Teaching

Nonlinear Dynamical Systems (AM 224)

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

COURSE NOTES:

  • One-dimensional dynamical systems (PDF) 
  • Introduction to n-dimensional systems (PDF)
  • Global theory for 2D systems (PDF)
  • Linear dynamical systems (PDF)
  • Stability analysis of equilibria in nonlinear dynamical systems (PDF) 
  • Liouville theorem (PDF)
  • Lyapunov exponents (PDF)
  • Conservative and gradient systems (PDF)
  • Lagrangian and Hamiltonian dynamics (PDF)
  • Bifurcation of equilibria in 1D systems (PDF)
  • Bifurcation of equilibria in n-D systems (PDF) 
  • Lorenz equations (PDF)

 

 Grading policy: CANVAS course webpage

 

Purpose of the course

This course is aimed at graduate students in engineering, physics, and mathematics. The goal is to introduce graduate students to nonlinear dynamical systems described by systems of nonlinear ordinary differential equations, and to show how this knowledge can be used to understand the wonders of the nonlinear world. 

Tentative syllabus

  • Week 1: Overview of nonlinear dynamical systems and their classification. Flows on the line. Geometric approach. Fixed points. Linear and nonlinear stability analysis of fixed points.
  • Week 2: Forward and inverse flow maps. One-dimensional systems evolving from random initial states and system driven by random processes: Liouville and Fokker-Planck equations.
  • Week 3:  Introduction to high-dimensional dynamical systems.  Geometric approach,  existence and uniqueness of solutions. Linear dynamical systems. Applications. 
  • Week 4: Classification of equilibria. Linearized dynamics, Hartman-Grobman theorem. Stable and unstable manifolds of fixed points. Center manifolds. Lyapunov stability theory, inverse Lyapunov theorems, La Salle invariance principle.
  • Week 5: Dynamics of volumes in phase space. Liouville theorem and Liouville equations for systems evolving from random initial states. Systems driven by random noise.  Fokker-Plank equations.
  • Week 6: Limit cycles. Criteria to rule out periodic orbits in 2D systems. Poincaré-Bendixson theorem.
  • Week 7: Conservative systems. Lagrangian and Hamiltonian dynamics. Geometric aspects of Hamiltonian dynamics. 
  • Week 8: Introduction to bifurcations in nonlinear dynamical systems. Bifurcation of equilibria in 1D systems (saddle-node, transcritical, pitchfork).
  • Week 9: Bifurcation analysis in higher dimensions. Fundamental concepts. Co-dimension one bifurcations  (saddle-node, transcritical, pitchfork) in high-dimensions. Hopf bifurcations. 
  • Week 10: Measures of chaos. Lyapunov exponents and their computation. Lyapunov time and predictability. Lorenz equations, strange attractors. Other topics. 

Textbooks   

  • S. Wiggins, ``Introduction to applied nonlinear dynamical systems'', Springer.
  • H. Khalil, ``Nonlinear control'', Pearson.  
  • L. Perko, ``Differential equations and dynamical systems'', Springer.
  • R. C. Robinson, ``An introduction to dynamical systems'', American Mathematical Society. 
  • S. H. Strogatz, ``Nonlinear Dynamics and Chaos'', CRC Press.
  • J. Guckenheimer and P. Holmes, ``Nonlinear oscillations, dynamical systems and bifurcations of vector fields'', Springer.
  • Y. A. Kuznetsov, ``Elements of applied bifurcation theory'', Springer.