E-grāmata: Chaos: An Introduction to Dynamical Systems

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 Poincare-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 Rossler 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.