Nonlinear Dynamical Systems (AM 114/214)

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Instructor: Prof. Daniele Venturi (venturi@ucsc.edu)

Teaching Assistant: Abram Rodgers (akrodger@ucsc.edu), Paul Wintz (pwintz@ucsc.edu)

Main Lectures:  Tu/Th 11:40AM -1:15PM, Physical Sciences, room 114

Discussion Section: Wed 1:20-2:25PMPhysical Sciences, room 110

Office Hours

  • Prof. Daniele Venturi, Wednesday 10AM-12PM, Room BE-361C
  • Abram Rodgers, Monday 5-6PM, BE-151
  • Paul Wintz, Thursday 10:00-11:00AM, BE-312 C/D.

CANVAS COURSE WEBPAGES:  AM 114, AM 214

LECTURE NOTES: PDF

Midterm: November 12 (Tuesday), 11:40AM -1:15PM, Physical Sciences, room 114

Final: December 11 (Wednesday), 8-11AM Physical Sciences, room 114.

 


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, forward and inverse flow maps. 
  • Two-Dimensional Dynamical Systems - Geometric approach, existence and uniqueness of solutions, linear systems (analytical solution, classification of fixed points), equilibria in nonlinear systems, Hartman-Grobman theorem, center manifolds, 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 - Existence and uniqueness of solutions, flow maps, linear systems, Liouville theorem, probability density function equations, Lagrangian and Hamiltonian dynamics, fixed points and stability analysis, center manifolds, bifurcation analysis of equilibria, Hopf bifurcation, 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, Newton's method), 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 Assignments 50%, Midterm 25%, Final 25%, Research Project (non-mandatory).
  • AMS 214: Homework Assignments 50%, Midterm 25%, Final 25%, Research Project (non-mandatory).

 Late homework policy: Homework assignments will be accepted up to three days past the deadline. A 10% grade point deduction is taken for each day the assignment is late.

Textbook: Steven H. Strogatz, ``Nonlinear Dynamics and Chaos'', CRC Press, 2018.

 

 

Other recommended textbooks

  • L. Perko, ``Differential equations and dynamical systems'', Springer
  • 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
  • Y. A. Kuznetsov, ``Elements of applied bifurcation theory'', Springer (PDF)
  • S. N. Rasband, ``Chaotic dynamics of nonlinear systems'', Dover. 
  • J. Guckenheimer and P. Holmes, ``Nonlinear oscillations, dynamical systems and bifurcations of vector fields'', Springer.