What is the role of Lyapunov Exponent in analyzing a dynamic system in the following article(PDF file)?

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Masoud Abbaszadeh

It's a measure of sensitivity of the trajectory to the initial condition perturbations. If Lyapunov exponent is negative the two trajectories resulted from two different initial conditions, exponentially merge in time. It can also be seen as a measure of stiffness of the underlying ODE.