SlideShare une entreprise Scribd logo
1  sur  5
Télécharger pour lire hors ligne
Dynamical Systems
Solved Exercises
14 de julho de 2020
Miguel Fernandes
1. Consider the doubling map E2 : R/Z → R/Z defined by E2(x + Z) = 2x + Z.
(a) Find the number of periodic points of period n.
Remember: Given a topological dynamical system f : X → X, we say that a
point x is periodic of period p if fp
(x) = x holds.
We have:
En
2 (x + Z) = 2n
x + Z = x + Z ⇔ (2n
− 1)x = k ⇔ x =
k
2n − 1
for some integer k. Thus the periodic points of period n are those for which k
varies from 0 to 2n
− 2 and hence we have 2n
− 1 periodic points of period n.
(b) Conclude that the periodic points of E2 are dense in S1
.
Following the previous exercise, we see that the periodic points of E2 are uni-
formly distributed in S1
. Consequentely, making n → ∞ we get the desired
result.
2. Consider the dynamical system f : R → R defined by f(x) = |x − 2|.
(a) Find the fixed point of f. Is it periodic? What is its period? Comment.
The computation of the fixed point of f is the following:
f(x) = x ⇔ |x − 2| = x ⇔ x − 2 = ±x ⇔ x = 1.
Clearly, it is a periodic point of period 1, which means that, a particle starting
its trajectory at x = 1 will remain in that position for all future time.
(b) Define the first three iterates of f and sketch them.
Clearly, f(x) = |x − 2|, f2
(x) = ||x − 2| − 2| and f3
(x) = |||x − 2| − 2| − 2|, all
1
defined in the whole real line (a piecewise definition would be too cumbersome,
a nice task for the reader!). The graphs of such iterations are shown below.
Figura 1: First three iterates of the transformation f.
(c) Find other periodic and pre–periodic points of f.
It is clear in the figure above that the points of the set [0, 2]  {1} are periodic
of period 2 and the points belonging to ]−∞, 0[ ∪ ]2, +∞[ are pre–periodic.
3. Show that expanding maps of S1
are topologically mixing.
Remember: A topological continuous map f : X → X is said to be topologically
mixing if, given two non–empty open sets U and V , there exists a time N ∈ N such
that for all n ≥ N we have fn
(U) ∩ V = ∅.
Let f : S1
→ S1
be an expanding map (and hence is continuous) and F : R → R such
that π◦F = f ◦π (the existence of such function F is guaranteed by the continuity of
f), where π : R → S1
is the projection defined by π(x) = [x], for all x ∈ R. (Remark:
[x] denotes the point of S1
subject to the identification S1
= R/Z). Notice that we
can also write f ([x]) = [F(x)]. Clearly, if f([x]) = [mx], for some 1 < m ∈ Z then
F(x) = mx + n, for some n ∈ Z. Hence |F (x)| = m > 1, for all x ∈ R, and by
the mean value property it follows that |F(x) − F(y)| = |F (c)(x − y)| = m|x − y|,
where x, y ∈ R and c ∈ (x, y) (or c ∈ (y, x)). Take an open set U ⊂ R and choose an
interval I = (a, b) ⊂ U. It is straightforward that Fn
, n ∈ N, increases the length of
I by a factor of mn
. That implies that we can choose n0 such that mn0
> 1/(b − a)
and consequently the length of Fn0
(I) will exceed 1. Thus π(Fn0
(I)) = S1
and
S1
⊂ fn
(π(I)), for all n ≥ n0. This means that, given two non–empty open sets U
and V in S1
, the set fn
(U) ∩ V is non–empty, and the same for all future time.
4. Consider the rotation of the circle Rα : S1
→ S1
defined by Rα(x + Z) = x + α + Z.
(a) Show that Rα has periodic points if and only if α is rational.
If α is rational then it can be writeen in the form α = p/q, for some integers p
and q, then all points are periodic with period q. Indeed:
Rq
p/q(x + Z) = x + q ∗ p/q + Z = x + Z.
Reciprocally,
2
Rn
α(x + Z) = x + nα + Z = x + Z ⇔ x + nα = x + p ⇔ α = p/n
for some integer p. Thus if we assume that x + Z is periodic of period n, then α
must equal p/n, for some p ∈ Z.
(b) Show that 0 is a recurrent point when α is irrational.
Remember: The orbit of a point x is said to be dense if, for all y ∈ R/Z and
all > 0, there exists a time n ∈ N such that d(Rn
α(x), y) < . In particular,
making y = x, we find that x is a recurrent point.
Let α be an irrational number. We first observe that the map n → Rn
α(0), for
each n ∈ N, is one–to–one. For that, let Rn
α(0) = Rm
α . Then nα+Z = mα+Z ⇔
(n−m)α = k, for some integer k. But, since α is irrational, we must have n = m.
Now, fix > 0. We find a time n for which we have 0 < nα+Z < +Z. The den-
