Lecture : MATH 130 Review
数学Midterm代考 The section is the focus of our final exam. More than half of the points will be distributed to the subject of this section.
- Below is a summary of the topics/skills you need to master from each of the sections. Mastering the topics below, you will be able to solve the majority of problems in the exam. There will be 10-20% of challenging problems that require you to have a comprehensive understanding of the materials, and be able to use them in clever ways.
For each section, there are some examples listed. The problems in the exam will be of similar flavor. There are certainly a lot more exercises in the textbook that are not listed here. It does NOT mean that they are irrelevant. Homework is a great recourse as well, and the problems there might not be listed again.
The solution to the problem with a * will be provided in a separate file later.
A sample exam can be made easily from this: sample 2-4 examples below from sections 2-5, and sample 1-3 examples from section 6 (in general there are more questions and tasks in the examples of section 6).
2 Flows on the line
Be able to
• find and classify fixed points;
• draw phase portrait (Examples: 2.2.5, 2.2.7);
• analyze the asymptotic behaviour of solutions including convergence, finite time blow up, and impossibility of oscillations (Examples: 2.4.9). An interesting Example: 2.2.10.
3 Bifurcations 数学Midterm代考
This is one of the most important materials in this course. Given a system, you should be able to
• tell if there is bifurcation or not;
• if bifurcation occurs, classify the type of bifurcation (saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork) and draw the bifurcation diagram (Examples: 3.1.1, 3.2.1, 3.4.1, 3.4.5–3.4.10, 3.5.7.);
• simplify a system to its dimensionless form.
• if there are more than one parameter, find bifurcations of one parameter by fixing other parameters in different regions, and find bifurcation curves (Examples: 3.6.3, 3.7.4);
4 Flows on the circle
Given a system, you should be able to
• find and classify all the fixed points, and sketch the phase portrait on the circle (Examples: 4.1.2–4.1.7);
• identify the asymptotic limit of trajectories;
• for a uniform or nonuniform oscillator, if the solution is periodic, know how to compute the period (Example: 4.3.2);
• analyze the bifurcation on the circle (Examples: 4.3.3–4.3.8).
• draw phase portrait on the circle (roughly speaking a circle, on which there are arrows indicating the direction of the vector field and the fixed points highlighted). An assorted Example: 4.5.1.
5 Linear systems 数学Midterm代考
Given a two-dimensional linear system with constant coefficients, you should be able to
• find the general solutions, classify the origin (the fixed point) in the classes of nodes, spirals, saddle points, non-isolated fixed points, stars, degenerate nodes, centers (in terms of shapes), stable fixed points, unstable fixed points, attracting, and Lyapunov stable fixed points (in terms of stability) (Examples: 5.1.10, 5.2.3–5.2.10);
• draw phase portrait (Example: 5.2.12).
6 Phase plane
The section is the focus of our final exam. More than half of the points will be distributed to the subject of this section.
Given a two-dimensional nonlinear system, you should be able to
• use nullclines to draw the phase portrait (Examples: 6.1.1–6.1.6);
• find all fixed points, and use the linearization technique to predict the shape and the stability of each fixed point (understand the robust cases and marginal cases) (Examples: 6.3.1–6.3.6);
• understand that the existence and uniqueness theorem implies no trajectories can across each other (Example: 6.2.2, 6.3.9);
• find the basin of attraction of a stable fixed point (Examples: 6.4.1–6.4.3);
• use polar coordinates to show a fixed point of marginal cases (Example: 6.3.13, 6.3.15);
• identify conservative systems, and prove nonlinear centers of a conservative system (Examples: 6.5.1–6.5.4, 6.5.6, 6.5.13);
• identify general reversible systems, and prove nonlinear centers of a reversible system (Examples: 6.6.1–6.6.3, 6.6.8*, 6.6.10);
• compute the index of a system, and use the index theory to prove the existence or non-existence of closed orbits (Examples: 6.8.1–6.8.5, also see example 6.0.1). More examples: 6.4.8, 6.8.14*(not require, but good to know).
Example 6.0.1. * Use index theory to show that the system
x ′ = x(4 − y − x 2 ), y′ = y(x − 1)
has no closed orbits. Suppose it is known that the unstable manifold of the saddle point (2, 0)
connects to the stable fixed point (1, 3).