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

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

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

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

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

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

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

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

Synthesis

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