Chaos and Dynamical Systems presents an accessible, clear introduction to dynamical systems and chaos theory, an important and exciting area that has shaped many scientific fields. While the rules governing dynamical systems are well-specified and simple, the behavior of many dynamical systems is remarkably complex. Of particular note, simple deterministic dynamical systems produce output that appears random and for which long-term prediction is impossible. Using little math beyond basic algebra, David Feldman gives readers a grounded, concrete, and concise overview.In initial chapters, Feldman introduces iterated functions and differential equations. He then surveys the key concepts and results to emerge from dynamical systems: chaos and the butterfly effect, deterministic randomness, bifurcations, universality, phase space, and strange attractors. Throughout, Feldman examines possible scientific implications of these phenomena for the study of complex systems, highlighting the relationships between simplicity and complexity, order and disorder.Filling the gap between popular accounts of dynamical systems and chaos and textbooks aimed at physicists and mathematicians, Chaos and Dynamical Systems will be highly useful not only to students at the undergraduate and advanced levels, but also to researchers in the natural, social, and biological sciences.

• Cover
• Contents
• Preface
• 1. Introducing Iterated Functions
  • 1.1 Iterated Functions
  • 1.2 Thinking Globally
  • 1.3 Stability: Attractors and Repellors
  • 1.4 Another Example
  • 1.5 One More Example
  • 1.6 Determinism
  • 1.7 Summary
• 2. Introducing Differential Equations
  • 2.1 Newton’s Law of Cooling
  • 2.2 Exact Solutions
  • 2.3 Calculus Puzzles
  • 2.4 Qualitative Solutions
  • 2.5 Numerical Solutions
  • 2.6 Putting It All Together
  • 2.7 More about Numerical Solutions
  • 2.8 Notes on Terminology and Notation
  • 2.9 Existence and Uniqueness of Solutions
  • 2.10 Determinism and Differential Equations
  • 2.11 Iterated Functions vs. Differential Equations
  • 2.12 Further Reading
• 3. Interlude: Mathematical Models and the Newtonian Worldview
  • 3.1 Why Isn’t This the End of the Book?
  • 3.2 Newton’s Mechanistic World
  • 3.3 Laplacian Determinism and the Aspirations of Science
  • 3.4 Styles of Mathematical Models
  • 3.5 Levels of Models
  • 3.6 Pluralistic View of Mathematical Models
  • 3.7 Further Reading
• 4. Chaos I: The Butterfly Effect
  • 4.1 The Logistic Equation
  • 4.2 Periodic Behavior
  • 4.3 Aperiodic Behavior
  • 4.4 The Butterfly Effect
  • 4.5 The Butterfly Effect Defined
  • 4.6 Chaos Defined
  • 4.7 Lyapunov Exponents
• 5. Chaos II: Deterministic Randomness
  • 5.1 Symbolic Dynamics
  • 5.2 As Random as a Coin Toss
  • 5.3 Deterministic Sources of Randomness
  • 5.4 Implications of the Butterfly Effect
  • 5.5 Further Reading
• 6. Bifurcations: Sudden Transitions
  • 6.1 Logistic Differential Equation
  • 6.2 Logistic Equation with Harvest
  • 6.3 Bifurcations and Bifurcation Diagrams
  • 6.4 General Remarks on Bifurcations
  • 6.5 Catastrophes and Tipping Points
  • 6.6 Hysteresis
  • 6.7 Further Reading
• 7. Universality in Chaos
  • 7.1 Logistic Equation Bifurcation Diagram
  • 7.2 Exploring the Bifurcation Diagram
  • 7.3 Some Words about Emergence
  • 7.4 The Period-Doubling Route to Chaos
  • 7.5 Universality in Maps
  • 7.6 Universality in Physics
  • 7.7 Renormalization
  • 7.8 Phase Transitions, Critical Phenomena, and Power Laws
  • 7.9 Conclusion: Lessons and Limits to Universality
  • 7.10 Further Reading
• 8. Higher-Dimensional Systems and Phase Space
  • 8.1 A Quick Review of One-Dimensional Differential Equations
  • 8.2 Lotka–Volterra Differential Equations
  • 8.3 The Phase Plane
  • 8.4 Phase Planes in General
  • 8.5 The Rössler Equations and Phase Space
  • 8.6 Further Reading
• 9. Strange Attractors
  • 9.1 Chaos in Three Dimensions
  • 9.2 The Rössler Attractor
  • 9.3 Strange Attractors
  • 9.4 Back to 1D: The Lorenz Map
  • 9.5 Stretching and Folding
  • 9.6 Poincaré Maps
  • 9.7 Delay Coordinates and Phase Space Reconstruction
  • 9.8 Determinism vs. Noise
  • 9.9 Further Reading
• 10. Conclusion
  • 10.1 Summary
  • 10.2 Complex Systems
  • 10.3 Emergence(?)
  • 10.4 But Not Everything Is Simple
  • 10.5 Further Reading
  • 10.6 Farewell
• Bibliography
• Index