You must have B or higher in MATH 1502 OR MATH 1512 OR MATH 1555 OR MATH 1504 ((MATH 1552 OR MATH 15X2 OR MATH 1X52) AND (MATH 1522 OR MATH 1553 OR MATH 1554 OR MATH 1564 OR MATH 1X53)) to enroll.

Differential Equations, Dynamical Systems, and an Introduction to Chaos, Hirsch, Smale, and Devaney

Ordinary Differential Equations: Basics and Beyind, Schaeffer and Cain (free download from GT library)

Nonlinear Dynamics And Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Strogatz

Final grades will be calculated as follows: A:90-100, B:80-89, C:70-79, D:60-69, F: <60

**Meeting time**: TuTh 3-4:15 in Skiles 249**Office hours:**TBA, or by appointment**Recitation**:**Required Texts:**Differential Equations, Dynamical Systems, and an Introduction to Chaos, Hirsch, Smale, and Devaney

Ordinary Differential Equations: Basics and Beyind, Schaeffer and Cain (free download from GT library)

**Suggested Reference: **Nonlinear Dynamics And Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Strogatz

**Grading:**Quizzes (20%), 2 exams (40%), final exam (30%), project (10%)**[still in flux]**Final grades will be calculated as follows: A:90-100, B:80-89, C:70-79, D:60-69, F: <60

Our quantitative understanding of the world is often expressed as relationships between rates of change of quantities, such as position and time, chemical concentrations and time, population densities and time, amplitude and space, energy and temperature, etc. Such relationships are expressed in the language of differential equations. In this course we will explore classical, qualitative, and numerical approaches to understanding solutions of ordinary differential equations.

Linear differential equations can be broken down into parts, each of which can be easily analyzed, and then the effects added together. Nonlinear systems are fundamentally different; their components interact in complicated and often unpredictable ways, Nonlinear systems can exhibit a rich and complicated “zoo” of behaviors, most of which are impossible in linear systems.

Since the vast majority of ODEs in science and engineering are nonlinear and do not possess classical (closed form analytical) solutions, qualitative and numerical methods are most often used in practice to analyze ODEs. In this honors course we will focus on studying linear systems and then applying this knowledge to study the qualitative behavior of non-linear systems. We will study qualitative methods in greater depth than in Math 2552, and be more rigorous in our study throughout the semester.

Students with disabilities who require reasonable accommodations to participate fully in the course activities or meet course requirement are encouraged to register with the ADAPTS Disability services at 404.894.2564 or http://adapts.gatech.edu. Please contact me ahead of time to discuss any issues related to disabilities.

Linear differential equations can be broken down into parts, each of which can be easily analyzed, and then the effects added together. Nonlinear systems are fundamentally different; their components interact in complicated and often unpredictable ways, Nonlinear systems can exhibit a rich and complicated “zoo” of behaviors, most of which are impossible in linear systems.

Since the vast majority of ODEs in science and engineering are nonlinear and do not possess classical (closed form analytical) solutions, qualitative and numerical methods are most often used in practice to analyze ODEs. In this honors course we will focus on studying linear systems and then applying this knowledge to study the qualitative behavior of non-linear systems. We will study qualitative methods in greater depth than in Math 2552, and be more rigorous in our study throughout the semester.

**Learning Outcomes**- Learn what are ODEs, famous examples from physics, chemistry and biology, classifications of ODEs.
- Learn to model real-world phenomena using ODEs
- Learn analytical (mainly separable) and graphical methods to analyze first order ODEs, as well as the concepts of phase flow and Poincar\’e map for an ODE. Learn to find analytical solutions using Matlab and Mathematica.
- Learn the statements (and perhaps proofs) of the fundamental existence and uniqueness of solutions of ODEs and the continuous dependence of solutions, variational equation.
- Learn to use Matlab’s suite of numerical ODE solvers and understand the sources of numerical errors, dangers, and best practices.
- Learn the linearity principle and to solve first order linear systems of ODEs, with emphasis on 2 and 3 dimensions. Sketch the phase plane, identify the equilibrium points, and determine their stability.
- Learn to solve second order linear ODEs with constant coefficients using 4. and variation of parameters. Learn how oscillations and resonance arise in these ODEs.
- Learn basic properties of Laplace Transforms and their application to solve 2nd order linear ODEs with constant coefficients with non-smooth forcing.
- Learn to find equilibrium points of nonlinear systems in the plane, use the Hartman-Grobman theorem to deduce stability and sketch the phase flow around the equilibrium points. Stable manifold theorem for nonlinear saddles in planar systems. Find conserved quantities, null-clines, and sketch as much of the phase plane as possible for nonlinear ODEs.
- Practice clear, concise communication of mathematical ideas.

Students with disabilities who require reasonable accommodations to participate fully in the course activities or meet course requirement are encouraged to register with the ADAPTS Disability services at 404.894.2564 or http://adapts.gatech.edu. Please contact me ahead of time to discuss any issues related to disabilities.

**Homework Exams Links Lecture Schedule**