單調動態系統及其應用
本課程是由 國立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.
參考書目：
Hal L. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (Mathematical Surveys & Monographs),Amer Mathematical Society, 1995.
SZE-BI HSU, Ordinary Differential Equations with Applications (Applied Mathematics), 2nd Ed. World Scientific Publishing Company, 2013.
Xiao-Qiang Zhao, Dynamical Systems in Population Biology, 2nd Ed, Springer, 2017.
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| 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 | 碩士生 |
| 預備知識 | 無 |
| 課程提供 | 課程影音 課程綱要 |
| 週次 | 課程內容 | 課程影音 |
|---|---|---|
| Review on General Dynamical System | 線上觀看 | |
| Monotone Dynamical System Definitions Monotone Strongly monotone Strongly order preserving (SOP) | 線上觀看 | |
| Hirsch’s Convergence Theorem No attracting periodic orbit in monotone system | 線上觀看 | |
| Non-ordering of limit sets Limit set dichotomy Generic quasi-convergence | 線上觀看 | |
| Competitive and Cooperative Differential Equations Kamke Condition | 線上觀看 | |
| Positively invariant sets and Monotone Solutions Non-oscillation principle | 線上觀看 | |
| Topological Equivalence of flow of competitive or cooperative n-d systems to a flow on (n-1)-d system of Lipschitz differential equations | 線上觀看 | |
| Poincare-Bendixson Theorem of 3-d competitive systems Alternative Cones The Field-Noyes Model | 線上觀看 | |
| Strong Monotonicity A Biochemical Control Circuit Stability and the Perron-Frobenius Theorem | 線上觀看 | |
| Irreducible Cooperative Systems Competition and Migration Smale’s Construction | 線上觀看 | |
| Quasimonotone Systems of Parabilic Equations The Basic Setup, Maximum Principles Positively Invariant Sets Comparison and Monotonicity The Strong Order Preserving Property | 線上觀看 | |
| Irreducible Cooperative Systems A Biochemical Control Circuit Two Patch Competitive Models | 線上觀看 | |
| Quasimonotone Systems of Parabilic Equations The Basic Setup Strong Parabilic Maximum Principles | 線上觀看 | |
| Comparison Principle Krein-Rutman Theorem Weak Maximum Principle Positively Invariant Sets Comparison and Monotonicity | 線上觀看 | |
| Krein-Rutman Theorem Reaction-Diffusion system Principle eigenvalue Two species competition in the flow reactor model | 線上觀看 |
課程目標
(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.
參考書目
Hal L. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (Mathematical Surveys & Monographs),Amer Mathematical Society, 1995.
SZE-BI HSU, Ordinary Differential Equations with Applications (Applied Mathematics), 2nd Ed. World Scientific Publishing Company, 2013.
Xiao-Qiang Zhao, Dynamical Systems in Population Biology, 2nd Ed, Springer, 2017.