Lyapunov exponent

A rate measuring the asymptotic exponential growth or decay of separation between nearby trajectories in a dynamical system. Its sign indicates local divergence, convergence, or neutral behavior.

Lyapunov characteristic exponent: A Lyapunov exponent computed along a particular trajectory, often over a finite time interval. Finite-time values reveal how local stretching varies before the asymptotic rate is reached.
Lyapunov time: A characteristic time over which exponential error growth substantially degrades a trajectory prediction. The largest positive exponent sets the approximate timescale for loss of predictability.
Dynamical system: A mathematical model describing how a state changes over time according to a rule. Lyapunov exponents characterize the evolution of nearby states in such models.
Eigenvalue: A scalar by which a linear transformation scales one of its eigenvectors. Eigenvalues describe fixed linear transformations, while exponents describe asymptotic growth along evolving trajectories.
Tangent space: A vector space of infinitesimal directions around a point on a differentiable manifold. Perturbations in this space are evolved to measure trajectory separation.
Lorenz system: A three-dimensional system of differential equations with chaotic trajectories and a strange attractor. Its positive largest exponent quantifies sensitive dependence in a classic chaotic model.
Phase space: A space whose points represent the possible states of a dynamical system. Trajectory separation is measured between nearby points in this space.
Spectral radius: The largest absolute value of the eigenvalues of a square matrix. It captures growth for repeated application of one matrix, not generally a changing sequence.
Jacobian matrix: A matrix of first derivatives describing the local linear approximation of a differentiable map. Its action propagates small perturbations in discrete-time systems.
Weather forecasting: The prediction of atmospheric conditions using observations and numerical models. Positive exponents help explain why small initial errors limit forecast horizons.

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