# Sitemas Dinamicos E01
Exercises
1. Consider the following vector fields.

a ) $\dot{x} = y$,
      $\dot{y} = −\delta y − \mu x$,     $(x, y) ∈ \mathbb{R}^2$

b) $\dot{x} = y$,
      $\dot{y} = −δy − μx − x^2$,     $(x, y) ∈ \mathbb{R}^2$

c) $\dot{x} = y$,
      $\dot{y} = −δy − μx − x^3$,     $(x, y) ∈ \mathbb{R}^2$

d) $\dot{x} = −δx − μy + xy$,
    $\dot{y} = μx − δy + \frac{1}{2} (x^2 − y^2)$,     $(x, y) ∈ \mathbb{R}^2$

e) $\dot{x} = −x + x^3$,
    $\dot{y} = x + y$,    $(x, y) ∈ \mathbb{R}^2$

f) $\dot{r} = r(1 − r^2)$,
    $\dot{θ} = \cos 4θ$,     $(r, θ) ∈ R^+ × S^1$

g) $\dot{r} = r(δ + μr^2 − r^4)$,
    $\dot{θ} = 1− r^2$,     $(r, θ) ∈ R^+ × S^1$

h)
˙θ
= v,
v˙ = −sin θ − δv + μ,
(θ, v) ∈ S
1 × R.
i)
˙θ
1 = ω1,
˙θ
2 = ω2 + θn
1, n ≥ 1,
(θ1, θ2) ∈ S
1 × S
1
.
j)
˙θ
1 = θ2 − sin θ1,
˙θ
2 = −θ2,
(θ1, θ2) ∈ S
1 × S
1
.
k)
˙θ
1 = θ2
1,
˙θ
2 = ω2,
(θ1, θ2) ∈ S
1 × S
1
.
Find all fixed points and discuss their stability.

b)
x˙ = y,
y˙ = −δy − μx − x2,
(x, y) ∈ R2
.
c)
x˙ = y,
y˙ = −δy − μx − x3,
(x, y) ∈ R2
.
d)
x˙ = −δx − μy + xy,
˙ y = μx − δy + 12
(x2 − y2),
(x, y) ∈ R2
.
e) x˙ = −x + x3,
y˙ = x + y,
(x, y) ∈ R2
.
f)
r˙ = r(1 − r2),
˙θ
= cos4θ,
(r, θ) ∈ R+ × S
1
.
g)
r˙ = r(δ + μr2 − r4),
˙θ
= 1− r2,
(r, θ) ∈ R+ × S
1
.
h)
˙θ
= v,
v˙ = −sin θ − δv + μ,
(θ, v) ∈ S
1 × R.
i)
˙θ
1 = ω1,
˙θ
2 = ω2 + θn
1, n ≥ 1,
(θ1, θ2) ∈ S
1 × S
1
.
j)
˙θ
1 = θ2 − sin θ1,
˙θ
2 = −θ2,
(θ1, θ2) ∈ S
1 × S
1
.
k)
˙θ
1 = θ2
1,
˙θ
2 = ω2,
(θ1, θ2) ∈ S
1 × S
1
.
Find all fixed points and discuss their stability.
2. Consider the following maps.
a) x  → x,
y  → x + y,
(x, y) ∈ R2
.
b) x  → x2,
y  → x + y,
(x, y) ∈ R2
.
c) θ1  → θ1,
θ2  → θ1 + θ2,
(θ1, θ2) ∈ S
1 × S
1
.
d) θ1  → sin θ1,
θ2  → θ1,
(θ1, θ2) ∈ S
1 × S
1
.
e)
x  → 2xy
x+y ,
y  →

2xy2
x+y
 1/2
,
(x, y) ∈ R2
.
f) x  → x+y
2 ,
y  → (xy)1/2,
(x, y) ∈ R2
.
g) x  → μ − δy − x2,
y  → x,
(x, y) ∈ R2
.
h) θ  → θ + v,
v  → δv − μ cos(θ + v),
(θ, v) ∈ S
1 × R1
.
Find all the fixed points and discuss their stability.
3. Consider a Cr (r ≥ 1) diffeomorphism
x  → f(x), x∈ Rn.
Suppose f has a hyperbolic periodic orbit of period k. Denote the orbit by
O(p) =

p, f(p), f
2(p), · · · , fk−1(p), fk(p) = p

.
Show that stability of O(p) is determined by the linear map
y  → Dfk(fj (p))y
for any j = 0, 1, · · · , k−1. Does the same result hold for periodic orbits of noninvertible
maps?

4. Formulate the definitions of Liapunov stability and asymptotic stability for maps.
5. Show that hyperbolic fixed points of maps which are asymptotically stable in the linear
approximation are nonlinearly asymptotically stable.
6. Give examples of fixed points of vector fields and maps that are stable in the linear
approximation but are nonlinearly unstable.
7. Consider the linear vector field
x˙ = Ax, x ∈ Rn,
where A is an n × n constant matrix. Suppose all the eigenvalues of A have negative
real parts. Then prove that x = 0 is an asymptotically stable fixed point for this linear
vector field. (Hint: utilize a linear transformation of the coordinates which transforms
A into Jordan canonical form.)
8. Suppose that the matrix A in Exercise 7 has some eigenvalues with zero real parts
(and the rest have negative real parts). Does it follow that x = 0 is stable? Answer
this question by considering the following example.

x˙ 1
x˙ 2

=

0 1
0 0

x1
x2

.
9. Consider the linear map
x  → Ax, x ∈ Rn,
where A is an n×n constant matrix. Suppose all of the eigenvalues of A have modulus
less than one. Then prove that x = 0 is an asymptotically stable fixed point for this
linear map (use the same hint given for Exercise 7).
10. Suppose that the matrix A in Exercise 9 has some eigenvalues having modulus one
(with the rest having modulus less than one). Does it follow that x = 0 is stable?
Answer this question by considering the following example.

x1
x2

 →

1 1
0 1

x1
x2

.
11. Consider a nonautonomous vector field
x˙ = f(x, t), x ∈ Rn,
and suppose that ¯x(t) is a function (defined for some interval of t) satisfying
f(¯x(t), t) = 0.
Prove that if ¯x(t) is a trajectory of the vector field then it must be constant in time.
12. Consider the following vector field (Yang [2001]):
x˙ = −x,
˙φ=
1,
˙θ
= ω, (x, φ, θ) ∈ R × S
1 × S
1
,
where ω is an irrational number. Show that every trajectory is asymptotically orbitally
stable.
13. Consider the following vector field (Yang [2001]):
˙θ
= sin2
θ + (1 − r)2
,
r˙ = r(1 − r), (θ, r) ∈ S
1 × R.
Show that every trajectory, except r = 0, is asymptotically orbitally stable.

14. Does Descartes’ rule of signs provide any information about the roots of the polynomial
p(−λ), where p(λ) is given by (1.2.12)?
15. Use the Routh-Hurwitz test to determine the location of the roots of the following
polynomials:
(a) λ3 − 3λ2 + 3λ − 1,
(b) λ3 + 3λ2 − 4,
(c) λ3 + λ2 + λ + 1.

Referencias.

1. Stephen-Wiggins, S., "Introduction to Applied Nonlinear Dynamical Systems and Chaos", 2E., Springer.