For each project below, I provide references based on the bibliography at the bottom of this page.

The suggested content of the project is not binding. You may decide to adjust the topic and the sources of the project. In that case, please let me know.

Do not hesitate to contact me if you have questions.

PROJECTS:

1. Parametric Dependence of the Rotation Number (Yuchu Duan)
   • DRAFT - Reviewers: Annika Fuernsinn and Natalie Ranta
   • FINAL PAPER - Reviewers: Quintin Trayling and Kevin Doutre
   • Main result: Proposition 4.4.13 of [HK03].
   • Necessary material: Sections 4.4.1 and 4.4.4 of [HK03]
   • Assume without proof: material from Sections 4.3.1 - 4.3.6 and 4.4.2 - 4.4.3 of [HK03].
2. Dulac’s Criterion : Relation to Non-Periodic Orbits Of Planar Systems (Jack McKinnon)
   • DRAFT - Reviewers: Yuchu Duan and Adam Paquette
   • FINAL PAPER - Reviewers: Landon McDougall and Sonja Ruzic
   • Main result: Corollary 5.4 and Lemma 5.7 of [Gle94].
   • Necessary material: parts of Section 4.2 and Sections 5.5 - 5.6 of [Gle94] .
3. The Global Attractor of the Lorenz Equations (Matt Spragge)
   • DRAFT - Reviewers: Quintin Trayling and Annika Fuernsinn
   • FINAL PAPER - Reviewers: Yuchu Duan and Kevin Doutre
   • Main results: material from Section 8.2 of [Tes12].
   • Possible additional material: Section 11.6 of [Tes12] and references for Chapter 8 on page 342 of [Tes12].
4. Stability via Lyapunov's method. (Annika Fuernsinn)
   • DRAFT - Reviewers: Natalie Ranta and Emma May
   • FINAL PAPER - Reviewers: Landon McDougall and Keenan McPhail
   • Main result: Theorem 2.9, Theorem 2.10 and Theorem 2.11 in Section 2.2 of [Gle94].
   • Additional material: Theorem 6.13 and Theorem 6.14 in Section 6.6 of [Tes12].
5. Conditions for Chaos in Tent Maps (Quintin Trayling)
   • DRAFT - Reviewers: Jack McKinnon and Landon McDougall
   • FINAL PAPER - Reviewers: Quentin Sanders and Natalie Ranta
   • Main result: Proposition 11.23 and Theorem 11.25 of [Gle94].
   • Necessary material: Section 11.4 of [Gle94].
   • Assume without proof: material from Sections 11.1 - 11.3 of [Gle94].
6. Characterizing Minimal Translations of the Torus (Alexander Zotine)
   • DRAFT - Reviewers: Matt Spragge and Keenan McPhail
   • FINAL PAPER - Reviewers: Jack McKinnon and Annika Fuernsinn
   • Main result: Proposition 5.1.2 in Section 5.1.5 of [KH03].
   • Possible additional result: Theorem 5.1.5 in Section 5.1.4 of [KH03].
   • Necessary material: Sections 5.1.1, 5.1.2, 5.1.4, 5.1.5 of [KH03].
7. Poincare' Recurrence and Image Scrambling (Landon McDougall)
   • DRAFT - Reviewers: Alexander Zotine and Kevin Doutre
   • FINAL PAPER - Reviewers: Jack McKinnon and Matt Spragge
   • Main results: Proposition 6.1.3 in Section 6.1.1, Theorem 6.1.6 and Corollary 6.1.7 in Section 6.1.2 of [KH03].
   • Necessary material: Sections 6.1.1 - 6.1.2 of [KH03].
8. Recurrence and uniform recurrence. (Quentin Sanders)
   • DRAFT - Reviewers: Kevin Doutre and Matt Spragge
   • FINAL PAPER - Reviewers: Alexander Zotine and Adam Paquette
   • Main results: Theorem 6.1.9 in Section 6.1.2 and Theorem 6.1.13 in Section 6.1.3 of [KH03]
   • Necessary Material: 6.1.2 - 6.1.3 of [KH03].
   • Assume without proof: material from Section 6.1.1 and Lemma A.1.15 (Baire Category Theorem) of [KH03].
9. Hyperbolic automorphisms of the torus. (Emma May)
   • DRAFT - Reviewers: Sonja Ruzic and Quentin Sanders
   • Main result: Proposition 7.1.10 of Section 7.1.4 of [KH03].
   • Necessary material: Section 7.1.4 of [KH03].
   • Possible additional material: Section 7.2.4 of [KH03].
10. Coding of the CAT map (Sonja Ruzic)
    • DRAFT - Reviewers: Adam Paquette and Yuchu Duan
    • FINAL PAPER - Reviewers: Alexander Zotine and Quentin Sanders
    • Main result: Theorem 7.4.9 and Theorem 7.4.10 in Section 7.4.5 of [KH03].
    • Necessary material: parts of Section 7.3 of [KH03].
    • Assume without proof: Proposition 7.1.10 in Section 7.1.4 of [KH03].
11. Properties of Topological Entropy in Dynamical Systems (Natalie Ranta)
    • DRAFT - Reviewers: Emma May and Jack McKinnon
    • FINAL PAPER - Reviewer: Keenan McPhail
    • Main result: Propositioni 8.2.9 in Section 8.2.6 of [KH03].
    • Necessary material: Sections 8.2.1, 8.2.2, 8.2.4, 8.2.5 of [KH03].
12. Noether's theorem and the Kepler problem (Kevin Doutre)
    • DRAFT - Reviewers: Keenan McPhail and Alexander Zotine
    • FINAL PAPER - Reviewers: Matt Spragge and Annika Fuernsinn
    • Main result: Theorem 8.9 in Section 8.3 of [Tes12] and its application to Kepler's problem (Section 8.5).
    • Assume without proof: necessary parts of Sections 8.1, 8.2 and 8.4 of [Tes12].
13. Stranger repeller in the logistic map for parameter values greater than 2 + √8 (Adam Paquette)
    • DRAFT - Reviewers: Quentin Sanders and Sonja Ruzic
    • FINAL PAPER - Reviewers: Quintin Trayling and Natalie Ranta
    • Main results: Theorem 11.21 and Theorem 11.22 in Section 11.7 of [Tes12].
    • Necessary material: parts of Sections 11.4 - 11.6 of [Tes12].
    • Assume without proof: material from Sections 11.4 - 11.5 of [Tes12].
14. The stable manifold theorem (Keenan McPhail)
    • DRAFT - Reviewers: Landon McDougall and Quintin Trayling
    • FINAL PAPER - Reviewers: Yuchu Duan and Sonja Ruzic
    • Main result: Theorem 4.7 in Sections 4.2 and 4.6 of [Gle94].
    • Needed material: pats of Section 4.5 of [Gle94].
    • Assume without proof: material from Sections 4.1 (Poincaré Linearization Theorem) of [Gle94].
    • Possible additional material: Section 9 of [Tes12].

BIBLIOGRAPHY:

• [HK03] Boris Hasselblatt and Anatole Katok. A first course in dynamics. With a panorama of recent developments. Cambridge University Press, 2003. ISBN 0-521-58304-7, 0-521-58750-6.
• [Gle94] Paul Glendinning. Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge University Press, 1994. ISBN 0-521-42566-2.
• [Tes12] Gerald Teschl. Ordinary differential equations and dynamical systems. Graduate Studies in Mathematics. Volume 140. American Mathematical Society, 2012. ISBN 978-0-8218-8328-0.

*** oral presentations have been cancelled ***