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Orbital Divergence Characterizes Expansive Dynamics on Biological Attractors

In the dynamical world of equilibria (fixed points in phase space) and periodic cycles (fixed points of a return map), a common concern involves their stability. What happens if an adventitious jiggle moves the orbit a little distance away from the fixed point? Would the wind wiggled suspension bridge start to flap with increasing amplitude or would it damp back down quickly to its stable state. A “Lyapounov functional,” L, is constructed which can be visualized like a smooth potential bowl around the fixed point such that any L stable solution that starts at its bottom tends to stay there or is asymptotically L stable if the solution converges to the fixed point at the bowl’s bottom as t → ∞. If the point is not L stable, it is L unstable.

The modern study of nonlinear systems have produced another kind of stability issue with a similar appellation yielding other direction specific indices, the Lyapounov characteristic exponents, λ (Oseledec, 1968; Eckmann and Ruelle, 1985; Ruelle, 1990; Ott et al, 1994). In this context, the instability is not one of perturbative escape from a fixed point, but of the average rate with which the (theoretically infinitesimal) distances among a handful of points representing a set of initial conditions (each a precision limited, hypothetical repetition of the same experiment), are being stretched apart by the expansive action of a strange attractor system. In three dimensions, one can envision a ball of initial conditions being elongated along the unstable direction and ironed down from both sides along the stable direction over time, transforming the ball into an ellipsoid and then into a (recurrent) curve. In simplest terms and thinking about a one dimensional scalar time series, the Lyapounov exponent reflects the multiplicative average (logarithmic addition) of the sequence of slopes of the series of straight lines connecting the points. An average slope of > 45° is expansive such that a linear distance on the x-axis is increased when mapped onto the y axis. A slope of < 45° is a contraction mapping reducing the linear distance of the x-axis when mapped to the y axis.

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