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Invitation to Dynamical Systems [Minkštas viršelis]

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  • Formatas: Paperback / softback, 408 pages, aukštis x plotis x storis: 232x191x18 mm, weight: 540 g
  • Serija: Dover Books on Mathema 1.4tics
  • Išleidimo metai: 27-Apr-2012
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486485943
  • ISBN-13: 9780486485942
Kitos knygos pagal šią temą:
Invitation to Dynamical SystemsDidesnis paveikslėlis
  • Formatas: Paperback / softback, 408 pages, aukštis x plotis x storis: 232x191x18 mm, weight: 540 g
  • Serija: Dover Books on Mathema 1.4tics
  • Išleidimo metai: 27-Apr-2012
  • Leidėjas: Dover Publications Inc.
  • ISBN-10: 0486485943
  • ISBN-13: 9780486485942
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780486485942.html
Raktažodžiai:
This text is designed for those who wish to study mathematics beyond linear algebra but are not ready for abstract material. Rather than a theorem-proof-corollary-remark style of exposition, it stresses geometry, intuition, and dynamical systems. An appendix explains how to write MATLAB, Mathematica, and C programs to compute dynamical systems. 1996 edition.


This text is designed for those who wish to study mathematics beyond linear algebra but are unready for abstract material. Rather than a theorem-proof-corollary exposition, it stresses geometry, intuition, and dynamical systems. 1996 edition.
Preface xi
1 Introduction
1(36)
1.1 What is a dynamical system?
1(7)
1.1.1 State vectors
1(1)
1.1.2 The next instant: discrete time
2(2)
1.1.3 The next instant: continuous time
4(2)
1.1.4 Summary
6(1)
Problems
6(2)
1.2 Examples
8(27)
1.2.1 Mass and spring
8(2)
1.2.2 RLC circuits
10(1)
1.2.3 Pendulum
11(5)
1.2.4 Your bank account
16(1)
1.2.5 Economic growth
17(2)
1.2.6 Pushing buttons on your calculator
19(3)
1.2.7 Microbes
22(2)
1.2.8 Predator and prey
24(2)
1.2.9 Newton's Method
26(2)
1.2.10 Euler's method
28(3)
1.2.11 "Random" number generation
31(1)
Problems
32(3)
1.3 What we want; what we can get
35(2)
2 Linear Systems
37(64)
2.1 One dimension
37(13)
2.1.1 Discrete time
37(8)
2.1.2 Continuous time
45(3)
2.1.3 Summary
48(1)
Problems
49(1)
2.2 Two (and more) dimensions
50(41)
2.2.1 Discrete time
51(6)
2.2.2 Continuous time
57(24)
2.2.3 The nondiagonalizable case
81(7)
Problems
88(3)
2.3 Examplification: Markov chains
91(10)
2.3.1 Introduction
91(2)
2.3.2 Markov chains as linear systems
93(3)
2.3.3 The long term
96(2)
Problems
98(3)
3 Nonlinear Systems 1: Fixed Points
101(52)
3.1 Fixed points
102(8)
3.1.1 What is a fixed point?
102(1)
3.1.2 Finding fixed points
103(1)
3.1.3 Stability
104(4)
Problems
108(2)
3.2 Linearization
110(18)
3.2.1 One dimension
110(7)
3.2.2 Two and more dimensions
117(9)
Problems
126(2)
3.3 Lyapunov functions
128(18)
3.3.1 Linearization can fail
128(2)
3.3.2 Energy
130(3)
3.3.3 Lyapunov's method
133(4)
3.3.4 Gradient systems
137(7)
Problems
144(2)
3.4 Examplification: Iterative methods for solving equations
146(7)
Problems
150(3)
4 Nonlinear Systems 2: Periodicity and Chaos
153(78)
4.1 Continuous time
154(14)
4.1.1 One dimension: no periodicity
154(1)
4.1.2 Two dimensions: the Poincare-Bendixson theorem
155(6)
4.1.3 The Hopf bifurcation
161(2)
4.1.4 Higher dimensions: the Lorenz system and chaos
163(4)
Problems
167(1)
4.2 Discrete time
168(50)
4.2.1 Periodicity
169(4)
4.2.2 Stability of periodic points
173(2)
4.2.3 Bifurcation
175(13)
4.2.4 Sarkovskii's theorem
188(14)
4.2.5 Chaos and symbolic dynamics
202(14)
Problems
216(2)
4.3 Examplification: Riffle shuffles and the shift map
218(13)
4.3.1 Riffle shuffles
218(1)
4.3.2 The shift map
219(4)
4.3.3 Shifting and shuffling
223(3)
4.3.4 Shuffling again and again
226(3)
Problems
229(2)
5 Fractals
231(86)
5.1 Cantor's set
231(11)
5.1.1 Symbolic representation of Cantor's set
232(1)
5.1.2 Cantor's set in conventional notation
233(3)
5.1.3 The link between the two representations
236(1)
5.1.4 Topological properties of the Cantor set
236(4)
