1. Introduction to Chaos Clint Sprott
1/4/2018
Introduction to Chaos
Clint Sprott
Department of Physics
University of Wisconsin – Madison USA
Presented at the Utrecht Physics Challenge
in Utrecht, Netherlands
on May 6, 2017
Entire presentation available on WWW
Workshop on Self-Organization

2. Abbreviated History Kepler (1605) Newton (1687) Poincare (1890)
1/4/2018
Abbreviated History
Kepler (1605)
Newton (1687)
Poincare (1890)
Lorenz (1963)
Workshop on Self-Organization

3. Johannes Kepler (1605) Assistant to Tycho Brahe
3 laws of planetary motion
Elliptical orbits
Repeatable
Predictable

4. Isaac Newton (1687) Invented calculus Derived 3 laws of motion F = ma
Proposed law of gravity
F = Gm1m2/r 2
Explained Kepler’s laws
Got headaches (3-body problem)

5. 3-Body Problem

6. Simplified Solar System

7. Henri Poincare (1890) 200 years later! King Oscar (Sweden, 1887)
Prize won – 200 pages
No analytic solution exists!
Sensitive dependence on initial conditions (Lyapunov exponent)
Chaos! (Li & Yorke, 1975)

8. Chaotic Double Pendulum

9. Sensitive Dependence on I. C.

10. Double Pendulum Simulation
1/4/2018
Double Pendulum Simulation
This models a driven pendulum with friction
Looks like those taffy machines at the fair
Show silly putty

11. Equations for Double Pendulum
1/4/2018
Equations for Double Pendulum
This models a driven pendulum with friction
Looks like those taffy machines at the fair
Show silly putty

12. Edward Lorenz (1963) Meteorologist at MIT
Had his own personal computer
Rediscovered chaos in a simple system of equations:
dx/dt = σ(y - x)
dy/dt = -xz + rx - y
dz/dt = xy - bz
3 variables (x, y, z)
2 nonlinearities (xz, xy)
3 parameters (σ, r, b)

13. Lorenz Attractor Strange attractor A fractal object Fractal dim ~ 2.05
Butterfly effect

14. Butterfly Effect

15. Sensitive Dependence on Init Cond
Initial conditions differ by 0.01%

16. Conditions for Chaos
At least 3 variables (to keep the orbits from intersecting)
At least one nonlinearity (to keep the orbits bounded)
A source of energy (to keep the system going)

17. Usual Route to Chaos Stable equilibrium (point attractor) Limit cycle
(periodic attractor)
Period doubling
Strange attractor

18. Period Doubling  Chaos
The simplest
chaotic flow!
Sprott (1997)
dx/dt = y
dy/dt= z
dz/dt= -az + y2 - x

19. Chaotic Circuit This models a driven pendulum with friction
1/4/2018
Chaotic Circuit
This models a driven pendulum with friction
Looks like those taffy machines at the fair
Show silly putty

20. Equations for Chaotic Circuit
1/4/2018
Equations for Chaotic Circuit
dx/dt = y
dy/dt = z
dz/dt = -az - by + c(sgn x - x)
Jerk system
Period doubling route to chaos
This models a driven pendulum with friction
Looks like those taffy machines at the fair
Show silly putty

21. Bifurcation Diagram for Chaotic Circuit
1/4/2018
Bifurcation Diagram for Chaotic Circuit

22. Applications for Chaos
Secure communications
Meteorology
Ecology
Economics
Sociology
Psychology
Politics
Philosophy

23. References
lectures/utrecht.pptx (this talk)
(my chaos textbook)
(contact me)

24. Questions Who won King Oscar’s Prize? Johannes Kepler Isaac Newton
Henri Poincare
Edward Lorenz
What is the 3-body problem?
Biological survival of the fittest
The motion of three bodies with mutual attraction
The social interactions of three friends
The riddle of a multiple homicide
What is a nonlinear system?
A system in which effects have multiple causes
A system whose whole is not equal to the sum of its parts
A system that exhibits chaos
A system with many variables
What is a strange attractor?
An object whose shape is unpredictable
An object that attracts another dissimilar object
An object that attracts another similar object
A fractal produced by a chaotic process
What is the butterfly effect?
Turbulence produced by fluctuating organisms
Irregular oscillations of a dynamical system
Behavior of a complex system
Sensitive dependence on initial conditions