sity of the irrational numbers on the real line allows us to assert that α belongs
to some interval [p/q, p/q + /2q], where p and q > 0 are integers. Consequently,
if we set n = q we have qα ∈ [p, p + /2] and thus qα + Z ≤ /2 + Z < + Z.
This implies that, for certain iterates of Rα, the trajectory is sufficiently close
to 0.
5. Consider the linear transformation T : T2
= R2
/Z2
→ R2
/Z2
defined by:
T(x, y) = (2x + y, x + y)(mod 1)1
.
(a) Show that T is well defined, i.e., if (x−x , y−y ) ∈ Z2
then L(x, y)−L(x , y ) ∈ Z2
.
It’s an immediate consequence of the linearity of T and the fact that Z2
is closed
under T.
(b) Compute the matrix of the linear transformation L : R2
→ R2
defined by
L(x, y) = (2x+y, x+y). Justify that L is invertible and compute its eigenvalues.
The matrix of L (relative to the canonical basis) is:
ML =
Å
2 1
1 1
ã
and is clearly non–singular (determinant equal to 1 and hence M−1
L has integer
entries). Furthermore, its eigenvalues are the inverses:
λ1 =
3 +
√
5
2
and λ2 = λ−1
1 =
3 −
√
5
2
.
(c) Show that the periodic points of T are dense in T2
.
Let x = a/b and y = c/b (same denominator) provided a, b, c, d ∈ Z. Then
T(x, y) = T(a/b, c/b) = 2a+c
b
, a+c
b
and so T(x, y) is also a rational point whose
coordinates have the same denominator as (x, y). But notice that we only have
b2
different choices for (x, y) in T2
(why is that) whose points shall form the set
A. Thus we have {Tn
(x, y)| n ∈ N} ⊂ A. Given T is invertible, we find that the
point (x, y) is periodic. Since b is arbitrary, such periodic points are dense in T2
(find the anology with the expanding map).
1
The so–called Arnold cat map.
3
(d) Show that the number of n−periodic points of T is given by | det(Mn
L − I2)|
(assuming this number is not zero).
The periodic points of T of period n are the solutions of:
Mn
L x = x (mod 1).
Notice that such equation is equivalent to (Mn
L −I2)x = 0 (mod 1) or more preci-
sely (Mn
L −I2)x = k, for some k ∈ Z. This makes us consider the transformation
L∗
: R2
→ R2
defined by L∗
(x) = (Mn
L −I2)x. According to the explained above,
it is clear that the periodic points of T are given by the integer points in the
set L∗
([0, 1[ × [0, 1[). Furthermore, we can easily verify that L∗
transforms the
square [0, 1[ × [0, 1[ into a paralelogram (without 2 edges). Making use of Pick’s
Theorem and regarding the fact that the area of the parallelogram is given by
| det(Mn
L − I2)|, we find the desired result.
(e) Compute the number of points of period 5.
Given the previous result, we have that the number os such periodic points is
given by | det(M5
L − I2)|. But | det(M5
L − I2)| = |(λ5
1 − 1)(λ−5
1 − 1)|, where λ1 is
the eigenvalue in (b). Since λ1 = (3 +
√
5)/2, we find |(λ5
1 − 1)(λ−5
1 − 1)| = 121.
6. Consider the alphabet β = {1, 2, 3} made of three letters. Let A be the transition
matrix defined as follows:
A =
Ñ
0 1 1
1 1 0
0 0 1
é
.
(a) Find elements of the set ΣA = x = x1x2... ∈ Σ| axnxn+1
= 1, n ≥ 1 (Σ is the
set of those infinite words whose letters belong to the alphabet β).
For instance, 221333... and 121333...
(b) How many words are there with with ’3’ as first letter?
One, 33333...
(c) Let α be a word of the form α = 1...3 in a total of 4 letters and hence finite.
Write the all the possibilities for α. How many are they?
There are only two possibilities: 1213 and 1333.
(d) Consider those finite words, of length n + 1, with first and last letters given by i
and j, respectively. Show that the number of words in such conditions is given
by the ij element of An
, where A is the transition matrix. For simplicity, denote
such number by Nn
(i, j). Verify the result for exercise (d).
We use induction on n for a proof. The case n = 1 is trivial (definition of
transition matrix). Now, using the induction hypothesis:
Nn+1
(i, j) = k Nn
(i, k)N1
(k, j) = k(An
)ikakj = (An+1
)ij.
In the case of the exercise (d), the power A3
is the matrix:
A3
=
Ñ
1 2 2
2 3 2
0 0 1
é
4
and the result follows.
(e) Conclude that the third row of A is invariant under the power operator.
Just remember exercises (b) and (d).
(f) Conclude that A is neither irreducible nor aperiodic.
Since the third row of A is invariant under the power operator and it contains
zeros, we conclude that A is not irreducible and hence cannot be aperiodic.
5

