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**Research Projects**** ****Additional Course Material ****Matlab Code**

**Instructor: **Prof. Daniele Venturi (venturi AT ucsc.edu)

**Teaching Assistant: **Tenavi Nakamura-Zimmerer (tenakamu AT ucsc.edu)

**Main Lectures: **Tu/Th 1:30-3:05PM, Soc. Sci 2 Room 075

**Discussion Session**: Tu 7:10-8:45PM, Soc. Sci 2 Room 075

**Office Hours**

- Prof. Daniele Venturi, Tuesdays 9-11AM, Room BE-353B
- Mr. Tenavi Nakamura-Zimmerer, Fridays 11AM-12PM BE-118, 3-4PM, Room BE312 C/D

**CANVAS COURSE WEBPAGE: **Click **HERE** and follow the instrcutions

**In Class Exams:** Tuesday Nov 7, 1:30-3:05PM, Soc. Sci 2 Room 075

**Final: **Wednesday Dec 13 4-7PM, Soc. Sci 2 Room 075

**Purpose of the course: **Many complex phenomena in physics, biology, and engineering can be modeled in terms of nonlinear dynamical systems. The purpose of the course is for students to learn how to extract quantitative and qualitative information from such models.

**Course content: **

*One-Dimensional Dynamical Systems -*Geometric approach, flows on the line, linear and nonlinear stability analysis of fixed points, potential systems, bifurcations (saddle-node, transcritical, pitchfork), normal forms.*Two-Dimensional Dynamical Systems -*Geometric approach, existence and uniqueness of solutions, linear systems (analytical solution, classification of fixed points), fixed points of linear systems in higher dimensions, equilibria in 2D nonlinear systems, Hartman-Grobman theorem, conservative systems, reversible systems, index theory, limit cycles (Liénard systems, criteria to rule out closed orbits), Poincaré-Bendixon theorem, Poincare maps, linear stability analysis of periodic orbits, bifurcations (normal form of co-dimension 1 bifurcations, heteroclinic and homoclinic bifurcations, Hopf bifurcations, global bifurcations of cycles).*High-Dimensional Dynamical Systems*- Liouville theorem, probability density function equations, Hamiltonian dynamical systems, Lorenz equations (symmetry, volume contraction, stability analysis of fixed points, trapping region, limit cycles, bifurcation analysis, transient chaos, strange attractor), Lyapunov exponents.*Discrete Dynamical Systems*: Examples (Poincaré map, numerical methods for ODEs, Henon map), n-dimensional maps (fixed points, linear stability analysis), one-dimensional maps (cobwebs, logistic map, tent map), analysis of the logistic map, Lyapunov exponents.

**Grading policy **

**AMS 114****:**Homework 50%, Midterm 25%, Final 25%, Research Project (non-mandatory).**AMS 214:**Homework 50%, Midterm 20% (in class), Research Project 30%.

**Textbook:** Steven H. Strogatz, ``Nonlinear Dynamics and Chaos''

**Other recommended textbooks**

- R. C. Robinson, ``An introduction to dynamical systems'', American Mathematical Society.
- S. Wiggins, ``Introduction to applied nonlinear dynamical systems'', Springer.
- M. W. Hirsch, S. Smale and R. L. Devaney, ``Differential equations, dynamical systems, and introdution to chaos'', Academic Press
- S. N. Rasband, ``Chaotic dynamics of nonlinear systems'', Dover.
- J. Guckenheimer and P. Holmes, ``Nonlinear oscillations, dynamical systems and bifurcations of vector fields'', Springer.
- Y. A. Kuznetsov, ``Elements of applied bifurcation theory'', Springer.