(

**a**) LEs,

*λ*_{0}, ...

*λ*_{n}, characterize the contraction/expansion of an initially small perturbation,

*ɛ*_{0}, to the system. (

**b**) The leading LE determines the principal dynamics and characteristics of the attractor of a dynamical system. For

*λ*_{0}<0, the attractor will be a stable fixed point; stable oscillating solutions will be obtained, if

*λ*_{0}=0; for

*λ*_{0}>0 we observe chaos and the system will exhibit a so-called strange attractor; if more than one LE is positive, then we speak of hyperchaos and the attractor will exhibit behaviour with similar statistical properties to white noise. (

**c**) Key steps in the UKF for qualitative inference. At the

*k*^{th} iteration, the current prior parameter distribution is formed by perturbing the previous posterior,

*θ*_{k}, with the process noise

*v*_{k}. The distribution of LEs for the model

*f* induced by the prior parameter distribution is calculated via the LE estimation routine L and the unscented transform. Comparing the mean LE,

, to the target LE,

*λ*_{target}, the prior parameters are updated using the UKF update equations. As the filter proceeds, parameters are found that locally minimise the sum of squared error between target and estimated LEs.

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