Contenu connexe

Tendances

Applied Calculus Chapter 1 polar coordinates and vector
Applied Calculus Chapter  1 polar coordinates and vectorApplied Calculus Chapter  1 polar coordinates and vector
Applied Calculus Chapter 1 polar coordinates and vectorJ C
 
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006Michael Soltys
 
Lattices and codes
Lattices and codesLattices and codes
Lattices and codesSpringer
 
Multiple integrals
Multiple integralsMultiple integrals
Multiple integralsTarun Gehlot
 
Integration and its basic rules and function.
Integration and its basic rules and function.Integration and its basic rules and function.
Integration and its basic rules and function.Kartikey Rohila
 
Lesson 27: Evaluating Definite Integrals
Lesson 27: Evaluating Definite IntegralsLesson 27: Evaluating Definite Integrals
Lesson 27: Evaluating Definite IntegralsMatthew Leingang
 
Number Theory for Security
Number Theory for SecurityNumber Theory for Security
Number Theory for SecurityAbhijit Mondal
 

Tendances (20)

Chemistry Assignment Help
Chemistry Assignment Help Chemistry Assignment Help
Chemistry Assignment Help
 
Fourier series 2
Fourier series 2Fourier series 2
Fourier series 2
 
Chapter 2 (maths 3)
Chapter 2 (maths 3)Chapter 2 (maths 3)
Chapter 2 (maths 3)
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli...
 Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli... Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Appli...
 
Number theory lecture (part 2)
Number theory lecture (part 2)Number theory lecture (part 2)
Number theory lecture (part 2)
 
Applied Calculus Chapter 1 polar coordinates and vector
Applied Calculus Chapter  1 polar coordinates and vectorApplied Calculus Chapter  1 polar coordinates and vector
Applied Calculus Chapter 1 polar coordinates and vector
 
Metric space
Metric spaceMetric space
Metric space
 
Number theory lecture (part 1)
Number theory lecture (part 1)Number theory lecture (part 1)
Number theory lecture (part 1)
 
plucker
pluckerplucker
plucker
 
Number theory
Number theoryNumber theory
Number theory
 
Fourier series 3
Fourier series 3Fourier series 3
Fourier series 3
 
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
The proof complexity of matrix algebra - Newton Institute, Cambridge 2006
 
Fourier series 1
Fourier series 1Fourier series 1
Fourier series 1
 
Lattices and codes
Lattices and codesLattices and codes
Lattices and codes
 
Multiple integrals
Multiple integralsMultiple integrals
Multiple integrals
 
Integration and its basic rules and function.
Integration and its basic rules and function.Integration and its basic rules and function.
Integration and its basic rules and function.
 
