Consider the general phenomena of life and consciousness itself. One definition could be related on its ability to sustain itself, a system of negative feedback that buffers homeostasis. I call that the ROBUSTNESS definition.
Another definition would focus on the FRAGILITY of life, on its sensitivity to initial conditions, on the presence of a 'sweet spot' that, if pushed at the right time, can make a large difference. Life is dynamic.
Lyapunov exponent measures whether the butterfly effect is possible: if a butterfly flaps its wings in China, could that cause it to rain in Arizona? If the equations that govern weather have a Lyapunov exponent greater that the absolute value of one, then two very similar initial conditions (the butterfly flaps, the butterfly doesn't flap) can have arbitrarily divergent outcomes (it rains or its sunny and clear). This is what the Mandlebrot set measures. If you make a splash in the ocean, odds are that it won't cause a tidal wave. But if you're mean to a Postal Worker on the wrong day of the month it could have disastrous consequences.
Entropy always increases on the whole, but for life order is maintained at the expense of the periphery. This has nothing to do with Lyapunov exponent because there is not necessarily more order in a system with negative feedback where small differences are ironed out than there is in system with positive feedback which is extremely sensitive to initial conditions. Order or disorder matters more in the second. Entropy is a property of a system, whereas Lyapunov exponent is a property of the equations of change. Entropy is one constraint in the equations of change, a constraint that must always be satisfied by increasing. The question of whether life is ordered or entropic can perhaps best be dealt with by noting that entropy is the inverse of information. ...