Hence the xed point (R, J) = (0.0) is a saddle point if a2 < b2 and a stable node
if a2 > b2 . The eigenvalues and corresponding eigenvectors are
A,=a+b, v,=(l,l), x12=ab, v2=(1,1).
Since a+b > ab, the eigenvector (1,1) spans the unstable manifold when th
w=1+e¢=l+%saz+0(2). (57)
Now for the physical interpretation. The Dufng equation describes the un-
damped motion of a unit mass attached to a nonlinear spring with restoring force
F(x)= x£x3. We can use our intuition about ordinary linear springs if we
wr
stepsize A: = 0.1. The solutions have the shape expected from Section 2.3. I
Computers are indispensable for studying dynamical systems. We will use them
liberally throughout this book, and you should do likewise.
EXERCISES FOR CHAPTER 2
2.1 A Geometric W
Math 3100 Dynamical Systems Winter 2010
Assignment 6.
Due Tuesday Mar. 23
1. Show that the function
1
1
1
F (x, y) = x2 + x4 + y 2
2
4
2
is an integral of motion for the system
y = x + x3 .
x = y,
Hint: use Chain Rule and compute
d
F (x(t), y(t).
dt
2. Fi
Math 3100!
Introduction to Dynamical Systems!
Winter, 2014!
Instructor: !
Dr. Xiaoqiang Zhao, HH-2009, [email protected]!
Time: ! Tuesday and Thursday, 14:00-15:15!
Classroom: ! A-1045!
Ofce Hours: ! Tu 10:30-12:00, Wed 13:00-14:30, or by appointment!
Course We
MATH 3100, Midterm Exam, Winter 2014
Name: Student No.2
Instruction: This exam includes 6 questions. All answers must be carefully justi-
ed. The value of each problem is marked on the left margin.
ll
[20] 1. Find all equilibrium points of equation I
dete
2-17 ,
« mg I X
let 1[130: e Cosx : 0 2
WM .5 e : cum.- i {c we;
\ /.- 0 S4. r4.an
from ?\gtu.el uQ, 9% I/ . = c we":
.\:._._._>._,;'.U.<-, 9. V. . v \,. . > ax
E 1
gm 0th mirteaa Mara
rJMSn mam-w, -axns, Jamaal La (3.4 (MW?
av:01an1l<ad,néN.
I M [we
A Ss-anmuw Z _ ( -
57m): me MM)
i gm = 308,1
J? a L _ W!
0h] y (I H. 50- 70
' _ + _ a .
. a lm : " : " m ' [.3 F K
S f 51x T'jwm m fm ii I t; lh/rjlés
«age, raJecfcru'cg Wlth- 'w_,-f0 be?va 'Ffditl'fv ét~JlfcrIww 41.5 'T«w 4-.3
Qwilh QM 2/0, ~ ermH : O «
7. S.olw4{0n )fO
f: rm) (Ml)
? L" V20
912- f
- 0%
<3) Fir1, ff; 1172'
J'éwwr'zm), F
H (L2) , rw
r:\ .s O. smbgg DAM. cjcle, (mt p). is Cm unmLL 01bit cacb.
01:74:11! [lo/1'0"; m CUMfGPg/ozjum).
9 (I I 0/71/91?!" [Hui/0)7 (1/0; Luge .
{
,.:.M W "m_
R
3J4 v . A
09 X- 2 golwhn 7h: Hwy
{PM =° :7 Fwy. We; am, mo) and o)
(5 r. ) , a J
f M a gala/1e, (/m is a m "01¢.
Cyjf/h w 5 halfmug.
G) If/A <0! (0,0) (5 q wade. "0:11, (y) is 0 Halo/11.
[6/0) 1') ch Nixie
(910}{5 01 SADIJL/
(0,0) r5 6- New v.11,
(Hing
slowly as the condition of the forest changes. According to Ludwig et al. (1978), r
increases as the forest grows, while I: remains fixed.
They reason as follows: let S denote the average size of the trees, interpreted as
the total surface area of the bra
EXAMPLE 6.3.6:
Show that the system it = 13' , j): l + x + y2 has no closed orbits.