Lesson 27: Evaluating Definite Integrals
Lesson 27: Evaluating Definite IntegralsLesson 27: Evaluating Definite Integrals
Lesson 27: Evaluating Definite Integrals
 
Number Theory for Security
Number Theory for SecurityNumber Theory for Security
Number Theory for Security
 
Lecture5
Lecture5Lecture5
Lecture5
 
Chapter 3 (maths 3)
Chapter 3 (maths 3)Chapter 3 (maths 3)
Chapter 3 (maths 3)
 

Similaire à Dynamical systems solved ex

Dynamical systems
Dynamical systemsDynamical systems
Dynamical systemsSpringer
 
Fourier 3
Fourier 3Fourier 3
Fourier 3nugon
 
Solution set 3
Solution set 3Solution set 3
Solution set 3慧环 赵
 
Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...
Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...
Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...Kelly Lipiec
 
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
 
Conjugate Gradient Methods
Conjugate Gradient MethodsConjugate Gradient Methods
Conjugate Gradient MethodsMTiti1
 
Fourier series of odd functions with period 2 l
Fourier series of odd functions with period 2 lFourier series of odd functions with period 2 l
Fourier series of odd functions with period 2 lPepa Vidosa Serradilla
 
Iterative procedure for uniform continuous mapping.
Iterative procedure for uniform continuous mapping.Iterative procedure for uniform continuous mapping.
Iterative procedure for uniform continuous mapping.Alexander Decker
 
Interpolation techniques - Background and implementation
Interpolation techniques - Background and implementationInterpolation techniques - Background and implementation
Interpolation techniques - Background and implementationQuasar Chunawala
 
Z transform and Properties of Z Transform
Z transform and Properties of Z TransformZ transform and Properties of Z Transform
Z transform and Properties of Z TransformAnujKumar734472
 
Polya recurrence
Polya recurrencePolya recurrence
Polya recurrenceBrian Burns
 

Similaire à Dynamical systems solved ex (20)

Dynamical systems
Dynamical systemsDynamical systems
Dynamical systems
 
Imc2017 day2-solutions
Imc2017 day2-solutionsImc2017 day2-solutions
Imc2017 day2-solutions
 
Number theory
Number theoryNumber theory
Number theory
 
Fourier 3
Fourier 3Fourier 3
Fourier 3
 
Solution set 3
Solution set 3Solution set 3
Solution set 3
 
Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...
Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...
Arbitrary Pole Assignability By Static Output Feedback Under Structural Contr...
 
02_AJMS_186_19_RA.pdf
02_AJMS_186_19_RA.pdf02_AJMS_186_19_RA.pdf
02_AJMS_186_19_RA.pdf
 
02_AJMS_186_19_RA.pdf
02_AJMS_186_19_RA.pdf02_AJMS_186_19_RA.pdf
02_AJMS_186_19_RA.pdf
 
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...
 
Fourier series
Fourier seriesFourier series
Fourier series
 
Imc2017 day1-solutions
Imc2017 day1-solutionsImc2017 day1-solutions
Imc2017 day1-solutions
 
math camp
math campmath camp
math camp
 
Signals Processing Homework Help
Signals Processing Homework HelpSignals Processing Homework Help
Signals Processing Homework Help
 
Conjugate Gradient Methods
Conjugate Gradient MethodsConjugate Gradient Methods
Conjugate Gradient Methods
 
Fourier series of odd functions with period 2 l
Fourier series of odd functions with period 2 lFourier series of odd functions with period 2 l
Fourier series of odd functions with period 2 l
 
Iterative procedure for uniform continuous mapping.
Iterative procedure for uniform continuous mapping.Iterative procedure for uniform continuous mapping.
Iterative procedure for uniform continuous mapping.
 
Interpolation techniques - Background and implementation
Interpolation techniques - Background and implementationInterpolation techniques - Background and implementation
Interpolation techniques - Background and implementation
 
Z transform and Properties of Z Transform
Z transform and Properties of Z TransformZ transform and Properties of Z Transform
Z transform and Properties of Z Transform
 
lec z-transform.ppt
lec z-transform.pptlec z-transform.ppt
lec z-transform.ppt
 
Polya recurrence
Polya recurrencePolya recurrence
Polya recurrence
 

Plus de Maths Tutoring

Identidades trigonométricas
Identidades trigonométricasIdentidades trigonométricas
Identidades trigonométricasMaths Tutoring
 
Trigonometria 12 ano revisoes
Trigonometria 12 ano revisoesTrigonometria 12 ano revisoes
Trigonometria 12 ano revisoesMaths Tutoring
 
Intervalos e propriedades de números reais - Grau de dificuldade elevado
Intervalos e propriedades de números reais - Grau de dificuldade elevadoIntervalos e propriedades de números reais - Grau de dificuldade elevado
Intervalos e propriedades de números reais - Grau de dificuldade elevadoMaths Tutoring
 