5.1.5 In what sense a fractal?
240(1)
Problems
241(1)
5.2 Biting out the middle in the plane
242(4)
5.2.1 Sierpinski's triangle
242(1)
5.2.2 Koch's snowflake
243(2)
Problems
245(1)
5.3 Contraction mapping theorems
246(14)
5.3.1 Contraction maps
246(1)
5.3.2 Contraction mapping theorem on the real line
247(2)
5.3.3 Contraction mapping in higher dimensions
249(1)
5.3.4 Contractive affine maps: the spectral norm
250(4)
5.3.5 Other metric spaces
254(1)
5.3.6 Compact sets and Hausdorff distance
255(3)
Problems
258(2)
5.4 Iterated function systems
260(16)
5.4.1 From point maps to set maps
260(2)
5.4.2 The union of set maps
262(3)
5.4.3 Examples revisited
265(5)
5.4.4 IFSs defined
270(1)
5.4.5 Working backward
271(5)
Problems
276(1)
5.5 Algorithms for drawing fractals
276(11)
5.5.1 A deterministic algorithm
277(2)
5.5.2 Dancing on fractals
279(4)
5.5.3 A randomized algorithm
283(3)
Problems
286(1)
5.6 Fractal dimension
287(20)
5.6.1 Covering with balls
287(3)
5.6.2 Definition of dimension
290(2)
5.6.3 Simplifying the definition
292(8)
5.6.4 Just-touching similitudes and dimension
300(6)
Problems
306(1)
5.7 Examplification: Fractals in nature
307(10)
5.7.1 Dimension of physical fractals
308(4)
5.7.2 Estimating surface area
312(2)
5.7.3 Image analysis
314(1)
Problems
314(3)
6 Complex Dynamical Systems
317(24)
6.1 Julia sets
317(10)
6.1.1 Definition and examples
317(7)
6.1.2 Escape-time algorithm
324(2)
6.1.3 Other Julia sets
326(1)
Problems
327(1)
6.2 The Mandelbrot set
327(6)
6.2.1 Definition and various views
327(5)
6.2.2 Escape-time algorithm
332(1)
Problems
332(1)
6.3 Examplification: Newton's method revisited
333(3)
Problems
336(1)
6.4 Examplification: Complex bases
336(5)
6.4.1 Place value revisited
336(2)
6.4.2 IFSs revisited
338(2)
Problems
340(1)
A Background Material
341(14)
A.1 Linear algebra
341(6)
A.1.1 Much ado about 0
341(1)
A.1.2 Linear independence
342(1)
A.1.3 Eigenvalues/vectors
342(1)
A.1.4 Diagonalization
343(1)
A.1.5 Jordan canonical form
343(1)
A.1.6 Basic linear transformations of the plane
344(3)
A.2 Complex numbers
347(2)
A.3 Calculus
349(2)
A.3.1 Intermediate and mean value theorems
349(1)
A.3.2 Partial derivatives
350(1)
A.4 Differential equations
351(4)
A.4.1 Equations
351(1)
A.4.2 What is a differential equation?
351(1)
A.4.3 Standard notation
352(3)
B Computing
355(14)
B.1 Differential equations
355(10)
B.1.1 Analytic solutions
356(1)
B.1.2 Numerical solutions
357(8)
B.2 Triangle Dance
365(1)
B.3 About the accompanying software
366(3)
Bibliography 369(2)
Index 371