VO 250082 Dynamical systems and nonlinear differential equations

Lecturer: Prof. Henk Bruin

Email H. Bruin for further information for this course.

Proseminar by: Dr. Henna Koivusalo

Announcements

Schedule

Exercise Schedule

Contents of the course

Basic notions for continuous and discrete dynamical systems; flows, attractors, and chaos; stability of stationary points by linearization and by Lyapunov functions, Poincaré-Bendixson theory, bifurcations.

The course will be given in English

Contents of the course (S = Schmeiser's notes, T = Teschl's book), St = Strogatz lecture on youtube.

• Bifurcations (S§5. T§6.5. T§11.1.)
• Chaos (T§10.3)
• Cusp bifurcation (S§5.4, St)
• Existence, uniqueness, continuity of solutions of ODEs (S§1. T§2.2. T§2.4.)
• Euler-Lagrange equations (T§8.3. S§10)
• Feigenbaum (bifurcation diagram) (S§6. T§11.1, St)
• Hamiltonian systems (S§10. T§8.3. St)
• Harmonic and other oscillators (S§8.3.)
• Homoclinic orbits and tangles (S Def 10. T§13.2)
• Hopf bifurcation (S§8.4. T p219-220)
• Inhomogeneous linear ODEs (S§2.1. T§3.2.)
• (In)stability of equilibria, sink, sources, saddles, centers (T§3.2.)
• (In)stability of fixed points (T§6.5, T§10.2.)
• Iteration of maps, cobweb diagrams (T§10.2.)
• Kepler problem = two-body problem (T§8.5. S§10)
• Lagrangian = Lagrange function (T§8.3. S§10)
• Legendre transform (T§8.3. S§10)
• Limit cycles (S§8), see also Van der Pol equation and Poincaré-Bendixson.
• Linear ODEs (S§2. T§3.2. T§3.3 St)
• Linearization, the Hartman-Grobman theorem (S Thm 8. T§9.3.)
• Logistic differential equation (T§10.1.)
• Logistic (= quadratic) map (T§10.1. St)
• Lorenz system (S§9. T§8.2., St and St)
• Lyapunov functions (S§7. T§6.6. St))
• Orbits, omega-limit sets (T§6.3 T§8.1. S Def 4.)
• Pendulum (T§6.7. S§10 (Example 8).)
• Phase portraits (T§3.2.)
• Poincaré maps (T§6.4. T§12.2)
• Poincaré-Bendixson Theorem (S§8.5. T§7.3, St)
• Sensitive dependence on initial conditions (T§10.3)
• Sharkovskiy's Theorem (T§11.2)
• Smale's horseshoe (T§13.1.)
• Stable/unstable manifolds (S Thm 7. T§9.2. T§12.3.)
• Structural stability
• Symbolic dynamics (T§11.5.)
• Strange attractors (T§11.6.)
• Van der Pol equations (S§8, T§7.2, St)

References/Background Reading

• Kathy Alligood, Tim Sauer and James Yorke, Chaos, an Introduction to Dynamical Systems, Springer 1996 ISBN 0-387-94677-2
• Michael Brin and Garrett Stuck, Introduction to Dynamical Systems, Cambridge University Press 2002 ISBN 0-521-80841-3
• Robert Devaney, An Introduction to Cahotic Dynamical Systems, Benjamin and Cummings Publishing 1986, ISBN 0-8053-1601-9
• Clark Robinson, Dynamical Systems (Stability, Symbolic Dynamics, and Chaos), CRC Press 1995 ISBN 0-8493-8493-1.
• Steven Strogatz, Nonlinear dynamics and chaos, with applications to physics, biology and engineering, CRC Press, 2015, ISBN-13: 978-0813349107 or ISBN-10: 0813349109
• ODE Classnotes (or in fact book) of Prof. G. Teschl.

Assessment

Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich).

Exam Material:

• Linear ODEs and phase portraits
• Existence, uniqueness and continuity of solutions of ODEs
• Iteration of maps on the line (cobweb diagrams), circle and plane
• Types of orbits (fixpoint/equilibria, periodic, saddle/sink/source/center) and their stability (hyperbolicity)
• Stable and unstable manifold and spaces - Hartman-Grobman Theorem
• Bifurcations (saddle-node/font, transcritical, pitchfork, period doubling, Hopf, cusp)
• Poincaré-Bendixson Theorem, alpha/omega-limit sets and limit cycles
• Lyapunov functions
• Definitions of chaos, sensitive dependence on initial conditions.
• Symbolic dynamics
• Smale's horseshoe
• Hamiltonian systems and first integrals
• Examples: logistic maps, logistic differential equations, Van der Pol equations, Lorenz equations, harmonic oscillator and pendulum

Course material (Hand-outs)

• Class notes in pdf (written by Christian Schmeiser)
• Exercises for the Proseminar.
• An Applet for cobweb diagrams for the logistic family.
• An Applet for the bifurcation diagram for the logistic family.
• Hand-out about structural stability
• Hand-out about symbolic itineraries
• Some bifurcation diagrams:

  Pitchfork bifurcation	Period doubling bifurcation	Transcritical bifurcation
  Spruce budworm (Caterpillar and Moth)	Two saddle node (= fold) bifurcations meeting in a cusp (x,k)-plane	Bifurcation surface for cusp (projecting to (r,k)-plane

  as well as a lecture by Strogatz on the same topic.
• Some phase portraits for the Van der Pol oscillator:
  and also some youtube videos here, here, and a lecture by Strogatz (Cornell University).
• Some youtube videos on Huygens resonance here, here, here, and here, and if you still didn't have enough, also here.
• Some youtube videos on Smale and his horseshoe: here and here.

Smale's horseshoe	A homoclinic tangle	Another homoclinic tangle

• Some youtube videos on Lorenz and the Lorenz attractor. here, here, and here and the Wikipedia page
• Written out class notes of June 18 in pdf.
• Some youtube videos on three-body choreographies, here, here, and here and here. Or this one.
• Some youtube videos on the Kepler problem (motion by the Earth and Sun), on gravitation, by Feynman (sort of), and general.
• Online Lecture by Anima Nagar
• Preview
• Lecture 1
• Lecture 2
• Lecture 3