程序代写CS代考 MAT332 – cscodehelp代写

MAT332
Introduction to Nonlinear Dynamical Systems and Chaos
Lecture 7
Qun WANG
University of Toronto Mississauga
October 1, 2021
(UTM)
MAT332 Intro Nonlinear Dyn and 1, 2021
1 / 10

Review
Example (Logistic Curve)
Consider the population growth model x ̇=x(1−x), x(0)=x0
Show that the flow exists locally and is unique.
Calculate the first variation equation at the fixed point x = 1 and show its stability.
Solution
(1)
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021
2 / 10

Learning Objective
For today’s lecture, students are expected to:
Objective
Practice Phase Portrait of Nonlinear 2D Problems
Get some feeling for the stability around Fixed Point (without rigorous definitions)
You don’t have to do difficult maths to follow today’s lecture—Drawing anime is enough!
…But before that, we somehow mention the following theorem guarantees the existence and uniqueness.
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021 3 / 10

Fundamental Theorem for system in Rn
Definition
Amapf :Rn →Rn issaidtobeLipschitzatx0 ∈Rn,ifthereexistsa constant K and an ε > 0 s.t. for any x,y in the ball Br(x0), one has that
|f (x) − f (y)| ≤ K|x − y|
Here for x = (x1, x2, …xn) ∈ Rn, |x| = 􏰤x12 + x2 + … + xn2
Theorem
Suppose that x ̇ = F(x) is a nonlinear system in Rn. Consider the initial condition x = x0. If F is Lipschitz continuous at x0, the flow exists locally and is unique.
Tips: In general, do you really need to prove the Lipschitz continuity each time you want to apply this theorem?
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021 4 / 10

Fundamental Theorem for System in Rn
Proposition
Suppose that F(x) is differentiable and all its partial derivative ∂Fi (x) is ∂xj
continuous at x0. Then there exists a r > 0 s.t. F(x) is Lipschitz in Br (x0).
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021 5 / 10

General Strategy
Strategy (Phase Portrait near Fixed points)
1 Solve the equation F(x) = 0 to find the nullclines;
2 Observe how the plane is divided into regions by these nullclines;
3 Based on x ̇ and y ̇ in these regions, associate the vector field;
4 Conclude if starting near a fixed point (or the nullclines), the solution will eventually leave it or approach it.
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021 6 / 10

Phase Portrait for 2D system Definition
The nullclines are the sets where the x ̇ = 0 or y ̇ = 0. Example
x ̇ = y + x(1 − x2 − y2)
y ̇ = −x + y(1 − x2 − y2)
Convince yourself that the flow exists at least locally and is unique
Solution
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021 7 / 10

Phase Portrait for 2D system
Example
Solution
x ̇ = y + x(1 − x2 − y2)(x2 + y2 − 4)
y ̇ = −x + y(1 − x2 − y2)(x2 + y2 − 4)
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021 8 / 10

Phase Portrait for 2D system
Example
Solution
x ̇ = − x
y ̇ = − 2 y + 2 x 3
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021
9 / 10

Phase Portrait for 2D system
Example
Solution
x ̇ = x − y
y ̇ = 1 − xy2
(UTM) MAT332 Intro Nonlinear Dyn and 1, 2021
10 / 10