2.2 Basic definitions
43
Next dropping the higher order terms of u results in the linear dierence equation
ut+1 (1
)ut + ut
T
(1
qz) .
This is a higher order linear dierence equation and the character
Fall 2017: CMDA 3605
Math. Modeling I
Homework 7
Dr. Matthias Chung
Submission deadline: 10/19/2017
1. The following epidemic model is referred to as a SIS epidemic model
St+1 = St It St + ( + b)It ,
Fall 2017: CMDA 3605
Math. Modeling I
Homework 4
Dr. Matthias Chung
Submission deadline: 09/28/2017
1. Find all of the equilibria for the system of dierence equations
(a) x(t + 1) = ax(t)e
(b) x(t + 1
Fall 2017: CMDA 3605
Math. Modeling I
Homework 10
Dr. Matthias Chung
Submission deadline: 11/09/2017
1. Consider the ODE system
x0 = x(4 x y),
y 0 = y(8 3x y).
Compute and graph the direction of flow
Fall 2017: CMDA 3605
Math. Modeling I
Homework 1
Dr. Matthias Chung
Submission deadline: 09/07/2017
1. Give an example of a mathematical model. Include in your narrative its classifications,
limitatio
Fall 2017: CMDA 3605
Math. Modeling I
Homework 8
Dr. Matthias Chung
Submission deadline: 10/26/2017
1. Models that are commonly used for population dynamics in fisheries are
dx
= g(x)x,
dt
where g(x)
Fall 2017: CMDA 3605
Math. Modeling I
Homework 9
Dr. Matthias Chung
Submission deadline: 11/02/2017
1. Consider the system of differential equations for a competitive pair of populations
x0 =x(K x ay)
Fall 2017: CMDA 3605
Math. Modeling I
Homework 3
Dr. Matthias Chung
Submission deadline: 09/21/2017
1. Let Un and Vn be the total amount of pollutant in lakes A and B respectively, in year
n, and that
2.2 Basic definitions
2.2.16
39
Bifurcations
Bifurcation points (r, x
(r) describe points where a change in the solution behavior occurs.
Four dierent types exist and occur when f 0 (
x(t) = 1.
1. Sad
Chapter 2. Difference Equations
16
Theorem 2.2.2 Gershgorins Theorem. Let A 2 Rnn and let Rk denote the circle in
the complex plane with center akk and radius rk =
n
X
j=1
j6=k
Rk =
8
>
>
<
>
>
:
z 2
Introductory Example
Basic definitions
First order linear equations
Cobwebbing method
Second and higher order equations
First-order linear systems
Approximating Eigenvalues
Inverse Power Method
QR dec
2.2 Basic definitions
33
of real positive roots is k including multiplicity). With Frobenius Theorem we know 1 is
dominant eigenvalue and 1 7 1, hence increases/decreases population. Therefore we just
What is Mathematical Modeling?
Computational Modeling
Exercises
Philosophy of
Mathematical Mo
1. Introduction
All models are wrong, but some are
useful.
George E. P. Box, 1919-2013
1.1
What is Mathema
Books
Articles
4. Markov Chains
Andrey Markov, Russian mathematician 1856-1922
Let Xt and Zt denote random variables and xt and xt their corresponding realizations,
e.g., Xt is the sum of two dices an
Chapter 3. Differential Equations
60
for some 2 (ti , ti+1 ). Since ti+1
ti = h and y 0 (ti ) = f (ti , yi ) we get
y(ti+1 ) = y(ti ) + hf (ti , y(ti ) +
h2 00
y (i )
2
2
Small step sizes justify the
Introduction
Local stability in first order equations
Phase planes
Linear delay differential equations
Growth Models
Local Stability in First Order Systems
Phase plane analysis
Numerical solvers for O
Fall 2017: CMDA 3605
Math. Modeling I
Homework 6
Dr. Matthias Chung
Submission deadline: 10/12/2017
1. For the nonlinear difference equation xt+1 = r xt x2t , show that there is a stable
1
2-cycle for
Fall 2017: CMDA 3605
Math. Modeling I
Homework 11
Dr. Matthias Chung
Submission deadline: 11/16/2017
1. Given the following differential equations
(a) y 0 = 1 + y 2 ,
(b) y 0 = t/y,
(c) y 0 + 2y = 4,
Fall 2017: CMDA 3605
Math. Modeling I
Homework 12
Dr. Matthias Chung
Submission deadline: 11/30/2017
1. Use Taylors method of order two to approximate the solution for each of the following
initial va
Fall 2017: CMDA 3605
Math. Modeling I
Homework 2
Dr. Matthias Chung
Submission deadline: 09/14/2017
1. The tent function T is given by
(
2x
for x 0.5,
T (x) =
2(1 x) for x > 0.5.
Determine the functio
Fall 2017: CMDA 3605
Math. Modeling I
Homework 5
Dr. Matthias Chung
Submission deadline: 10/05/2017
1. Consider the difference equation
x(t + 1) = 6 x(t)2
Find all equilibria and 2-cycle solution of t