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Discovering Dynamical Systems Through Experiment and Inquiry [Kõva köide]

, (Gustavus Adolphus Collete)
  • Formaat: Hardback, 216 pages, kõrgus x laius: 234x156 mm, kaal: 449 g, 40 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 23-Mar-2021
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 0367903946
  • ISBN-13: 9780367903947
Teised raamatud teemal:
Discovering Dynamical Systems Through Experiment and InquirySuurem pilt
  • Formaat: Hardback, 216 pages, kõrgus x laius: 234x156 mm, kaal: 449 g, 40 Illustrations, black and white
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 23-Mar-2021
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 0367903946
  • ISBN-13: 9780367903947
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780367903947.html
Märksõnad:

Discovering Dynamical Systems Through Experiment and Inquiry differs from most texts on dynamical systems by blending the use of computer simulations with inquiry-based learning (IBL). IBL is an excellent tool to move students from merely remembering the material to deeper understanding and analysis. This method relies on asking students questions first, rather than presenting the material in a lecture.

Another unique feature of this book is the use of computer simulations. Students can discover examples and counterexamples through manipulations built into the software. These tools have long been used in the study of dynamical systems to visualize chaotic behavior.

We refer to this unique approach to teaching mathematics as ECAP—Explore, Conjecture, Apply, and Prove. ECAP was developed to mimic the actual practice of mathematics in an effort to provide students with a more holistic mathematical experience. In general, each section begins with exercises guiding students through explorations of the featured concept and concludes with exercises that help the students formally prove the results.

While symbolic dynamics is a standard topic in an undergraduate dynamics text, we have tried to emphasize it in a way that is more detailed and inclusive than is typically the case. Finally, we have chosen to include multiple sections on important ideas from analysis and topology independent from their application to dynamics.

