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單調動態系統及其應用 Monotone Dynamical Systems with Applications in Biology

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本課程是由陽明交通大學應用數學系提供。

 

(1) Review on General Dynamical System, topological property of w-limit set, alpha limit set, 2-d Poincare Bendixson Theorem , Stable Manifold Theorem, Hopf Bifurcation, Chaotic Dynamical Systems.
 
(2) Monotone Dynamical System
Definitions, Monotone, Strongly monotone, Strongly order preserving (SOP)
Hirsch’s Convergence Theorem, No attracting periodic orbit in monotone system. Non-ordering of limit sets, limit set dichotomy , generic quasi-convergence.
 
(3) Competitive and Cooperative Differential Equations
Kamke Condition, Positively invariant sets and Monotone Solutions, Non-oscillation principle (Ito’s Lemma), Topological Equivalence of flow of competitive or cooperative n-d systems to a flow on (n-1)-d system of Lipschitz differential equations, Poincare-Bendixson Theorem of 3-d competitive systems, Alternative Cones, The Field-Noyes Model.
 
(4) Irreducible Cooperative Systems
Strong Monotonicity, A Biochemical Control Circuit, Stability and the Perron-Frobenius Theorem, Competition and Migration, Smale’s Construction
 
(5) Quasimonotone Systems of Parabilic Equations
Parabolic Systems: The Basic Setup, Maximum Principles, Positively Invariant Sets , Comparison and Monotonicity, The Strong Order Preserving Property, The Biochemical Control Circuit with Diffusion.
 
(6) Two species competition in the flow reactor model in a river system.  
 
(7) Sublinearity of Monotone Dynamical System, Butler McGhee Lemma, Uniform Persistence Theory.
 
(8) Jiang’s global stability theorem for cooperative systems
 
(9) PDE, Reaction-Diffusion system, Principle eigenvalue
 
(10) Krein-Rutman Theorem , Systems with coupled PDE and ODE, Generalized Krein-Rutman’s Theorem.
 
(11) Abstract theory of two species competition in ordered Banach Spaces, Dancer-Hess Lemma. 
Slower Diffuser vs Fast Diffuser, Two species Competition in an unstirred chemostat.
 
(12) Poincare Bendixson Theorem for Monotone Cyclic Feedback System.
授課教師 應用數學系 許世壁老師
課程學分 2學分
授課年度 107學年度
授課對象 研究生
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