Introduction to Modern Dynamics Chaos Networks Space and Time 2nd Edition by David Nolte 0192583166 9780192583161

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ISBN-10 :  0192583166 

ISBN-13 :  9780192583161

Author : David D. Nolte 

The best parts of physics are the last topics that our students ever see. These are the exciting new frontiers of nonlinear and complex systems that are at the forefront of university research and are the basis of many high-tech businesses. Topics such as traffic on the World Wide Web, the spread of epidemics through globally-mobile populations, or how the synchronization of global economies are governed by universal principles just as profound as Newton’s laws. Nonetheless, the conventional university physics curriculum reserves most of these topics for graduate study because of the assumed need for advanced mathematics. However, by using only linear algebra and calculus, combined with exploratory computer simulations, all of these topics become accessible to advanced undergraduate students. The structure of this book combines the three main topics of modern dynamics – chaos theory, dynamics on complex networks, and general relativity – into a coherent framework. By taking a geometric view of physics, concentrating on the time evolution of physical systems as trajectories through abstract spaces, these topics share a common and simple mathematical language through which any student can gain a unified physical intuition. Given the growing importance of complex dynamical systems in many areas of science and technology, this text provides students with an up-to-date foundation for their future careers. This second edition has an updated introductory chapter and has added key topics to help students prepare for their GRE physics subject exam. It also has expanded chapters on Hamiltonian dynamics, Hamiltonian chaos, and Econophysics, while increasing the number of homework problems at the end of each chapter. The second edition is designed to fulfill the textbook needs of any advanced undergraduate course in mechanics.

Introduction to Modern Dynamics: Chaos, Networks, Space, and Time 2nd Table of contents:

