An Optimization Approach for Finding a Spectrum of Lyapunov Exponents - Computational Neuroscience - page 287

on Lyapunov exponents , as this metric has many important characteristics such as
invariance to transformations and computability directly from data, without solv-
ing the differential or difference equations describing the corresponding dynamical
system.
Let us consider any two nearby divergent trajectories originating from a 1D flow
(i.e., trajectories in continuous time as described by differential equations). The
growth of the difference

δ t between the two nearby trajectories over a time period

Δ t = t 1 − t 0 can be described by

δ t ∼ δ 0 e λΔ t , (16.1)

where denotes the systems Lyapunov exponent.

An important reason for using the Lyapunov exponent as a characteristic mea-
sure of a dynamical system is its invariance 1 to rescaling, shifts and other transfor-
mations of data such as the imprecise reconstruction of a strange attractor from a
time series. The fact that trajectories diverge over the course of time would not in
itself be very dramatic if it was only very slow, thus we speak of chaos only if this
separation is exponentially fast. There are n different Lyapunov exponents for an n -
dimensional system, defined as follows: Consider the evolution of an infinitesimal
sphere of perturbed initial conditions. During its evolution along the reference tra-
jectory, the sphere will become deformed into an infinitesimal ellipsoid. Let

λ

δ ( t ) ,

k

k n , denote the length of the k th principal axis of the ellipsoid. Thus, the

deformation of the sphere corresponds to the stretching, contraction, and rotation of
the principal directions. For large t , the diameter of the ellipsoid is controlled by the
most positive

= 1 , 2 , 3 ,...,

depends slightly on which trajectory
we study, so we should average over many different points on the same trajectory to
get the true value of

λ k . As we shall see later on, λ

(see Table 16.1).
In dissipative systems one can also find a negative maximal Lyapunov exponent
which reflects the existence of a stable fixed point. Two trajectories which approach
the fixed point also approach each other exponentially fast. If the motion settles
down onto a limit cycle, two trajectories can only separate or approach each other
slower than exponentially. In this case the maximal Lyapunov exponent is 0 and the
motion is called marginally stable. If a deterministic system is perturbed by random
noise, on the s mall scales it can be characterized by a diffusion process, with

λ

δ t

growing as √ t . Thus the maximal Lyapunov exponent is infinite. According to the

Table 16.1: P ossible types of motion and the corresponding Lyap unov exponents
Type of motion

Maximal Lyapunov exponent

Stable fixed point λ < 0

Stable limit cycle λ = 0

Chaos 0 < λ < ∞

Noise λ = ∞

1

See Oseledec's theorem in [12].