Lyapunov Exponent Calculator

Initial Condition (x₀):

Parameter (r):

Number of Iterations:

Lyapunov Exponent: -

System Behavior: -

Understanding Lyapunov Exponents

What are Lyapunov Exponents?

Lyapunov exponents quantify the rate at which nearby trajectories in phase space diverge (or converge), providing a measure of a system's sensitivity to initial conditions.

Key Formulas

λ = lim(n→∞) (1/n)∑ln|f'(xᵢ)|

For logistic map: f(x) = rx(1-x)

where:

λ is the Lyapunov exponent
n is the number of iterations
f'(x) is the derivative of the map
xᵢ are the orbit points

Properties and Interpretations

Positive Exponent (λ > 0)

Chaotic behavior

Exponential divergence

Sensitive dependence

Unpredictable long-term

Zero Exponent (λ = 0)

Marginally stable

Periodic behavior

Conservative systems

Critical transitions

Negative Exponent (λ < 0)

Stable behavior

Attracting fixed points

Predictable dynamics

Dissipative systems

Advanced Topics

Spectrum Analysis

Multiple exponents

Kaplan-Yorke dimension

Entropy connection

Fractal properties

Applications

Weather prediction

Financial markets

Population dynamics

Neural networks

Related Concepts

Bifurcation theory

Strange attractors

Ergodic theory

Information theory