Part I: Geometric Mechanics
1 Physics and Geometry
1.1 State space and dynamical flows
1.1.1 State space
1.1.2 Dynamical flows
1.1.2 Dynamical flows
1.2 Coordinate representation of dynamical systems
1.2.1 Coordinate notation and configuration space
1.2.2 Trajectories in 3D configuration space
1.2.3 Generalized coordinates
1.3 Coordinate transformations
1.3.1 Jacobian matrix
1.3.2 Metric spaces and basis vectors
1.3.3 Metric tensor
1.3.4 Two-dimensional rotations
1.3.5 Three-dimensional rotations of coordinate frames
1.4 Uniformly rotating frames
1.4.1 Motion relative to the Earth
1.4.2 Foucault’s pendulum
1.5 Rigid-body motion
1.5.1 Inertia tensor
1.5.2 Parallel axis theorem
1.5.3 Angular momentum
1.5.4 Euler’s equations
1.5.5 Force-free top
1.5.6 Top with fixed tip (precession with no wobble)
1.6 Summary
1.7 Bibliography
1.8 Homework problems
2 Lagrangian Mechanics
2.1 Calculus of variations
2.1.1 Variational principle
2.1.2 Stationary action
2.2 Lagrangian applications
2.2.1 Mass on a spring
2.2.2 Simple pendulum
2.2.3 Symmetric top with fixed tip
2.3 Dissipation in Lagrangian systems
2.4 Lagrange undetermined multipliers
2.5 Examples of Lagrangian applications with constraints
2.5.1 Massive pulley
2.5.2 Atwood machine with massive pulley
2.5.3 Cylinder rolling down an inclined plane
2.6 Conservation laws
2.6.1 Conservation of energy
2.6.2 Ignorable (cyclic) coordinates
2.7 Central force motion
2.7.1 Reduced mass for the two-body problem
2.7.2 Effective potential energy
2.7.3 Kepler’s laws
2.8 Virial theorem
2.9 Summary
2.10 Bibliography
2.11 Homework problems
3 Hamiltonian Dynamics and Phase Space
3.1 The Hamiltonian function
3.1.1 Legendre transformations and Hamilton’s equations
3.1.2 Canonical transformations
3.2 Phase space
3.2.1 Liouville’s theorem and conservation of phase space volume
3.2.2 Poisson brackets
3.3 Integrable systems and action-angle variables
3.4 Adiabatic invariants
3.5 Summary
3.6 Bibliography
3.7 Homework problems
Part II: Nonlinear Dynamics
4 Nonlinear Dynamics and Chaos
4.1 One-variable dynamical systems
4.2 Two-variable dynamical systems
4.2.1 Two-dimensional fixed points
4.2.2 Phase portraits
4.2.3 Types of 2D fixed points
4.2.4 Separatrices: stable and unstable manifolds
4.3 Limit cycles
4.3.1 Van der Pol oscillator
4.3.2 Poincaré sections (first-return map)
4.3.3 Homoclinic orbits
4.4 Discrete iterative maps
4.4.1 One-dimensional logistic map
4.4.2 Feigenbaum number and universality
4.5 Three-dimensional state space and chaos
4.5.1 Stability and fixed-point classification
4.5.1.1 3D fixed-point classifications
4.5.2 Limit cycles and Poincaré sections
4.5.3 Autonomous dynamical models in 3D
4.5.4 Evolving volumes in state space
4.6 Non-autonomous (driven) flows
4.7 Summary and glossary
4.8 Bibliography
4.9 Homework problems
Analytic problems
Computational projects
5 Hamiltonian Chaos
5.1 Perturbed Hamiltonian systems and separatrix chaos
5.1.1 Perturbed pendulum and double well
5.1.2 Perturbed action-angle dynamics
5.2 Nonintegrable Hamiltonian systems
5.2.1 The Hénon–Heiles Hamiltonian
5.2.2 Liouville’s theorem and area-preserving maps
5.3 The Chirikov Standard Map
5.3.1 Invariant tori and the Poincaré–Birkhoff theorem
5.3.2 The Standard Map: kicked rotator
5.4 KAM theory
5.5 Degeneracy and the web map
5.6 Quantum chaos [optional]5.7 Summary
5.8 Bibliography
5.9 Homework problems
Analytic problems
Computational projects
6 Coupled Oscillators and Synchronization
6.1 Coupled linear oscillators
6.2 Simple models of synchronization
6.2.1 Integrate-and-fire oscillators
6.2.1.1 Phase locking
6.2.1.2 Frequency locking
6.2.2 Frequency-locked phase oscillators
6.3 Rational resonances
6.3.1 The sine-circle map
6.3.2 Resonances
6.4 External synchronization
6.4.1 External synchronization of an autonomous phase oscillator
6.4.2 External synchronization of a van der Pol oscillator
6.4.3 Mutual synchronization of two van der Pol oscillators
6.5 Synchronization of chaos
6.6 Summary
6.7 Bibliography
6.8 Homework problems
Analytic problems
Numerical projects
Part III: Complex Systems
7 Network Dynamics
7.1 Network structures
7.1.1 Types of graphs
7.1.2 Statistical properties of networks
7.1.2.1 Degree and moments
7.1.2.2 Adjacency matrix
7.1.2.3 Graph Laplacian
7.1.2.4 Distance matrix
7.2 Random network topologies
7.2.1 Erdös–Rényi graphs
7.2.2 Small-world networks
7.2.3 Scale-free networks
7.3 Synchronization on networks
7.3.1 Kuramoto model of coupled phase oscillators on a complete graph