Teste 11ano produto interno e vetores
Teste 11ano produto interno e vetoresTeste 11ano produto interno e vetores
Teste 11ano produto interno e vetoresMaths Tutoring
 
Teste eqs e intervalos com res
Teste eqs e intervalos com resTeste eqs e intervalos com res
Teste eqs e intervalos com resMaths Tutoring
 
Teste equações e intervalos
Teste equações e intervalosTeste equações e intervalos
Teste equações e intervalosMaths Tutoring
 
Sucessoes e series com res
Sucessoes e series com resSucessoes e series com res
Sucessoes e series com resMaths Tutoring
 
Sucessoes, séries 20/21
Sucessoes, séries 20/21Sucessoes, séries 20/21
Sucessoes, séries 20/21Maths Tutoring
 
Ano 20/21 - Ficha 9ano - Intervalos
Ano 20/21 - Ficha 9ano - IntervalosAno 20/21 - Ficha 9ano - Intervalos
Ano 20/21 - Ficha 9ano - IntervalosMaths Tutoring
 
Fluid Mechanics Exercises
Fluid Mechanics ExercisesFluid Mechanics Exercises
Fluid Mechanics ExercisesMaths Tutoring
 
Worksheet - Differential Equations
Worksheet - Differential EquationsWorksheet - Differential Equations
Worksheet - Differential EquationsMaths Tutoring
 
Numeros complexos ficha
Numeros complexos fichaNumeros complexos ficha
Numeros complexos fichaMaths Tutoring
 
Teste polinómios 2 - 10.º ano Fev2020
Teste polinómios 2 - 10.º ano Fev2020Teste polinómios 2 - 10.º ano Fev2020
Teste polinómios 2 - 10.º ano Fev2020Maths Tutoring
 

Plus de Maths Tutoring (20)

O que é a pedagogia
O que é a pedagogiaO que é a pedagogia
O que é a pedagogia
 
Teste Derivadas
Teste DerivadasTeste Derivadas
Teste Derivadas
 
Ficha2 Derivadas
Ficha2 DerivadasFicha2 Derivadas
Ficha2 Derivadas
 
Teste 12ano
Teste 12ano Teste 12ano
Teste 12ano
 
Identidades trigonométricas
Identidades trigonométricasIdentidades trigonométricas
Identidades trigonométricas
 
limite sinx/x 12 ano
limite sinx/x 12 anolimite sinx/x 12 ano
limite sinx/x 12 ano
 
Trigonometria 12 ano revisoes
Trigonometria 12 ano revisoesTrigonometria 12 ano revisoes
Trigonometria 12 ano revisoes
 
Teorema de Bolzano
Teorema de BolzanoTeorema de Bolzano
Teorema de Bolzano
 
Intervalos e propriedades de números reais - Grau de dificuldade elevado
Intervalos e propriedades de números reais - Grau de dificuldade elevadoIntervalos e propriedades de números reais - Grau de dificuldade elevado
Intervalos e propriedades de números reais - Grau de dificuldade elevado
 
Teste algebra linear
Teste algebra linearTeste algebra linear
Teste algebra linear
 
Teste 11ano produto interno e vetores
Teste 11ano produto interno e vetoresTeste 11ano produto interno e vetores
Teste 11ano produto interno e vetores
 
Teste eqs e intervalos com res
Teste eqs e intervalos com resTeste eqs e intervalos com res
Teste eqs e intervalos com res
 
Teste equações e intervalos
Teste equações e intervalosTeste equações e intervalos
Teste equações e intervalos
 
Sucessoes e series com res
Sucessoes e series com resSucessoes e series com res
Sucessoes e series com res
 
Sucessoes, séries 20/21
Sucessoes, séries 20/21Sucessoes, séries 20/21
Sucessoes, séries 20/21
 
Ano 20/21 - Ficha 9ano - Intervalos
Ano 20/21 - Ficha 9ano - IntervalosAno 20/21 - Ficha 9ano - Intervalos
Ano 20/21 - Ficha 9ano - Intervalos
 
Fluid Mechanics Exercises
Fluid Mechanics ExercisesFluid Mechanics Exercises
Fluid Mechanics Exercises
 
Worksheet - Differential Equations
Worksheet - Differential EquationsWorksheet - Differential Equations
Worksheet - Differential Equations
 
Numeros complexos ficha
Numeros complexos fichaNumeros complexos ficha
Numeros complexos ficha
 
