Differential equations are utilized to analyze the mathematical structure of nonlinear phenomena, In this lecture we introduce methods to handle differential equations. In the first half of lectures, we study basic theories such existence theorem. In the second half, we study more advanced theories to analyze the large time behavior of solutions.
In this lecture, we study the basic concepts and methods to study the mathematical structure of nonlinear phenomena, We show the existence of solutions to ordinary differential equations. Then we discuss asymptotic analysis, bifurcation theory and limit cycle as special topics.
Ordinary differential equations
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
The lectures provide the fundamentals of ordinary differential equations.
Course schedule | Required learning | |
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Class 1 | Existence of time local solution | Understand the contents covered by the lecture. |
Class 2 | Uniqueness of solution | Understand the contents covered by the lecture. |
Class 3 | Dependence of solution on parameter | Understand the contents covered by the lecture. |
Class 4 | Existence of time global solution | Understand the contents covered by the lecture. |
Class 5 | Linear approximation of autonomous system | Understand the contents covered by the lecture. |
Class 6 | Stability and instability of equilibrium point | Understand the contents covered by the lecture. |
Class 7 | Asymptotic analysis by linearization | Understand the contents covered by the lecture. |
Class 8 | Lyapunov’ｓ method | Understand the contents covered by the lecture. |
Class 9 | Asymptotic analysis by Lyapunov’ｓ method | Understand the contents covered by the lecture. |
Class 10 | Stable, instable and center manifolds | Understand the contents covered by the lecture. |
Class 11 | Stable, unstable and center manifolds | Understand the contents covered by the lecture. |
Class 12 | Asymptotic analysis by center manifold theorem | Understand the contents covered by the lecture. |
Class 13 | Introduction to bifurcation theory | Understand the contents covered by the lecture. |
Class 14 | Limit Cycle | Understand the contents covered by the lecture. |
Class 15 | Poincaré–Bendixson theorem | Understand the contents covered by the lecture. |
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