UBC Mathematics 345: Nonlinear Dynamics and Chaos

Section 201; Room: LSK 460; Time: TTh 2:00-3:30

Website for this course: Visit http://www.math.ubc.ca/~yxli/m345_18.html
Contact:

logbif_vs_r.ode Ode file for Logistice bifurcation diagram.
logbif_vs_r_BlowUP1.ode Ode file for Logistice bifurcation diagram blow-up 1.
logbif_vs_r_BlowUP2.ode Ode file for Logistice bifurcation diagram blow-up 2.

Some Fun Links:


Homework Assignments: Important Links:
Rules for Exams and Final Grade Evaluation: Calendar: Text Book and Course Outline (Subject to changes as the course progresses!):

Recommended textbook (required): S. Strogatz, Nonlinear Dynamics and Chaos 2nd Ed. (2015)

Chapter:
Corresponding Textbook Sections
Hrs Topics
I. One-Dimensional Flows:
2.0-2.5, 2.8, 3.0-3.5, 4.0-4.4.
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  • A geometric way of thinking (the phase line)
  • Fixed points and stability
  • Population growth
  • Nondimensionalization
  • Linear stability analysis
  • Existence and uniqueness
  • Bifurcations
  • Saddle-node bifurcation, normal form
  • Transcritical bifurcation, normal form
  • Chemical kinetics
  • Pitchfork bifurcation, normal form
  • Overdamped bead on a rotating hoop
  • Dimensional analysis and scaling
  • Overdamped pendulum (with steady applied torque)
II. Two-Dimensional Flows:
5.0-5.2, 6.0-6.5, 6.7, 7.0-7.3, 8.0-8.3, 8.7
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  • Linear systems
  • Classification of linear systems
  • Stability language
  • The phase plane
  • Existence and uniqueness
  • Fixed points and linearization
  • Two competing species ("rabbits versus sheep")
  • The effect of small nonlinear terms
  • Conservative systems
  • Pendulum: undamped and damped
  • Limit cycles
  • Ruling out closed orbits
  • Poincare-Bendixson Theorem
  • Bifurcations in two dimensions: saddle-node, transcritical and pitchfork bifurcations
  • Hopf bifurcations
  • Oscillating chemical reactions
  • Poincare maps
III. Chaos:
10.0-10.3,9.2-9.5,10.4-10.5
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  • One-dimensional maps
  • Fixed points and linear stability analysis
  • The logistic map
  • Invariant sets, attractors, chaos
  • Liapunov exponents
  • Ruelle plots
  • Lorenz equations
  • Doubling map