##  [Lyapunov Exponent](/lyapunov-exponent-0) 

 Definition

A real number quantifying the average exponential rate at which nearby trajectories of a dynamical system separate (positive) or converge (negative) along a given direction.

 

 

 

 

 

 





## Principle

Principle

Linearize the flow along a trajectory and measure long-time growth rates of perturbations via the logarithmic growth of tangent vectors divided by time.

 

 

 

 

 





## Demonstration

Demonstration

For the logistic map at parameter values exhibiting chaos, numerically computed largest exponent is positive, indicating sensitive dependence on initial conditions over long iterations.

 

 

 

 

## Misapplication

Misapplication

Estimating a global exponent from short finite-time data without accounting for nonstationarity or failing to distinguish the largest exponent from transient expansion rates.

 

 

 

 

 





## Consequence

Consequence

A positive maximal exponent implies exponential sensitivity and typical unpredictability for nearby initial conditions; negative exponents indicate local asymptotic attraction.

 

 

 

 

## Reversal

Reversal

If all Lyapunov exponents are strictly negative the trajectory is uniformly attracting; changing sign of a leading exponent can mark a route to instability or chaos.

 

 

 

 

 





## Boundary

Boundary

Defined for trajectories of continuous or discrete dynamical systems where linearization and Oseledec-type limits exist; may be undefined or path-dependent for non-smooth or non-autonomous systems without ergodic hypotheses.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often compared with global complexity measures: an exponent quantifies local exponential rates along directions, not the total information growth of the system.

 

 

 

 

 





## Synthesis

Synthesis

A metric of linearized long-time growth or decay of infinitesimal perturbations along a trajectory that diagnoses stability and routes to chaotic behavior.