Chaos, Networks, Space and Time

By David D. Nolte

Part I Geometric Mechanics

Chapter 1.  Physics and Geometry

            1.1 State space and dynamical flows

            1.2 Coordinate representation of dynamical systems

            1.3 Coordinate transformation

            1.4 Uniformly rotating frames

            1.5 Rigid-body motion

            1.6 Summary

            1.7 Bibliography

            1.8 Homework problems

Chapter 2.  Lagrangian Mechanics

            2.1 Calculus of variations

            2.2 Lagrangian applications

            2.3 Dissipation in Lagrangian systems

            2.4 Lagrange undetermined multipliers

            2.5 Examples of Lagrangian applications with constraints

            2.6 Conservation laws

            2.7 Central force motion

            2.8 Virial theorem

            2.9 Summary

            2.10 Bibliography

            2.11 Homework problems

Chapter 3.  Hamiltonian Dynamics and Phase Space

            3.1 The Hamiltonian function

            3.2 Phase space

            3.3 Integrable systems and action-angle variables

            3.4 Adiabatic invariants 1

            3.5 Summary

            3.6 Bibliography

            3.7 Homework problems

Part II Nonlinear Dynamics

Chapter 4.  Nonlinear Dynamics and Chaos

            4.1 One-variable dynamical systems

            4.2 Two-variable dynamical systems

            4.3 Limit cycles

            4.4 Discrete iterative maps

            4.5 Three-dimensional state space and chaos

            4.6 Non-autonomous (driven) flows

            4.7 Summary and glossary

            4.8 Bibliography

            4.9 Homework problems

Chapter 5.  Hamiltonian Chaos

            5.1 Perturbed Hamiltonian systems and separatrix chaos

            5.2 Nonintegrable Hamiltonian systems

            5.3 The Chirikov Standard Map

            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

Chapter 6.  Coupled Oscillators and Synchronization

            6.1 Coupled linear oscillators

            6.2 Simple models of synchronization

            6.3 Rational resonances

            6.4 External synchronization

            6.5 Synchronization of chaos

            6.6 Summary

            6.7 Bibliography

            6.8 Homework problems

Part III Complex Systems

Chapter 7. Network Dynamics

            7.1 Network structures

            7.2 Random network topologies

            7.3 Synchronization on networks

            7.4 Diffusion on networks

            7.5 Epidemics on networks

            7.6 Summary

            7.7 Bibliography

            7.8 Homework problems

Chapter 8.  Evolutionary Dynamics

            8.1 Population dynamics

            8.2 Viral infection and acquired resistance

            8.3 Replicator dynamics

            8.4 Quasispecies

            8.5 Game theory and evolutionary stable solutions

            8.6 Summary

            8.7 Bibliography

            8.8 Homework problems

Chapter 9.  Neurodynamics and Neural Networks

            9.1 Neuron structure and function

            9.2 Neuron dynamics

            9.3 Network nodes: artificial neurons

            9.4 Neural network architectures

            9.5 Hopfield neural network

            9.6 Content-addressable (associative) memory

            9.7 Summary

            9.8 Bibliography

            9.9 Homework problems

Chapter 10.  Economic Dynamics

            10.1 Microeconomics and equilibrium

            10.2 Macroeconomics

            10.3 Business cycles

            10.4 Random walks and stock prices [optional]

            10.5 Summary

            10.6 Bibliography

            10.7 Homework problems

Part IV Relativity and Space-Time

Chapter 11.  Metric Spaces and Geodesic Motion

            11.1 Manifolds

            11.2 Derivative of a tensor

            11.3 Geodesic curves in configuration space

            11.4 Geodesic motion

            11.5 Summary

            11.6 Bibliography

            11.7 Homework problems

Chapter 12.  Relativistic Dynamics

            12.1 The special theory

            12.2 Lorentz transformations

            12.3 Metric structure of Minkowski space

            12.4 Relativistic trajectories

            12.5 Relativistic dynamics

            12.6 Linearly accelerating frames (relativistic)

            12.7 Summary

            12.8 Bibliography

            12.9 Homework problems

Chapter 13.  The General Theory of Relativity and Gravitation

            13.1 The Newtonian correspondence

            13.2 Riemann curvature tensor

            13.3 Einstein’s field equations

            13.4 Schwarzschild space-time

            13.5 Kinematic consequences of gravity

            13.6 The deflection of light by gravity

            13.7 Planetary orbits

            13.8 Black holes

            13.9 Gravitational waves

            13.10 Summary

            13.11 Bibliography

            13.12 Homework problems

Appendix

A.1 Index notation: rows, columns, and matrices

A.2 The complex plane

A.3 Solution of linear and linearized ODEs

A.4 Runge–Kutta numerical solvers for ODEs

A.5 Tangents and normals to a curve in the plane

A.6 Elliptic integrals

A.7 MATLAB and Python programs for numerical homework