Teaching

Nonlinear Dynamical Systems (AM 114/214)

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

COURSE NOTES:

  • One-dimensional dynamical systems (PDF) 
  • Bifurcations of equilibria in one-dimensional systems (PDF)
  • Introduction to n-dimensional systems (PDF)
  • Linear dynamical systems (PDF)
  • Stability analysis of equilibria in nonlinear dynamical systems (PDF) 
  • Conservative and gradient systems (PDF)
  • Lagrangian and Hamiltonian dynamics (PDF)
  • Global theory for two-dimensional systems (PDF)
  • Bifurcations in n-dimensional nonlinear systems (PDF) 
  • Lorenz equations (PDF)

 

 Grading policy: CANVAS course webpage

 

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 (Neural networks, 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.

 

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

  

Other recommended textbooks

  • L. Perko, ``Differential equations and dynamical systems'', Springer
  • J. Guckenheimer and P. Holmes, ``Nonlinear oscillations, dynamical systems and bifurcations of vector fields'', Springer.
  • S. Wiggins, ``Introduction to applied nonlinear dynamical systems'', Springer.