Chaos: An Introduction to Dynamical Systems (Textbooks in Mathematical Sciences)

Kathleen T. Alligood

ISBN 10: 354078036X / ISBN 13: 9783540780366

Editore: Springer, 1997

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Dati bibliografici

• Titolo: Chaos: An Introduction to Dynamical Systems (Textbooks in Mathematical Sciences)
• Editore: Springer
• Data di pubblicazione: 1997
• Lingua: Inglese
• ISBN 10: 354078036X
• ISBN 13: 9783540780366
• Rilegatura: paperback
• Condizione: Like New
• Tipo di libro: book

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Riassunto

BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ- ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time.

Contenuti

1 One-Dimensional Maps.- 1.1 One-Dimensional Maps.- 1.2 Cobweb Plot: Graphical Representation of an Orbit.- 1.3 Stability of Fixed Points.- 1.4 Periodic Points.- 1.5 The Family of Logistic Maps.- 1.6 The Logistic Map G(x) = 4x(l ? x).- 1.7 Sensitive Dependence on Initial Conditions.- 1.8 Itineraries.- Challenge l: Period Three Implies Chaos.- Exercises.- Lab Visit 1: Boom, Bust, and Chaos in the Beetle Census.- 2 Two-Dimensional Maps.- 2.1 Mathematical Models.- 2.2 Sinks, Sources, and Saddles.- 2.3 Linear Maps.- 2.4 Coordinate Changes.- 2.5 Nonlinear Maps and the Jacobian Matrix.- 2.6 Stable and Unstable Manifolds.- 2.7 Matrix Times Circle Equals Ellipse.- Challenge 2: Counting the Periodic Orbits of Linear Maps on a Torus.- Exercises.- Lab Visit 2: Is the Solar System Stable?.- 3 Chaos.- 3.1 Lyapunov Exponents.- 3.2 Chaotic Orbits.- 3.3 Conjugacy and the Logistic Map.- 3.4 Transition Graphs and Fixed Points.- 3.5 Basins of Attraction.- Challenge 3: Sharkovskii’s Theorem.- Exercises.- Lab Visit 3: Periodicity and Chaos in a Chemical Reaction.- 4 Fractals.- 4.1 Cantor Sets.- 4.2 Probabilistic Constructions of Fractals.- 4.3 Fractals from Deterministic Systems.- 4.4 Fractal Basin Boundaries.- 4.5 Fractal Dimension.- 4.6 Computing the Box-Counting Dimension.- 4.7 Correlation Dimension.- Challenge 4: Fractal Basin Boundaries and the Uncertainty Exponent.- Exercises.- Lab Visit 4: Fractal Dimension in Experiments.- 5 Chaos in Two-Dimensional Maps.- 5.1 Lyapunov Exponents.- 5.2 Numerical Calculation of Lyapunov Exponents.- 5.3 Lyapunov Dimension.- 5.4 A Two-Dimensional Fixed-Point Theorem.- 5.5 Markov Partitions.- 5.6 The Horseshoe Map.- Challenge 5: Computer Calculations and Shadowing.- Exercises.- Lab Visit 5: Chaos in Simple Mechanical Devices.- 6 Chaotic Attractors.- 6.1 Forward Limit Sets.- 6.2 Chaotic Attractors.- 6.3 Chaotic Attractors of Expanding Interval Maps.- 6.4 Measure.- 6.5 Natural Measure.- 6.6 Invariant Measure for One-Dimensional Maps.- Challenge 6: Invariant Measure for the Logistic Map.- Exercises.- Lab Visit 6: Fractal Scum.- 7 Differential Equations.- 7.1 One-Dimensional Linear Differential Equations.- 7.2 One-Dimensional Nonlinear Differential Equations.- 7.3 Linear Differential Equations in More than One Dimension.- 7.4 Nonlinear Systems.- 7.5 Motion in a Potential Field.- 7.6 Lyapunov Functions.- 7.7 Lotka-Volterra Models.- Challenge 7: A Limit Cycle in the van der Pol System.- Exercises.- Lab Visit 7: Fly vs. Fly.- 8 Periodic Orbits and Limit Sets.- 8.1 Limit Sets for Planar Differential Equations.- 8.2 Properties of ?-Limit Sets.- 8.3 Proof of the Poincaré-Bendixson Theorem.- Challenge 8: Two Incommensurate Frequencies Form a Torus.- Exercises.- Lab Visit 8: Steady States and Periodicity in a Squid Neuron.- 9 Chaos in Differential Equations.- 9.1 The Lorenz Attractor.- 9.2 Stability in the Large, Instability in the Small.- 9.3 The Rössler Attractor.- 9.4 Chua’s Circuit.- 9.5 Forced Oscillators.- 9.6 Lyapunov Exponents in Flows.- Challenge 9: Synchronization of Chaotic Orbits.- Exercises.- Lab Visit 9: Lasers in Synchronization.- 10 Stable Manifolds and Crises.- 10.1 The Stable Manifold Theorem.- 10.2 Homoclinic and Heteroclinic Points.- 10.3 Crises.- 10.4 Proof of the Stable Manifold Theorem.- 10.5 Stable and Unstable Manifolds for Higher Dimensional Maps.- Challenge 10: The Lakes of Wada.- Exercises.- Lab Visit 10: The Leaky Faucet: Minor Irritation or Crisis?.- 11 Bifurcations.- 11.1 Saddle-Node and Period-Doubling Bifurcations.- 11.2 Bifurcation Diagrams.- 11.3 Continuability.- 11.4 Bifurcations of One-Dimensional Maps.- 11.5 Bifurcations in Plane Maps: Area-Contracting Case.- 11.6 Bifurcations in Plane Maps: Area-Preserving Case.- 11.7 Bifurcations in Differential Equations.- 11.8 Hopf Bifurcations.- Challenge 11: Hamiltonian Systems and the Lyapunov Center Theorem.- Exercises.- Lab Visit 11: Iron + Sulfuric Acid ? Hopf Bifurcation.- 12 Cascades.- 12.1 Cascades and 4.669201609.- 12.2 Schematic Bifurcation Diagrams.- 12.3 Generic Bifurcations.- 12.4 The Cascade Theorem.- Challenge 12: Universality in Bifurcation Diagrams.- Exercises.- Lab Visit 12: Experimental Cascades.- 13 State Reconstruction from Data.- 13.1 Delay Plots from Time Series.- 13.2 Delay Coordinates.- 13.3 Embedology.- Challenge 13: Box-Counting Dimension and Intersection.- A Matrix Algebra.- A.1 Eigenvalues and Eigenvectors.- A.2 Coordinate Changes.- A.3 Matrix Times Circle Equals Ellipse.- B Computer Solution of Odes.- B.1 ODE Solvers.- B.2 Error in Numerical Integration.- B.3 Adaptive Step-Size Methods.- Answers and Hints to Selected Exercises.