AMATH 402/502 Homework 3 DUE at time and place posted on website
1 In this problem we will study the integrate-and-re model of a neuron. The membrane potential V (t) of the neuron obeys the following dierential equation: dV = Vrest V + RI (t) dt
where Vre
AMATH 402/502 Homework DUE at time and place posted on website
1 Problem 7.3.1 2 Problem 7.3.4 3 Problem 8.1.6 For (c), choose values of that demonstrate dierent behavior on all sides of the bifurcation(s). That is, just choose the appropriate number of v
AMATH 402/502 Homework DUE at time and place posted on website
1 Consider the 2-D linear map xn+1 = Axn , where A =
0 1/2 1/2 a
a. Find the xed point(s) for any value of the real parameter a. b. Say whether each xed point(s) is stable or unstable when a =
AMATH 402
Winter 2006
Solution for Homework 10
1.(2.2.2)
x = 1 x14
x = 0
x =1, 1
x =1 is a stable fixed point.
x =1 is an unstable fixed point.
1
2.(2.3.3(b)
N = aN ln (bN )
N = 0
There is a stable fixed point at N =b1
vector field( b = .5)
2
N (t)
a woul
AMATH 402
Winter 2006
Solution for Homework 3
2.4.1
x = x (1 x)
x (1 x) = 0
x =0, 1
f 0 (x) = 1 2 x
x =0 f(x)= 1 Unstable fixed point
x =1 f(x)= 1 Stable fixed point
2.4.5
2
x = 1 ex
2
1 ex = 0
x =0
2
f(x)=2 xex
x =0 f(x)= 0 Linear stability analysis fail
AMATH 402
Winter 2006
Solution for Homework 7
1.(7.1.1)
r = r3 4 r
= 1
We should pay attention to the following facts
r > 0, r > 2
r < 0, r < 2
convert to xy form
x = rr x y
x = x3 + xy 2 4 x y
y = rr y + x
y = yx2 + y 3 4 y + x
use pplane
1
2.(7.1.3)
2
AMATH 402
Winter 2006
Solution for Homework 5
5.1.10
a)
x = y
y = 4 x
x = x0 cos (2 t) + 1/2 y0 sin (2 t)
y = 2 x0 sin (2 t) + y0 cos (2 t)
It is Not attracting.
It is Liapunov Stable.
Hence, it is not asymptotically stable.
b)
x = 2 y
y = x
x =
AMATH 402
Winter 2006
Solution for Homework 4
3.4.12
There exist infinite solutions.
Quadfurcation x = (x2 r)(x2 2r)
expand our result
(2n)-furcation x = (x2 r)(x2 2r) (x2 nr)
(2n+1)-furcation x = x(x2 r)(x2 2r) (x2 nr)
3.4.14
x = r x + x3 x5
r x + x3 x5
AMATH 402
Winter 2006
Solution for Homework 6
1.(6.5.2)
(a)
x = y
y = x x2
Fixed points: (0,
0), (1, 0)
0
1
Jacobian: A =
1 2x 0
= 1 2 x, 1 2 x
at fixed point: (0, 0) =1, 1, so it is a saddle point
at fixed point: (1, 0) =i, i, so it is a center
(b)
1
AMATH 402
Winter 2006
Solution for Homework 2
2.1
2.1.1
Fixed Points: sin(x) = 0, x = n,n = 0, 1, 2, . . .
2.1.2
Maximum velocity to the right : x get its minimum
sin(x) get its minimum, sin(x) = 1
x = 2n /2, n = 0, 1, 2, . . .
2.1.3
a)
dx
x = d(dtx)
= d
AMATH 402/502 Homework DUE at time and place posted on website
1 Problem 6.3.1 2 Problem 6.3.10, (a-c) only. For the context of part (c) of this problem, we will say that a xed point is a saddle regardless of linearization if you can (1) nd a direction al
AMATH 402/502 Homework DUE at time and place posted on website
1 Problem 5.2.1 2 Problem 5.2.2 3 Problem 5.2.9 4 Problem 5.2.12. For (c), doing one qualitative sketch of phase portraits is just ne for each of the three cases. 5 Problem 5.3.4 6 Problem 5.3
AMATH 402/502 Homework 2 DUE at time and place posted on website
1 Problem 3.1.3 2 Problem 3.2.3 3 Problem 3.2.4 4 Problem 3.3.2 5 Problem 3.4.2 6 Problem 3.4.8 7 Problem 3.4.15 The homework will be graded statistically. Late homework is not accepted. You