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Math 388: Introduction to Chaos Theory
Fall 2017 Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

Course Information

Course Description: An elementary treatment of chaos theory and its applications concentrating on discrete dynamical systems. Uses theory and applications illustrated by computer experiments to develop such topics as bifurcation, attractors, the logistic map, period-doubling routes to chaos, symbolic dynamics, Sarkovskii's theorem, fractals, and Julia and Mandelbrot sets for complex dynamics.

Number of Credits: 3

Prerequisites: MATH 211 with a grade of C or better or MATH 213 with a grade of C or better.

Course-Section and Instructors

Course-Section Instructor
Math 388-001 Professor D. Blackmore

Office Hours for All Math Instructors: Fall 2017 Office Hours and Emails

Required Textbook:

Title Chaos: An Introduction to Dynamical Systems + Notes
Author K. Alligood, T. Sauer and J. Yorke
Edition 1st
Publisher Springer
ISBN # 0-387-94677-2

Required Software: MATLAB with dfield and pplane (tutoring available).

University-wide Withdrawal Date: The last day to withdraw with a W is Monday, November 6, 2017. It will be strictly enforced.

Course Goals/ Miscellaneous

Course Objectives

Course Assessment: The assessment of objectives achieved through homework assignments, projects, regular in-class quizzes, and the midterm and final examinations.


DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

Homework and Quizzes 10%
Project 20%
Midterm Ecam 30%
Final Exam 40%

Your final letter grade will be based on the following tentative curve.

A 88 - 100 C 62 - 67
B+ 82 - 87 D 55 - 61
B 75 - 81 F 0 - 54
C+ 68 - 74

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced. Students accumulating more than three absences will have their grade reduced.

Homework and Quiz Policy: Homework is due on the assigned date; late homework will reduce the number of points awarded, and will only be accepted at discretion of the instructor. Quizzes are given on an announced topic at times specified by the instructor.

Exams: There will be one midterm exam held in class during the semester and one comprehensive final exam. Exams are held on the following days:

Midterm Exam November 1, 2017
Final Exam Period December 15 - 21, 2017

The final exam will test your knowledge of all the course material taught in the entire course. Make sure you read and fully understand the Math Department's Examination Policy. This policy will be strictly enforced.

Makeup Exam Policy: There will be No make-up QUIZZES OR EXAMS during the semester. In the event an exam is not taken under rare circumstances where the student has a legitimate reason for missing the exam, the student should contact the Dean of Students office and present written verifiable proof of the reason for missing the exam, e.g., a doctor’s note, police report, court notice, etc. clearly stating the date AND time of the mitigating problem. The student must also notify the Math Department Office/Instructor that the exam will be missed.

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

Additional Resources

Math Tutoring Center: Located in the Central King Building, Lower Level, Rm. G11 (See: Fall 2017 Hours)

Further Assistance: For further questions, students should contact their instructor. All instructors have regular office hours during the week. These office hours are listed on the Math Department's webpage for Instructor Office Hours and Emails.

All students must familiarize themselves with and adhere to the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. The Department of Mathematical Sciences takes these policies very seriously and enforces them strictly.

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at The office is located in Fenster Hall Room 260. A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Fall 2017 Academic Calendar, Registrar)

Date Day Event
September 5, 2017 T First Day of Classes
September 11, 2017 M Last Day to Add/Drop Classes
November 6, 2017 M Last Day to Withdraw
November 21, 2017 T Thursday Classes Meet
November 22, 2017 W Friday Classes Meet
November 23 - 26, 2017 R - Su Thanksgiving Break - University Closed
December 13, 2017 W Last Day of Classes
December 14, 2017 R Reading Day
December 15 - 21, 2017 F - R Final Exam Period

Course Outline

Date Sections Topic Assignment
9/6 1.1 -1.3 1D Maps, Cobweb Maps, Stability of Fixed Points Selected Probs.
9/8 1.4 -1.6 Periodic Points, Logistic Map Selected Probs.
9/13 1.7,1.8 Sensitive Dependence and Itineraries Selected Probs.
9/15 2.1 2D Map Models Selected Probs.
9/20 2.2 – 2.4 Hyperbolic Fixed Points, Linear Maps, Coordinate Changes Selected Probs.
9/22 2.5, 2.6 Jacobian, Stable and Unstable Manifolds Selected Probs.
9/27 3.1,3.2 Lyapunov Exponents, Chaotic Orbits Selected Probs.
9/29 3.3   Conjugacy and Logistic Map                                          Selected Probs.
10/4 3.4, 3.5 Transition Graphs, Basins of Attraction, Sharkovskii’s Theorem Selected Probs.
10/6 Labs + Chapters 1 , 2 and 3 Selected Probs.
10/11 4.1, 4.2 Cantor Sets, Probabilistic Construction of Fractals Selected Probs.
10/13 4.3, 4.4 Deterministic Construction of Fractals, Mandelbrot and Julia Sets Selected Probs.
10/18 4.5 – 4.7 Fractal Dimensions: Box-Counting and Correlation Dimensions Selected Probs.
10/20 Labs + Chapter 4 Selected Probs.
10/25 5.1, 5.2  2D Chaos: Lyapunov Exponents and Their Calculation
10/27 5.3 Lyapunov Dimension +  REVIEW for MIDTERM EXAM  Selected Probs.
11/1 ----------- MIDTERM EXAM  ----------------------
11/3 5.4 Brouwer Fixed Point Theorem Selected Probs.
11/8 5.5 Markov Partitions, Shift Map, Baker Map Selected Probs.
11/10 5.6 Smale Horseshoe Map Selected Probs.
11/15 5.6, Notes Smale Horseshoe and Symbolic Dynamics  Selected Probs.
11/17 Labs + Chapter 5 Selected Probs.
11/22 6.1, 6.2 Limit Sets and Chaotic Attractors -------------------
11/29 6.2 Chaotic Attractors Selected Probs.
12/1 7.1 – 7.3 Sytems of ODEs, Poincaré Maps (Sections) Selected Probs.
12/6 7.4 – 7.7 Nonlinear Systems and Examples Selected Probs.
12/8 8.1,9.1-9.3 Poincaré-Bendixson Theorem, Lorenz, Rössler, and Chua Attractors Selected Probs.
12/13 --------- REVIEW FOR FINAL EXAM  ------------------

Updated by Professor D. Blackmore - 9/1/2017
Department of Mathematical Sciences Course Syllabus, Fall 2017