Teste polinómios 2 - 10.º ano Fev2020
Teste polinómios 2 - 10.º ano Fev2020Teste polinómios 2 - 10.º ano Fev2020
Teste polinómios 2 - 10.º ano Fev2020
 

Dernier

Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxDr. Asif Anas
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxheathfieldcps1
 
Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....Riddhi Kevadiya
 
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINTARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINTDR. SNEHA NAIR
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfMohonDas
 
Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...
Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...
Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...Dr. Asif Anas
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...raviapr7
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxiammrhaywood
 
Vani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational TrustVani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational TrustSavipriya Raghavendra
 
10 Topics For MBA Project Report [HR].pdf
10 Topics For MBA Project Report [HR].pdf10 Topics For MBA Project Report [HR].pdf
10 Topics For MBA Project Report [HR].pdfJayanti Pande
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfMohonDas
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptxraviapr7
 
Quality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICEQuality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICESayali Powar
 
Department of Health Compounder Question ‍Solution 2022.pdf
Department of Health Compounder Question ‍Solution 2022.pdfDepartment of Health Compounder Question ‍Solution 2022.pdf
Department of Health Compounder Question ‍Solution 2022.pdfMohonDas
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxraviapr7
 
Optical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptxOptical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptxPurva Nikam
 
3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptx3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptxmary850239
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxraviapr7
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptxSOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptxSyedNadeemGillANi
 

Dernier (20)

Ultra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptxUltra structure and life cycle of Plasmodium.pptx
Ultra structure and life cycle of Plasmodium.pptx
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
 
Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....Riddhi Kevadiya. WILLIAM SHAKESPEARE....
Riddhi Kevadiya. WILLIAM SHAKESPEARE....
 
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINTARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
ARTICULAR DISC OF TEMPOROMANDIBULAR JOINT
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdf
 
Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...
Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...
Unveiling the Intricacies of Leishmania donovani: Structure, Life Cycle, Path...
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...
 
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptxAUDIENCE THEORY -- FANDOM -- JENKINS.pptx
AUDIENCE THEORY -- FANDOM -- JENKINS.pptx
 
Vani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational TrustVani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
Vani Magazine - Quarterly Magazine of Seshadripuram Educational Trust
 
10 Topics For MBA Project Report [HR].pdf
10 Topics For MBA Project Report [HR].pdf10 Topics For MBA Project Report [HR].pdf
10 Topics For MBA Project Report [HR].pdf
 
HED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdfHED Office Sohayok Exam Question Solution 2023.pdf
HED Office Sohayok Exam Question Solution 2023.pdf
 
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptxClinical Pharmacy  Introduction to Clinical Pharmacy, Concept of clinical pptx
Clinical Pharmacy Introduction to Clinical Pharmacy, Concept of clinical pptx
 
Quality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICEQuality Assurance_GOOD LABORATORY PRACTICE
Quality Assurance_GOOD LABORATORY PRACTICE
 
Department of Health Compounder Question ‍Solution 2022.pdf
Department of Health Compounder Question ‍Solution 2022.pdfDepartment of Health Compounder Question ‍Solution 2022.pdf
Department of Health Compounder Question ‍Solution 2022.pdf
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptx
 
Optical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptxOptical Fibre and It's Applications.pptx
Optical Fibre and It's Applications.pptx
 
3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptx3.26.24 Race, the Draft, and the Vietnam War.pptx
3.26.24 Race, the Draft, and the Vietnam War.pptx
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptx
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptxSOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
SOLIDE WASTE in Cameroon,,,,,,,,,,,,,,,,,,,,,,,,,,,.pptx
 

