MAT332
Introduction to Nonlinear Dynamical Systems and Chaos
Lecture 4
Qun WANG
University of Toronto Mississauga
September 21, 2021
(UTM)
MAT332 Intro Nonlinear Dyn and 21, 2021
1 / 9
Review
Study the following system and find out (0, 0) is which kind of fixed point.
1
2
3 5
x ̇=Ax,A= 2 0 (1)
3 6
x ̇=Ax,A= 0 3 (2)
Discuss the behaviour of the vector field around the point (0, 0, 0, 0) for
30 x ̇=Ax,A=0 2
12 −2 1
(UTM)
MAT332 Intro Nonlinear Dyn and 21, 2021
2 / 9
Learning Objective
For today’s lecture, students are expected to:
Objective
Make the phase portrait for 1D linear system near fixed Points; Analyze the behaviour of the system based on the phase portrait.
You don’t need maths to follow today’s lecture—as long as you can draw anime!
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021 3 / 9
General Strategy
Strategy (Phase Portrait near Fixed points)
1 Solve the equation f (x) = 0 to find the fixed point(s);
2 For each fixed point x0, study the value of f(x) outside x0;
3 Conclude if starting near x0, the solution will eventually leave x0 or approach x0.
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021 4 / 9
Phase Portrait near Fixed Points
We will explore the models via the population model.
Example (Normal Reproduction)
Suppose that the population growth rate is proportional to the total population size, then we have the linear model
Solution
x ̇(t) = kx(t),k > 0 (3)
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021
5 / 9
Phase Portrait near Fixed Points
Example (Explosive Reproduction)
Suppose that the population growth rate is proportional to the square of total population size, then we have the non-linear model
Solution
x ̇(t) = kx2(t),k > 0. (4)
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021 6 / 9
Phase Portrait near Fixed Points
Example (Reproduction with Competition)
Suppose that the population is not small, then the previous model are no longer ideal, as food, water and other resources are quite limited, and people need to go through competition to ensure the survival. As a result, a more practical model could be
Solution
x ̇(t) = kx(t)(1 − x(t)) (5)
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021 7 / 9
Phase Portrait near Fixed Points
Example (Harvest Quotas)
Suppose that besides the limited resource, there is also a harvest behaviours by predator. As a result, a more practical model could be
Solution
x ̇(t) = kx(t)(1 − x(t)) − c (6)
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021
8 / 9
Phase Portrait near Fixed Points
Example (Relative Harvest Quotas)
Suppose furthermore the predator is somehow “reasonable”, As a result, the model could be
Solution
x ̇(t) = kx(t)(1 − x(t)) − px(t), p < k (7)
(UTM) MAT332 Intro Nonlinear Dyn and 21, 2021 9 / 9