Instructor:	Dr. Ricardo Carretero
Lectures:	Mo, We: 2:00 - 3:15 @ Phys-147
Office Hours:	Mo, We: 12:00 - 1:00 @ GMCS-591 (or by appointment)
E-mail:

Syllabus:

Important dates:
Mon, Aug. 29: First lecture.
Mon, Sep. 12: Last day to adjust schedule (adds, drops, etc.)
Wed, Dec. 7: Last lecture before finals.

Description:
Dynamical Systems is the study of phenomena that evolve in space and/or time. Whether a particular system comes from Biology, Physics, Chemistry, or even the Social Sciences, Dynamical Systems is the subject that provides the mathematical tools for its analysis. The seminal work by Lorenz in 1963, gave scientists insight to recognize a new type of motion called Chaos. Chaotic systems, which can be very simple, are capable of generating erratic behavior that is different from the one produced by quasi-periodic systems with a large number of frequencies of oscillations. Chaotic systems and their applications are the subjects of this course. The course is intended for senior undergraduate and first year graduate students in Applied Mathematics, Computational Science, Engineering, Physics, Chemistry, Biology, etc. Examples from interdisciplinary areas will be covered. Most of the concepts and examples will be supplemented with Matlab-based codes. As part of the course, students will be given access to a computer laboratory to complete the computer-based coursework.

Textbook:

Chaos: An Introduction to Dynamical Systems. Authors: Kathleen T. Alligood, Tim D. Sauer, James A. Yorke. Publisher: Springer 1996. You can buy the textbook from the Aztec shop . In addition to a textbook, some material will be drawn from several reference books and a number of journal articles.

Prerequisites:
Good knowledge of Calculus is the minimum requirement. Familiarity with elementary Differential Equations or Linear Algebra can be helpful but it is not necessary. Some computer experience is also desirable.

Outline:
The course will be based on the first six chapters (plus chap. 10 if time allows) of the textbook:

CHAPTER 1. One-Dimensional Maps 1.1 One-dimensional maps 1.2 Cobweb plot: Graphical representation of an orbit 1.3 Stability of fixed points 1.4 Periodic points 1.5 The family of logistic maps 1.6 The logistic map G(x)=4x(1-x) 1.7 Sensitive dependence on initial conditions 1.8 Itineraries CHAPTER 2. Two-Dimensional Maps 2.1 Mathematical models 2.2 Sinks, sources, and saddles 2.3 Linear maps 2.4 Coordinate changes 2.5 Nonlinear maps and the Jacobian matrix 2.6 Stable and unstable manifolds 2.7 Matrix times circle equals ellipse CHAPTER 3. Chaos 3.1 Lyapunov Exponents 3.2 Chaotic orbits 3.3 Conjugacy and the logistic map 3.4 Transition graphs and fixed points 3.5 Basins of attraction

CHAPTER 4. Fractals 4.1 Cantor sets 4.2 Probabilistic constructions of fractals 4.3 Fractals from deterministic systems 4.4 Fractal basin boundaries 4.5 Fractal dimension 4.6 Computing the box-counting dimension 4.7 Correlation dimension CHAPTER 5. Chaos in Two-Dimensional Maps 5.1 Lyapunov exponents 5.2 Numerical calculation of Lyapunov exponents 5.3 Lyapunov dimension 5.4 A two-dimensional fixed-point theorem 5.5 Markov partitions 5.6 The horseshoe map CHAPTER 6. Chaotic Attractors 6.1 Forward limit sets 6.2 Chaotic attractors 6.3 Chaotic attractors of expanding interval maps 6.4 Measure 6.5 Natural measure 6.6 Invariant measure for one-dimensional maps CHAPTER 10. Stable Manifolds and Crises 10.1 The Stable Manifold Theorem 10.2 Homoclinic and heteroclinic points 10.3 Crises 10.4 Proof of the Stable Manifold Theorem 10.5 Stable and unstable manifolds for higher dimensional maps

References:

	[1]	Chaos: An Introduction to Dynamical Systems. K.T. Alligood, T.D. Sauer & J.A. Yorke. Springer 1996.
	[2]	Nonlinear Dynamics and Chaos, Steven H. Strogatz. Perseus Books.

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