Sciatica: This system has no xed points: if j: = 0, then x = 0 and so
52: 1+ )22 at 0. By Theorem 6.8.2, closed orbits cannot exist. I
EXERCISES FOR CHAPTER 6
6.1 Phase Po
where weve used (2) to eliminate dt. (That was the second trick.)
Now we compute the change in the perturbations after one circuit around the
closed orbit Q) *:
ida._ 2w [acoswidw
g o Q+(a+1)sin¢*
_i 3E< 2? ._
=> lné(0)a+1ln[§l+(a+l)sm¢l ]o 0.
Hence
Math 3100 Dynamical Systems Winter 2010
Assignment 3.
Due Thursday Feb. 25
1. Write the general solution to the system of two uncoupled harmonic oscillators
x1 = 1 y1
y1 = 1 x1
x2 = 2 y2
y2 = 2 x2
in the amplitude-phase form. Make sure that your sol
Math 3100 Dynamical Systems Winter 2010
Assignment 2.
Due Thursday Feb. 4
1. Use the denition of stability, instability, and asymptotic stability of a
xed point to prove that x = 0 is, respectively, AS, U, and S but not AS for
the following dynamical syst
Math 3100 Dynamical Systems Winter 2010
Assignment 1.
Due Thursday Jan. 14
1. Solve the following triangular system:
x = xy
y =y+z
z=1
(Assume the general initial condition: x(0) = x0 , y(0) = y0 , z(0) = z0 .)
2. Simplify the following system by changin
Math 3100 Dynamical Systems Winter 2010
Assignment 4.
Due Thursday Mar. 4
1. Determine the type of the xed point (attracting/repelling node, center,
x
x
stable/unstable focus, saddle) for the system
=A
for each of
y
y
the following matrices A:
(a) A =
1 3
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 1 (part 1).
1. Solve the following triangular system:
x = xy
y =y+z
z=1
(Assume the general initial condition: x(0) = x0 , y(0) = y0 , z(0) = z0 .)
Solution. The last equation is independent
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 2.
1. Use the denition of stability, instability, and asymptotic stability of a
xed point to prove that x = 0 is, respectively, AS, U, and S but not AS for
the following dynamical system (DS)
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 3.
1. Write the general solution to the system of two uncoupled harmonic oscillators
x1 = 1 y1
y1 = 1 x1
x2 = 2 y2
y2 = 2 x2
in the amplitude-phase form. Make sure that your solution conta
Math 3100 Dynamical Systems Winter 2010
Assignment 7.
Due Tuesday Apr. 6
1. (a) For the scalar ow x = (x a)(x 2a), determine the xed points
x (a) and their stability, where possible, by a linear approximation. Here a
is a real parameter that can take on a
Math 3100 Dynamical Systems Winter 2010
Assignment 5.
Due Tuesday Mar. 16
In addition to the following, Question 3 from Assignment 4 is due.
1. The following is a mathematical model for combat where x(t) represents
the number of tanks used by the aggressi
MATHEM ATICS 3100
INTRODUCTION TO DYNAM ICAL SYSTEM S
MATH 3100: Introduction to Dynamical Systems
If a differential equation (flow) or difference equation (map) is nonlinear, It usually cannot be solved in exact
form. Therefore, we focus instead on the g
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 7.
1. (a) For the scalar ow x = (x a)(x 2a), determine the xed points x (a) and their
stability, where possible, by a linear approximation. Here a is a real parameter that can
take on any rea
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 6.
1. Show that the function
1
1
1
F (x, y) = x2 + x4 + y 2
2
4
2
is an integral of motion for the system
y = x + x3 .
x = y,
Hint: use Chain Rule and compute
d
F (x(t), y(t).
dt
Solution.
d
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 4.
1. Determine the type of the xed point (attracting/repelling node, center,
x
x
stable/unstable focus, saddle) for the system
=A
for each of
y
y
the following matrices A:
(a) A =
1 3
3 1
(d
Math 3100 Dynamical Systems Winter 2010
Solutions to Assignment 5.
1. The following is a mathematical model for combat where x(t) represents
the number of tanks used by the aggressing army, and y(t) represents the
number of antitank weapons employed by th