Dynamical systems solved ex

  • 1. Dynamical Systems Solved Exercises 14 de julho de 2020 Miguel Fernandes 1. Consider the doubling map E2 : R/Z → R/Z defined by E2(x + Z) = 2x + Z. (a) Find the number of periodic points of period n. Remember: Given a topological dynamical system f : X → X, we say that a point x is periodic of period p if fp (x) = x holds. We have: En 2 (x + Z) = 2n x + Z = x + Z ⇔ (2n − 1)x = k ⇔ x = k 2n − 1 for some integer k. Thus the periodic points of period n are those for which k varies from 0 to 2n − 2 and hence we have 2n − 1 periodic points of period n. (b) Conclude that the periodic points of E2 are dense in S1 . Following the previous exercise, we see that the periodic points of E2 are uni- formly distributed in S1 . Consequentely, making n → ∞ we get the desired result. 2. Consider the dynamical system f : R → R defined by f(x) = |x − 2|. (a) Find the fixed point of f. Is it periodic? What is its period? Comment. The computation of the fixed point of f is the following: f(x) = x ⇔ |x − 2| = x ⇔ x − 2 = ±x ⇔ x = 1. Clearly, it is a periodic point of period 1, which means that, a particle starting its trajectory at x = 1 will remain in that position for all future time. (b) Define the first three iterates of f and sketch them. Clearly, f(x) = |x − 2|, f2 (x) = ||x − 2| − 2| and f3 (x) = |||x − 2| − 2| − 2|, all 1
  • 2. defined in the whole real line (a piecewise definition would be too cumbersome, a nice task for the reader!). The graphs of such iterations are shown below. Figura 1: First three iterates of the transformation f. (c) Find other periodic and pre–periodic points of f. It is clear in the figure above that the points of the set [0, 2] {1} are periodic of period 2 and the points belonging to ]−∞, 0[ ∪ ]2, +∞[ are pre–periodic. 3. Show that expanding maps of S1 are topologically mixing. Remember: A topological continuous map f : X → X is said to be topologically mixing if, given two non–empty open sets U and V , there exists a time N ∈ N such that for all n ≥ N we have fn (U) ∩ V = ∅. Let f : S1 → S1 be an expanding map (and hence is continuous) and F : R → R such that π◦F = f ◦π (the existence of such function F is guaranteed by the continuity of f), where π : R → S1 is the projection defined by π(x) = [x], for all x ∈ R. (Remark: [x] denotes the point of S1 subject to the identification S1 = R/Z). Notice that we can also write f ([x]) = [F(x)]. Clearly, if f([x]) = [mx], for some 1 < m ∈ Z then F(x) = mx + n, for some n ∈ Z. Hence |F (x)| = m > 1, for all x ∈ R, and by the mean value property it follows that |F(x) − F(y)| = |F (c)(x − y)| = m|x − y|, where x, y ∈ R and c ∈ (x, y) (or c ∈ (y, x)). Take an open set U ⊂ R and choose an interval I = (a, b) ⊂ U. It is straightforward that Fn , n ∈ N, increases the length of I by a factor of mn . That implies that we can choose n0 such that mn0 > 1/(b − a) and consequently the length of Fn0 (I) will exceed 1. Thus π(Fn0 (I)) = S1 and S1 ⊂ fn (π(I)), for all n ≥ n0. This means that, given two non–empty open sets U and V in S1 , the set fn (U) ∩ V is non–empty, and the same for all future time. 4. Consider the rotation of the circle Rα : S1 → S1 defined by Rα(x + Z) = x + α + Z. (a) Show that Rα has periodic points if and only if α is rational. If α is rational then it can be writeen in the form α = p/q, for some integers p and q, then all points are periodic with period q. Indeed: Rq p/q(x + Z) = x + q ∗ p/q + Z = x + Z. Reciprocally, 2
  • 3. Rn α(x + Z) = x + nα + Z = x + Z ⇔ x + nα = x + p ⇔ α = p/n for some integer p. Thus if we assume that x + Z is periodic of period n, then α must equal p/n, for some p ∈ Z. (b) Show that 0 is a recurrent point when α is irrational. Remember: The orbit of a point x is said to be dense if, for all y ∈ R/Z and all > 0, there exists a time n ∈ N such that d(Rn α(x), y) < . In particular, making y = x, we find that x is a recurrent point. Let α be an irrational number. We first observe that the map n → Rn α(0), for each n ∈ N, is one–to–one. For that, let Rn α(0) = Rm α . Then nα+Z = mα+Z ⇔ (n−m)α = k, for some integer k. But, since α is irrational, we must have n = m. Now, fix > 0. We find a time n for which we have 0 < nα+Z < +Z. The den- sity of the irrational numbers on the real line allows us to assert that α belongs to some interval [p/q, p/q + /2q], where p and q > 0 are integers. Consequently, if we set n = q we have qα ∈ [p, p + /2] and thus qα + Z ≤ /2 + Z < + Z. This implies that, for certain iterates of Rα, the trajectory is sufficiently close to 0. 