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E-grāmata: Chaos: An Introduction to Dynamical Systems

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Chaos: An Introduction to Dynamical SystemsPalielināt
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Chaos: An Introduction to Dynamical Systems, was developed and class-tested by a distinguished team of authors at two universities through their teaching of courses based on the material. Intended for courses in nonlinear dynamics offered either in Mathematics or Physics, the text requires only calculus, differential equations, and linear algebra as prerequisites. Along with discussions of the major topics, including discrete dynamical systems, chaos, fractals, nonlinear differential equations and bifurcations, the text also includes Lab Visits, short reports that illustrate relevant concepts from the physical, chemical and biological sciences. There are Computer Experiments throughout the text that present opportunities to explore dynamics through computer simulations, designed to be used with any software package. And each chapter ends with a Challenge, which provides students a tour through an advanced topic in the form of an extended exercise.

Spanning the wide reach of nonlinear dynamics throughout mathematics and the natural and physical sciences, this book develops and explains the most intriguing and fundamental elements of the topic. Among the major topics included are discrete dynamical systems, chaos, fractals, nonlinear differential equations, and bifurcations. Developed and class-tested by a distinguished team of authors, this text is ideal for courses in nonlinear dynamics in either mathematics or physics.

Recenzijas

From the reviews:



" Written by some prominent contributors to the development of the field With regard to both style and content, the authors succeed in introducing junior/senior undergraduate students to the dynamics and analytical techniques associated with nonlinear systems, especially those related to chaos There are several aspects of the book that distinguish it from some other recent contributions in this area The treatment of discrete systems here maintains a balanced emphasis between one- and two- (or higher-) dimensional problems. This is an important feature since the dynamics for the two cases and methods employed for their analyses may differ significantly. Also, while most other introductory texts concentrate almost exclusively upon discrete mappings, here at least three of the thirteen chapters are devoted to differential equations, including the Poincare-Bendixson theorem. Add to this a discussion of $\omega$-limit sets, including periodic and strange attractors, as well as a chapter on fractals, and the result is one of the most comprehensive texts on the topic that has yet appeared." Mathematical Reviews



 

