Chaotic Dynamics
eBook - PDF

Chaotic Dynamics

Fractals, Tilings, and Substitutions

  1. English
  2. PDF
  3. Available on iOS & Android
eBook - PDF

Chaotic Dynamics

Fractals, Tilings, and Substitutions

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Table of contents
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About This Book

This undergraduate textbook is a rigorous mathematical introduction to dynamical systems and an accessible guide for students transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises that range from straightforward to more difficult with hints, and includes concrete applications of real analysis and metric space theory to dynamical problems. Proofs are complete and carefully explained, and there is opportunity to practice manipulating algebraic expressions in an applied context of dynamical problems. After presenting a foundation in one-dimensional dynamical systems, the text introduces students to advanced subjects in the latter chapters, such as topological and symbolic dynamics. It includes two-dimensional dynamics, Sharkovsky's theorem, and the theory of substitutions, and takes special care in covering Newton's method. Mathematica code is available online, so that students can see implementation of many of the dynamical aspects of the text.

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Information

Publisher
Cambridge University Press
Year
2016
ISBN
9781316944356
Topic
Mathematics
Subtopic
Mathematics General
Index
Mathematics
13
1.2
Newtonā€™s
Method
and
Fixed
Points
0.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
The
tent
map
intersects
y
=
x
at
x
=
0
and
x
=
2
/
3.
Example
1.2.5
Some
maps
do
not
have
ļ¬xed
points.
For
example,
f
:
R
ā†’
R
,
f
(
x
)
=
x
2
+
1
has
no
ļ¬xed
points,
since
the
equation
x
2
+
1
=
x
has
no
real
solution.
f
(
x
)
=
x
2
+
1
does
not
intersect
the
line
y
=
x
.
Example
1.2.6
Let
c
āˆˆ
R
and
f
c
:
R
ā†’
R
be
deļ¬ned
by
f
c
(
x
)
=
x
2
+
c
.
We
ask
for
which
values
of
c
does
f
c
have
a
ļ¬xed
point?
What
are
the
corresponding
values
of
the
ļ¬xed
point(s)?
If
we
graph
f
c
using
a
computer
algebra
system
and
a
ā€œmanipulateā€
type
plot,
we
can
see
that
when
c
=
1
there
are
no
ļ¬xed
points
(Example
1.2.5
),
but,
as
c
decreases,
at
some
value
of
c
,
f
c
intersects
the

Table of contents

  1. Cover
  2. Half-title page
  3. Series page
  4. Title page
  5. Copyright page
  6. Dedication
  7. Contents
  8. Preface
  9. 1 The Orbits of One-Dimensional Maps
  10. 2 Bifurcations and the Logistic Family
  11. 3 Sharkovsky's Theorem
  12. 4 Dynamics on Metric Spaces
  13. 5 Countability, Sets of Measure Zero and the Cantor Set
  14. 6 Devaney's Definition of Chaos
  15. 7 Conjugacy of Dynamical Systems
  16. 8 Singer's Theorem
  17. 9 Conjugacy, Fundamental Domains and the Tent Family
  18. 10 Fractals
  19. 11 Newton's Method for Real Quadratics and Cubics
  20. 12 Coppel's Theorem and a Proof of Sharkovsky's Theorem
  21. 13 Real Linear Transformations, the HĆ©non Map and Hyperbolic Toral Automorphisms
  22. 14 Elementary Complex Dynamics
  23. 15 Examples of Substitutions
  24. 16 Fractals Arising from Substitutions
  25. 17 Compactness in Metric Spaces and an Introduction to Topological Dynamics
  26. 18 Substitution Dynamical Systems
  27. 19 Sturmian Sequences and Irrational Rotations
  28. 20 The Multiple Recurrence Theorem of Furstenberg and Weiss
  29. Appendix A Theorems from Calculus
  30. Appendix B The Baire Category Theorem
  31. Appendix C The Complex Numbers
  32. Appendix D Weyl's Equidistribution Theorem
  33. References
  34. Index