MTH 442
Introduction to Difference Equations

Course Information and Syllabus, Spring 2020

Text:   M. R. S. Kulenovic and Orlando Merino,  Discrete Dynamical Systems and Difference Equations with Mathematica
Chapman&Hall/CRC Press, 2002.    

Lecture Notes  by   M. R. S. Kulenovic and O. Merino 

Prerequisites:  MTH 142, 243

About the Course: This course is an introduction to the basic concepts and techniques of difference equations for advanced undergraduates and beginning graduate students. Difference equations appear in situations where the (n + 1)st generation (or state) of a system depends upon some previous generations (or states). Such equations also appear naturally as discrete analogues of differential equations, and as numerical solutions of differential equations that model various diverse phenomena in biology, ecology, physiology, physics, engineering, economics, and other areas. In addition to performing mathematical analysis of difference equations, with the aid of a Computer Algebra System (Mathematica) you will experiment with difference equations, and discover that such equations possess fascinating properties with a great amount of structure. Some of these computer observations may be cast as theorems that you discover and prove!

Electronic Devices:

The exams and quizzes are of open book type. Cell phones, ipads, ipods, etc. should be

turned o during the quizzes and exams. Excepted from this are electronic pads and laptops

used for notetaking. In particular laptops with electronic version of the book are allowed.

Calculators are permitted in this class.


1.      Introduction to Difference Equations

2.      First order Difference Equations

3.      Linear equations with constant coefficients and variable coefficients

4.      Stability in both hyperbolic and non-hyperbolic cases, bifurcations  

5.      Linear theory for two dimensional systems of difference equations,

6.      Equilibrium solutions, stability, periodic solutions

7.      Period-doubling bifurcation

8.      Global dynamics for second order difference equations.


At the end of the semester, the student will be able to

1. Classify a given difference equation according to its type.

2. Investigate numerically, graphically and analytically, properties of solutions to difference equations

such as convergence to equilibria or periodic points, boundedness, chaotic behavior.

3. Find equilibrium and periodic solutions to autonomous scalar or planar difference equations, and

investigate their stability properties.

4. Analyze difference equations models by using computational and analytic tools.

5. Find and classify by type the bifurcation points of difference equations models.

6. Use computer simulations to make conjectures on the properties of solutions to difference equations,

and use mathematical analysis and other techniques to prove or disprove conjectures or claims about

difference equations.

Exams and Grading:

Course grade will be determined based on homework, quizzes, mid-term and final exams, Mathematica  projects and the final project. 

The exams could be in part of the take home variety.

TWO TESTS: 50 percent (25 percent each)

Mathematica PROJECTS, QUIZZES, AND HOMEWORK: 30 percent

FINAL PROJECT: 20 percent 

Computer Requirements

The use of computer algebra systems Mathematica is required. Mathematica worksheets dealing with the different problems in theory will be provided and demonstrated in the class. The textbooks comes with the simulation package Dynamica  which will be used for all simulation purposes. No programming is needed.

Instructor: Dr. M. Kulenovic                               Office: Lippitt 202D

Phone: 874-4436                                           
Online information: or

Office hours: MW: 9-10:30  and by appointment. 

Time:  MWF: 11-11:50

Place: Avedisan Hall 105

Illness Due to Flu 

The nation is experiencing widespread influenza-like illness. If any of us develop flu-like symptoms, we are being advised to stay home until the fever has subsided for 24 hours.  So, if you exhibit such symptoms, please do not come to class. Notify me at 874-xxxx or of your status, and we will communicate through the medium we have established for the class. We will work together to ensure that course instruction and work is completed for the semester.  

The Centers for Disease Control and Prevention have posted simple methods to avoid transmission of illness.  These include: covering your mouth and nose with tissue when coughing or sneezing; frequent washing or sanitizing your hands; avoiding touching your eyes, nose, and mouth; and staying home when you are sick.  For more information please view or .  URI Health Services web page, , will carry advice and local updates.

Special Needs:

Section 504 of the Rehabilitation act of 1973 and the Americans with Disabilities Act of 1990

require the University of Rhode Island to provide academic adjustments or the accommodations

for students with documented disabilities. The student with a disability shall be responsible for

self-identi cation to the Disability Services for Students in the Oce of Student Life, provid-

ing appropriate documentation of disability, requesting accommodation in a timely manner, and

follow-through regarding accommodations requested. It is the students responsibility to make ar-

rangements for any special needs and the instructors responsibility to accommodate them with

the assistance of the Oce of Disability Services for Students. Any student with a documented

disability is welcome to contact me as early in the semester as possible so that we may arrange

reasonable accommodations. As part of this process, please be in touch with Disability Services for

Students Oce at 330 Memorial Union, 401-874-2098,

 Useful links: 

1.  Difference Equations at URI

2.  The Dynamical Systems at Boston University

3.  Sprott's gateway

4.  Fractint