Math 6104
Infinite Dimensional Dynamical Systems
Fall, 2018
| Instructor: |
Dr. Xiaoqiang Zhao, HH-2009 |
| Time: |
Monday and Wednesday, 9:00-10:15 |
| Classroom: |
HH-3013 |
| Office Hours: |
Tu 10:30-12:00, Wed 13:00-14:30 |
Text:
X. Zhao, "Dynamical Systems in Population Biology",
second edition, Springer-Verlag, New York, 2017.
Evaluation:
| Assignments |
50% |
| Project(written submission and oral presentation) |
50% |
Course Contents
- Introduction: The dynamical systems approach to evolution equations
- Dissipative Dynamical Systems
- Limit sets and global attractors
- Chain transitive sets
- Uniform persistence
- Coexistence states
- Monotone Dynamics
- Attracting order intervals and connecting orbits
- Global attractivity and convergence
- Subhomogeneous maps
- Periodic semiflows
- Traveling Waves and Spreading Speeds
- Motivations
- Monostable waves and spreading speeds
- Bistable waves and global stability
- Applications to biological invasions
References
- J. K. Hale, "Asymptotic Behavior of Dissipative Systems",
Amer. Math. Soc., Providence, 1988.
- G. R. Sell and Y. You, "Dynamics of Evolutionary Equations",
Applied Math Sciences 143, Springer-Verlag, New York, 2002.
- H. L. Smith, "Monotone Dynamical Systems, An Introduction to the
Theory of Competitive and Cooperative Systems", Mathematical Surveys and
Monographs 41, Amer. Math. Soc., Providence, RI,1995.
- R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics
and Physics", Applied Math Sciences 68, Springer-Verlag,
New York, 1988.
-
P. Magal and X. Zhao,
Global attractors and steady states for uniformly persistent dynamical systems,
SIAM J. Math. Anal., 37(2005), 251-275.
-
X. Liang and X. Zhao,
Asymptotic speeds of spread
and traveling waves for monotone semiflows with
applications, Communications on Pure and Applied Math.,
60(2007), 1-40.
- J. Fang and X. Zhao,
Bistable traveling waves for monotone semiflows with applications,
Journal of European Mathematical Society, 17(2015), 2243-2288.
Assignments:
Reading Materials (for assignments):
Project Topics (for final exam):
-
Persistence and attractors (Section 1.3.3 in Zhao's 2017 book)
-
Asymptotically periodic semiflows (Section 3.2 in Zhao's 2017 book)
-
A discrete-time Chemostat model (Sections 4.1 and 4.2 in Zhao's 2017 book)
-
The comparison principle for ODEs (Section 3.1 in Smith's 1995 book)
-
Fisher waves in an epidemic model
Presentation Schedule: Dec. 3, Monday, 9:00-12:00, SN4040