Lyapunov Exponents

Chaos and Time-Series Analysis

10/3/00 Lecture #5 in Physics 505

* Comments on Homework #3 (Van der Pol Equation)

* Review (last week) - Dynamical Systems Theory

* General Properties of Lyapunov Exponents

* Lyapunov Exponent for 1-D Maps

* Lyapunov Exponents for 2-D Maps

* Lyapunov Exponents for 3-D Flows

* Numerical Calculation of Largest Lyapunov Exponent

  1. Start with any initial condition in the basin of attraction
  2. Iterate until the orbit is on the attractor
  3. Select (almost any) nearby point (separated by d0)
  4. Advance both orbits one iteration and calculate new separation d1
  5. Evaluate log |d1/d0| in any convenient base
  6. Readjust one orbit so its separation is d0 in same direction as d1
  7. Repeat steps 4-6 many times and calculate average of step 5
  8. The largest Lyapunov exponent is l1 = <log |d1/d0|>
  9. If map approximates an ODE, then l1 = <log |d1/d0|> / h
  10. A positive value of l1 indicates chaos

* General character of exponents in 3-D flows:

* Kaplan-Yorke (Lyapunov) Dimension

* Precautions

J. C. Sprott | Physics 505 Home Page | Previous Lecture | Next Lecture