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hw26

The document outlines Homework 26 for Math 21b: Linear Algebra, focusing on the analysis of nonlinear systems through various differential equations. It includes tasks such as analyzing species interaction models, a frictionless pendulum, and systems with friction, emphasizing the use of phase portraits and Jacobian matrices to determine stability and equilibrium points. The homework is due on April 13 and 14, 2016.

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2 views2 pages

hw26

The document outlines Homework 26 for Math 21b: Linear Algebra, focusing on the analysis of nonlinear systems through various differential equations. It includes tasks such as analyzing species interaction models, a frictionless pendulum, and systems with friction, emphasizing the use of phase portraits and Jacobian matrices to determine stability and equilibrium points. The homework is due on April 13 and 14, 2016.

Uploaded by

moien
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Math 21b: Linear Algebra Spring 2016

Homework 26: Nonlinear systems


This homework is due on Wednesday, April 13, respectively on Thursday, April 14, 2016. These
problems are adapted from the problems in the handout.

1 Analyze the system


dx
= 2x − x2 + xy
dt
dy
= 4y − xy − y 2
dt
It is an interaction model of species so that we only look at x ≥
0, y ≥ 0.
2 We analyze the system
dx
= x(1 − x + ky − k)
dt
dy
= y(1 − y + kx − k)
dt
in the cases k = 2 and k = 0. Again, as this is a population
model, we only look at x ≥ 0, y ≥ 0.
3 Analyze the frictionless pendulum
dx
= y
dt
dy
= −2 sin(x) ,
dt

4 Analyze the system


dx
= x2 + y 2 − 1
dt
dy
= xy
dt
5 Analyze the pendulum with friction
dx
= y
dt
dy
= − sin(x) − y .
dt

Nonlinear systems

Differential equations x0 = f (x, y), y 0 = g(x, y) generalize the


linear case x0 = ax + by, y 0 = cx + dy. To analyze such sys-
tems when f, g are not linear, we draw phase portraits. The
curves where f (x, y) = 0 or g(x, y) = 0 are called nullclines.
They intersect in equilibrium points. These are points
where x0 = 0, y 0 = 0. We can use linear algebra to analyze
the system

near such an 
equilibrium point (a, b). The matrix
 f (a, b) fy (a, b) 
A =  x  is called the Jacobian matrix. The
gx(a, b) gy (a, b)

linear system v 0 = Av is called the linearization at (x0, y0).


If this linear system is stable, the equilibrium point is stable.
In terms of the original nonlinear system, an equilibrium point
(x0, y0) is stable if all trajectories starting sufficiently close to
(x0, y0) tend to it as t → ∞.
Making an analysis of the system consists of 1) finding the
nullclines and equilibria 2) determine the stability of the equi-
libria 3) drawing the phase portrait of the system 4) analyzing
the possible behaviors of the trajectories.

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