 we analyzed the experimental data assessing the ability of a plant species to be grown in different
concentrations of saline. The experimental description is repeated here for your benefit.
A
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2 Linear Systems
In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation.
2.1 Matrix ODEs
Let
and
is a scalar. A linear function
Linear superposition
Linear scaling:
satisfies
)
Example:
is not a linear function, inst
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Chapter 2 part B
2.5 Complex Eigenvalues
Real Canonical Form
A semisimple matrix
with complex conjugate eigenvalues can be diagonalized using
the procedure previously described. However, the eigenvectors corresponding to the conjugate
eigenvalues are th
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Ch8Lecs
8.1 Bifurcations of Equilibria
Bifurcation theory studies qualitative changes in solutions as a parameter varies. In general,
one could study the bifurcation theory of ODEs, PDEs, integro -differential equations, discrete
mappings etc. Of course
Review for Exam 1
Variables:
Quantitative and Qualitative (discrete)
Qualitative variables include the Bernoulli (two categories), the binomial (two
categories and n independent trials) and the Poisson.
Quantitative variable will be normally distributed v
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Ch5Lecs
5.1 Stable and Unstable Sets
Recall the definition of the stable set (basin of attraction), ! ! () and unstable set (backwards
basin), ! ! (), of an invariant set .
! ! = cfw_! : ! ! as !
! ! = cfw_!
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Chapter 4B
4.7 Topological Conjugacy and Equivalence
This section is concerned with classification of dynamical systems. First we need some notions
from analysis and topology.
A map
is
surjective or onto if for all
there is at least one
injective or one
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Chapter 3
3.1 Set and Topological Preliminaries
Notation
natural numbers
integers
reals
element of
for all
there exists
such that
subset of (whether proper or not)
union
intersection
set subtraction
implies
if and only if
Euclidean norm of .
Notions fro
Homework 1
Due in Laboratory on February 7, 2014
An experiment was designed to explore the growth of the plant (Bienertia Sinuspersici), a species with C4 photosynthesis and is salt
tolerant. Four levels of salt concentration were of interest to the resea