7.3.2 Synchronization and topology
7.3.3 Synchronization of chaotic oscillators on networks
7.4 Diffusion on networks
7.4.1 Percolation
7.4.2 Diffusive flow equations
7.4.3 Discrete map for diffusion on networks
7.5 Epidemics on networks
7.5.1 Epidemic models: SI/SIS/SIR/SIRS
7.5.2 SI model: logistic growth
7.5.3 SIS/SIR/SIRS
7.5.4 Discrete infection on networks
7.6 Summary
7.7 Bibliography
7.8 Homework Problems
Analytic problems
Numerical projects
8 Evolutionary Dynamics
8.1 Population dynamics
8.1.1 Species competition and symbiosis
8.1.2 Predator–prey models
8.1.3 Stability of multispecies ecosystems
8.2 Viral infection and acquired resistance
8.2.1 Viral infection
8.2.2 Cancer chemotherapy and acquired resistance
8.3 Replicator dynamics
8.3.1 The replicator equation
8.3.2 Hypercycles
8.4 Quasispecies
8.4.1 The quasispecies equation
8.4.2 Hamming distance and binary genomes
8.4.3 The replicator–mutator equation
8.5 Game theory and evolutionary stable solutions
8.5.1 Evolutionary game theory: hawk and dove
8.6 Summary
8.7 Bibliography
8.8 Homework problems
Analytic problems
Numerical projects
9 Neurodynamics and Neural Networks
9.1 Neuron structure and function
9.2 Neuron dynamics
9.2.1 The Fitzhugh–Nagumo model
9.2.2 The NaK model
9.3 Network nodes: artificial neurons
9.4 Neural network architectures
9.4.1 Single-layer feedforward (perceptron)
9.4.2 Multilayer feedforward
9.5 Hopfield neural network
9.6 Content-addressable (associative) memory
9.6.1 Storage phase
9.6.2 Retrieval phase
9.6.3 Example of a Hopfield pattern recognizer
9.7 Summary
9.8 Bibliography
9.9 Homework problems
Analytic problems
Numerical projects
10 Economic Dynamics
10.1 Microeconomics and equilibrium
10.1.1 Barter, utility, and price
10.1.2 Supply and demand
10.1.3 Cournot duopoly
10.1.4 Walras’ Law
10.2 Macroeconomics
10.2.1 IS-LM models
10.2.2 Inflation and unemployment
10.3 Business cycles
10.3.1 Delayed adjustments and the cobweb model
10.3.2 Lotka–Volterra-type investment-savings model
10.3.3 Business cycles as limit cycles
10.3.4 Chaotic business cycles
10.4 Random walks and stock prices [optional]10.4.1 Random walk of a stock price: efficient market model
10.4.2 Hedging and the Black–Scholes equation
10.4.3 Econophysics
10.5 Summary
10.6 Bibliography
10.7 Homework problems
Analytic problems
Numerical projects
Part IV: Relativity and Space-Time
11 Metric Spaces and Geodesic Motion
11.1 Manifolds
11.2 Derivative of a tensor
11.2.1 Derivative of a vector
11.2.2 Christoffel symbols
11.2.3 Derivative notations
11.2.4 The connection of Christoffel symbols to the metric
11.3 Geodesic curves in configuration space
11.3.1 Variational approach to the geodesic curve
11.3.2 Parallel transport
11.3.3 Examples of geodesic curves
11.3.4 Optics, ray equation, and light “orbits”
11.4 Geodesic motion
11.4.1 Force-free motion
11.4.2 Geodesic motion through potential energy landscapes
11.5 Summary
11.6 Bibliography
11.7 Homework problems
12 Relativistic Dynamics
12.1 The special theory
12.2 Lorentz transformations
12.2.1 Standard configuration (SC)
12.2.2 Minkowski space
12.2.3 Kinematic consequences of the Lorentz transformation
12.2.3.1 Time dilation
12.2.3.2 Length contraction
12.2.3.3 Velocity addition
12.2.3.4 Lorentz boost
12.2.3.5 Doppler effect
12.3 Metric structure of Minkowski space
12.3.1 Invariant interval
12.3.2 Proper time and dynamic invariants
12.4 Relativistic trajectories
12.4.1 Null geodesics
12.4.2 Trajectories as world lines
12.5 Relativistic dynamics
12.5.1 Relativistic energies
12.5.2 Momentum transformation
12.5.3 Invariant mass
12.5.4 Force transformation
12.6 Linearly accelerating frames (relativistic)
12.6.1 Equivalence Principle
12.6.2 Gravitational time dilation
12.7 Summary
12.8 Bibliography
12.9 Homework problems
13 The General Theory of Relativity and Gravitation
13.1 The Newtonian correspondence
13.2 Riemann curvature tensor
13.3 Einstein’s field equations
13.3.1 Einstein tensor
13.3.2 Newtonian limit
13.4 Schwarzschild space-time
13.5 Kinematic consequences of gravity
13.5.1 Time dilation
13.5.2 Length contraction
13.5.3 Redshifts
13.6 The deflection of light by gravity
13.6.1 Refractive index of general relativity
13.6.2 Gravitational lensing
13.7 Planetary orbits
13.8 Black holes
13.8.1 Orbits
13.8.2 Falling into a black hole
13.9 Gravitational waves
13.10 Summary
13.11 Bibliography
13.12 Homework problems

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