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Lyapunov Exponents

Lyapunov exponents characterize the rate of divergence of nearby trajectories in phase space, indicating the system's sensitivity to initial conditions and potential for chaotic behavior.
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The statement of the theorem

Consider a dynamical system defined by dxdt=F(x,t)\frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}, t). The evolution of a small perturbation δx(t)\delta\mathbf{x}(t) is governed by the linearized equation: \nddtδx=J(x,t)δx\frac{d}{dt} \delta\mathbf{x} = \mathbf{J}(\mathbf{x}, t) \delta\mathbf{x} \nwhere J\mathbf{J} is the Jacobian matrix of F\mathbf{F}. The Lyapunov exponents λk\lambda_k are defined by the asymptotic growth rate of the magnitude of the perturbation: \nλk=lim⁡t→∞1tln⁡(∣δx(t)∣∣δx(0)∣)\lambda_k = \lim_{t \to \infty} \frac{1}{t} \ln \left( \frac{|\delta\mathbf{x}(t)|}{|\delta\mathbf{x}(0)|} \right) \nPositive exponents indicate exponential divergence and chaotic behavior.
Source: Wikipedia