Preface xi
1 An Introduction to Dynamical Systems
1(10)
1.1 What Is a Dynamical System
1(1)
1.2 Numerical Iteration and Orbits
2(4)
1.3 Graphical Iteration
6(2)
1.4 Modeling Using Discrete Dynamical Systems
8(3)
2 Sequences
11(22)
2.1 Introduction to Sequences
11(1)
2.2 Convergence of Sequences
12(3)
2.3 The Squeeze Theorem
15(3)
2.4 Arithmetic Limit Theorems
18(1)
2.5 Bounded and Unbounded Sequences
19(4)
2.6 Subsequences
23(1)
2.7 Liminfs and Limsups
24(5)
2.8 Cauchy Sequences
29(4)
3 Fixed Points and Periodic Points
33(20)
3.1 Fixed Points
33(15)
3.1.1 Fixed Points of Linear Systems
34(3)
3.1.2 Attracting Fixed Points of Nonlinear Systems
37(4)
3.1.3 Repelling Fixed Points of Nonlinear Systems
41(3)
3.1.4 Neutral Fixed Points of Nonlinear Systems
44(4)
3.2 Periodic Points
48(5)
3.2.1 Stability of Periodic Points
48(2)
3.2.2 New Periodic Orbits from Old
50(3)
4 Analysis of Fixed Points
53(16)
4.1 Fixed Point Existence Theorems
53(6)
4.2 The Inverse and Implicit Function Theorems
59(5)
4.2.1 The Inverse Function Theorems
59(3)
4.2.2 The Implicit Function Theorem
62(2)
4.3 Hyperbolic Periodic Points
64(5)
5 Bifurcations
69(12)
5.1 What is a Bifurcation?
69(2)
5.2 Introduction to Bifurcation Diagrams
71(1)
5.3 The Tangent Bifurcation
72(4)
5.4 The Period Doubling Bifurcation
76(5)
6 Examples of Global Dynamics
81(12)
6.1 Local Dynamics vs. Global Dynamics
81(2)
6.2 The Logistic Map with a = 4 (Part 1)
83(2)
6.3 The Doubling Map
85(6)
6.3.1 Basic Dynamics of the Doubling Map
86(2)
6.3.2 The Doubling Map in Binary
88(3)
6.4 The Logistic Map with a > 4 (Part 1)
91(2)
7 The Tools of Global Dynamics
93(10)
7.1 How to study Global Dynamics
93(1)
7.2 The Cantor Set
94(2)
7.3 The Shift Map (Part 1)
96(7)
7.3.1 The Sequence Space on 2 Symbols
96(3)
7.3.2 Dynamics on the Sequence Space on 2 Symbols
99(4)
8 Examples of Chaos
103(14)
8.1 Introduction: The Definition of Chaos
103(4)
8.2 The Shift Map (Part 2)
107(1)
8.3 Topological Conjugacy
108(3)
8.4 Return to The Doubling Map
111(1)
8.5 The Logistic Map with a > 4 (Part 2)
111(4)
8.6 The Logistic Map with a = 4 (Part 2)
115(2)
9 Prom Fixed Points to Chaos
117(8)
9.1 Introduction
117(2)
9.2 Computing a Bifurcation Diagram
119(1)
9.3 Period-doubling to Chaos
120(2)
9.4 Windows of Stable Periodic Behavior
122(3)
10 Sarkovskii's Theorem
125(10)
10.1 Introduction
125(1)
10.2 The Intermediate Value Theorem
125(2)
10.3 Review of Two Fixed Point Theorems
127(1)
10.4 Sarkovskii's Theorem
127(8)
10.4.1 Discovering Sarkovskii's Theorem
128(2)
10.4.2 Using Sarkovskii's Theorem
130(5)
11 Dynamical Systems on the Plane
135(24)
11.1 Linear Algebra Foundations
135(3)
11.2 Linear Systems with Real Eigenvalues
138(6)
11.3 Linear Systems with Complex Eigenvalues
144(4)
11.4 Fixed Points of Nonlinear Systems
148(5)
11.5 Periodic Points
153(2)
11.6 Chaos in the Henon map
155(4)
12 The Smale Horseshoe
159(10)
12.1 Motivating the Horseshoe Map
159(1)
12.2 The Horseshoe Map
160(3)
12.3 More Symbolic Dynamics
163(3)
12.3.1 Two-Sided Sequence Space
163(1)
12.3.2 The Two-Sided Shift Map
164(2)
12.4 A Horseshoe in the Henon Map
166(3)
13 Generalized Symbolic Dynamics
169(24)
13.1 Topology Foundations
169(2)
13.2 Shift Dynamical Systems
171(10)
13.2.1 One-Sided Shift Spaces
171(1)
13.2.2 Two-Sided Shift Spaces
172(6)
13.2.3 Shifts of Finite Type
178(3)
13.3 Representing Shift Spaces with Graphs
181(7)
13.3.1 Higher Edge Graphs
184(4)
13.4 Markov Partitions
188(5)
Bibliography 193(2)
Index 195
Tom LoFaro: Tom LoFaro is the Clifford M. Swanson Professor of Mathematics at Gustavus Adolphus College. He earned his Bachelor's and Master's degrees in mathematics from the University of Missouri. He then earned his Ph.D. from Boston University in mathematics under the direction of Nancy Kopell. Tom's research interests are in the applications of dynamical systems to mathematical biology in general and neuroscience in particular and is an Affiliated Faculty Member of the Institute for the Study of Decision Making at New York University.

Tom has been active in the differential equations and dynamical systems education community and in the past worked closely with the Community for Ordinary Differential Equations Educators where he has published several articles their journal.

Jeff Ford: Jeff Ford is a Visiting Assistant Professor of Mathematics at Gustavus Adolphus College. He earned his Bachelors degree from Gustavus Adolphus College, his Masters degree in mathematics from Minnesota State University Mankato, and his Ph.D. in mathematics from Auburn University, studying under Dr. Krystyna Kuperberg. Jeff is interested in the existence volume-preserving dynamical systems with unique properties.

Jeff uses and assesses a variety of active learning techniques in his class including inquiry-based learning and team-based learning. His scholarship in this area centers on understanding how active learning techniques improve confidence and reduce anxiety in undergraduate students.