5. Consider the linear transformation T : T2 = R2 /Z2 → R2 /Z2 defined by: T(x, y) = (2x + y, x + y)(mod 1)1 . (a) Show that T is well defined, i.e., if (x−x , y−y ) ∈ Z2 then L(x, y)−L(x , y ) ∈ Z2 . It’s an immediate consequence of the linearity of T and the fact that Z2 is closed under T. (b) Compute the matrix of the linear transformation L : R2 → R2 defined by L(x, y) = (2x+y, x+y). Justify that L is invertible and compute its eigenvalues. The matrix of L (relative to the canonical basis) is: ML = Å 2 1 1 1 ã and is clearly non–singular (determinant equal to 1 and hence M−1 L has integer entries). Furthermore, its eigenvalues are the inverses: λ1 = 3 + √ 5 2 and λ2 = λ−1 1 = 3 − √ 5 2 . (c) Show that the periodic points of T are dense in T2 . Let x = a/b and y = c/b (same denominator) provided a, b, c, d ∈ Z. Then T(x, y) = T(a/b, c/b) = 2a+c b , a+c b and so T(x, y) is also a rational point whose coordinates have the same denominator as (x, y). But notice that we only have b2 different choices for (x, y) in T2 (why is that) whose points shall form the set A. Thus we have {Tn (x, y)| n ∈ N} ⊂ A. Given T is invertible, we find that the point (x, y) is periodic. Since b is arbitrary, such periodic points are dense in T2 (find the anology with the expanding map). 1 The so–called Arnold cat map. 3
  • 4. (d) Show that the number of n−periodic points of T is given by | det(Mn L − I2)| (assuming this number is not zero). The periodic points of T of period n are the solutions of: Mn L x = x (mod 1). Notice that such equation is equivalent to (Mn L −I2)x = 0 (mod 1) or more preci- sely (Mn L −I2)x = k, for some k ∈ Z. This makes us consider the transformation L∗ : R2 → R2 defined by L∗ (x) = (Mn L −I2)x. According to the explained above, it is clear that the periodic points of T are given by the integer points in the set L∗ ([0, 1[ × [0, 1[). Furthermore, we can easily verify that L∗ transforms the square [0, 1[ × [0, 1[ into a paralelogram (without 2 edges). Making use of Pick’s Theorem and regarding the fact that the area of the parallelogram is given by | det(Mn L − I2)|, we find the desired result. (e) Compute the number of points of period 5. Given the previous result, we have that the number os such periodic points is given by | det(M5 L − I2)|. But | det(M5 L − I2)| = |(λ5 1 − 1)(λ−5 1 − 1)|, where λ1 is the eigenvalue in (b). Since λ1 = (3 + √ 5)/2, we find |(λ5 1 − 1)(λ−5 1 − 1)| = 121. 6. Consider the alphabet β = {1, 2, 3} made of three letters. Let A be the transition matrix defined as follows: A = Ñ 0 1 1 1 1 0 0 0 1 é . (a) Find elements of the set ΣA = x = x1x2... ∈ Σ| axnxn+1 = 1, n ≥ 1 (Σ is the set of those infinite words whose letters belong to the alphabet β). For instance, 221333... and 121333... (b) How many words are there with with ’3’ as first letter? One, 33333... (c) Let α be a word of the form α = 1...3 in a total of 4 letters and hence finite. Write the all the possibilities for α. How many are they? There are only two possibilities: 1213 and 1333. (d) Consider those finite words, of length n + 1, with first and last letters given by i and j, respectively. Show that the number of words in such conditions is given by the ij element of An , where A is the transition matrix. For simplicity, denote such number by Nn (i, j). Verify the result for exercise (d). We use induction on n for a proof. The case n = 1 is trivial (definition of transition matrix). Now, using the induction hypothesis: Nn+1 (i, j) = k Nn (i, k)N1 (k, j) = k(An )ikakj = (An+1 )ij. In the case of the exercise (d), the power A3 is the matrix: A3 = Ñ 1 2 2 2 3 2 0 0 1 é 4
  • 5. and the result follows. (e) Conclude that the third row of A is invariant under the power operator. Just remember exercises (b) and (d). (f) Conclude that A is neither irreducible nor aperiodic. Since the third row of A is invariant under the power operator and it contains zeros, we conclude that A is not irreducible and hence cannot be aperiodic. 5