Papildus informācija

Springer Book Archives
INTRODUCTION v
1 ONE-DIMENSIONAL MAPS
1(42)
1.1 One-Dimensional Maps
2(3)
1.2 Cobweb Plot: Graphical Representation of an Orbit
5(4)
1.3 Stability of Fixed Points
9(4)
1.4 Periodic Points
13(4)
1.5 The Family of Logistic Maps
17(5)
1.6 The Logistic Map G(x) = 4x(1 - x)
22(3)
1.7 Sensitive Dependence on Initial Conditions
25(2)
1.8 Itineraries
27(5)
CHALLENGE 1: PERIOD THREE IMPLIES CHAOS
32(4)
EXERCISES
36(3)
LAB VISIT 1: BOOM, BUST, AND CHAOS IN THE BEETLE CENSUS
39(4)
2 TWO-DIMENSIONAL MAPS
43(62)
2.1 Mathematical Models
44(14)
2.2 Sinks, Sources, and Saddles
58(4)
2.3 Linear Maps
62(5)
2.4 Coordinate Changes
67(1)
2.5 Nonlinear Maps and the Jacobian Matrix
68(10)
2.6 Stable and Unstable Manifolds
78(9)
2.7 Matrix Times Circle Equals Ellipse
87(5)
CHALLENGE 2: COUNTING THE PERIODIC ORBITS OF LINEAR MAPS ON A TORUS
92(6)
EXERCISES
98(1)
LAB VISIT 2: IS THE SOLAR SYSTEM STABLE?
99(6)
3 CHAOS
105(44)
3.1 Lyapunov Exponents
106(3)
3.2 Chaotic Orbits
109(5)
3.3 Conjugacy and the Logistic Map
114(10)
3.4 Transition Graphs and Fixed Points
124(5)
3.5 Basins of Attraction
129(6)
CHALLENGE 3: SHARKOVSKII'S THEOREM
135(5)
EXERCISES
140(3)
LAB VISIT 3: PERIODICITY AND CHAOS IN A CHEMICAL REACTION
143(6)
4 FRACTALS
149(44)
4.1 Cantor Sets
150(6)
4.2 Probabilistic Constructions of Fractals
156(5)
4.3 Fractals from Deterministic Systems
161(3)
4.4 Fractal Basin Boundaries
164(8)
4.5 Fractal Dimension
172(5)
4.6 Computing the Box-Counting Dimension
177(3)
4.7 Correlation Dimension
180(3)
CHALLENGE 4: FRACTAL BASIN BOUNDARIES AND THE UNCERTAINTY EXPONENT
183(3)
EXERCISES
186(2)
LAB VISIT 4: FRACTAL DIMENSION IN EXPERIMENTS
188(5)
5 CHAOS IN TWO-DIMENSIONAL MAPS
193(38)
5.1 Lyapunov Exponents
194(5)
5.2 Numerical Calculation of Lyapunov Exponents
199(4)
5.3 Lyapunov Dimension
203(4)
5.4 A Two-Dimensional Fixed-Point Theorem
207(5)
5.5 Markov Partitions
212(4)
5.6 The Horseshoe Map
216(6)
CHALLENGE 5: COMPUTER CALCULATIONS AND SHADOWING
222(4)
EXERCISES
226(2)
LAB VISIT 5: CHAOS IN SIMPLE MECHANICAL DEVICES
228(3)
6 CHAOTIC ATTRACTORS
231(42)
6.1 Forward Limit Sets
233(5)
6.2 Chaotic Attractors
238(7)
6.3 Chaotic Attractors of Expanding Interval Maps
245(4)
6.4 Measure
249(4)
6.5 Natural Measure
253(3)
6.6 Invariant Measure for One-Dimensional Maps
256(8)
CHALLENGE 6: INVARIANT MEASURE FOR THE LOGISTIC MAP
264(2)
EXERCISES
266(1)
LAB VISIT 6: FRACTAL SCUM
267(6)
7 DIFFERENTIAL EQUATIONS
273(56)
7.1 One-Dimensional Linear Differential Equations
275(3)
7.2 One-Dimensional Nonlinear Differential Equations
278(6)
7.3 Linear Differential Equations in More than One Dimension
284(10)
7.4 Nonlinear Systems
294(6)
7.5 Mortion in a Potential Field
300(4)
7.6 Lyapunov Functions
304(5)
7.7 Lorka-Volterra Models
309(7)
CHALLENGE 7: A LIMIT CYCLE IN THE VAN DER POL SYSTEM
316(5)
EXERCISES
321(4)
LAB VISIT 7: FLY VS. FLY
325(4)
8 PERIODIC ORBITS AND LIMIT SETS
329(30)
8.1 Limit Sets for Planar Differential Equations
331(6)
8.2 Properties of w-Limit Sets
337(4)
8.3 Proof of the Poincare-Bendixson Theorem
341(9)
CHALLENGE 8: TWO INCOMMENSURATE FREQUENCIES FORM A TORUS
350(3)
EXERCISES
353(2)
LAB VISIT 8: STEADY STATES AND PERIODICITY IN A SQUID NEURON
355(4)
9 CHAOS IN DIFFERENTIAL EQUATIONS
359(40)
9.1 The Lorenz Attractor
359(7)
9.2 Stability in the Large, Instability in the Small
366(4)
9.3 The Rossler Attractor
370(5)
9.4 Chua's Circuit
375(1)
9.5 Forced Oscillators
376(3)
9.6 Lyapunov Exponents in Flows
379(8)
CHALLENGE 9: SYNCHRONIZATION OF CHAOTIC ORBITS
387(6)
EXERCISES
393(1)
LAB VISIT 9: LASERS IN SYNCHRONIZATION
394(5)
10 STABLE MANIFOLDS AND CRISES
399(48)
10.1 The Stable Manifold Theorem
401(8)
10.2 Homoclinic and Heteroclinic Points
409(4)
10.3 Crises
413(9)
10.4 Proof of the Stable Manifold Theorem
422(8)
10.5 Stable and Unstable Manifolds for Higher Dimensional Maps
430(2)
CHALLENGE 10: THE LAKES OF WADA
432(8)
EXERCISES
440(1)
LAB VISIT 10: THE LEAKY FAUCET: MINOR IRRITATION OR CRISIS?
441(6)
11 BIFURCATIONS
447(52)
11.1 Saddle-Node and Period-Doubling Bifurcations
448(5)
11.2 Bifurcation Diagrams
453(7)
11.3 Continuability
460(4)
11.4 Bifurcations of One-Dimensional Maps
464(4)
11.5 Bifurcations in Plane Maps: Area-Contracting Case
468(3)
11.6 Bifurcations in Plane Maps: Area-Preserving Case
471(7)
11.7 Bifurcations in Differential Equations
478(5)
11.8 Hopf Bifurcations
483(8)
CHALLENGE 11: HAMILTONIAN SYSTEMS AND THE LYAPUNOV CENTER THEOREM
491(3)
EXERCISES
494(2)
LAB VISIT 11: IRON + SULFURIC ACID XXX HOPF BIFURCATION
496(3)
12 CASCADES
499(38)
12.1 Cascades and 4.669201609...
500(4)
12.2 Schematic Bifurcation Diagrams
504(6)
12.3 Generic Bifurcations
510(8)
12.4 The Cascade Theorem
518(7)
CHALLENGE 12: UNIVERSALITY IN BIFURCATION DIAGRAMS
525(6)
EXERCISES
531(1)
LAB VISIT 12: EXPERIMENTAL CASCADES
532(5)
13 STATE RECONSTRUCTION FROM DATA
537(20)
13.1 Delay Plots from Time Series
537(4)
13.2 Delay Coordinates
541(4)
13.3 Embedology
545(8)
CHALLENGE 13: BOX-COUNTING DIMENSION AND INTERSECTION
553(4)
A MATRIX ALGEBRA
557(10)
A.1 Eigenvalues and Eigenvectors
557(4)
A.2 Coordinate Changes
561(2)
A.3 Matrix Times Circle Equals Ellipse
563(4)
B COMPUTER SOLUTION OF ODES
567(10)
B.1 ODE Solvers
568(2)
B.2 Error in Numerical Integration
570(4)
B.3 Adaptive Step-Size Methods
574(3)
ANSWERS AND HINTS TO SELECTED EXERCISES 577(10)
BIBLIOGRAPHY 587(8)
INDEX 595


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