The largest LCE can be defined as

λ(z0, v0)=
 
lim
n → ∞
1
n
ln(||DΦn(z0) v0 ||).     (47)

To calculate the largest LCE we thus have to determine ||DΦn(z0) v0 ||. We set the initial perturbation vector v0 with ||v0||=ε, where ε is some small number. We define
Φ(zi + vi)-Φ(zi) ≈ DΦ(zi) vi = vi+1' = fi+1 vi+1,     (48)
where vi is a perturbation vector on the i-th iterate of Φ (i.e. zi=Φi(z0)) and fi+1 is a scalar. We define
vi+1=
vi+1'
||vi+1'||
ε, so that ||vi+1||=||vi||=...=||v0||=ε.     (49)
Using the chain rule for DΦn(z0) we get
DΦn(z0) v0 = DΦ(zn-1)...DΦ(z1) DΦ(z0) v0.
Using (6.48) recursively this relation transforms to
DΦn(z0) v0 = fn...f2 f1 vn.
Because
 
lim
n → ∞
1
n
ln(||vn||)=0,
the LCE in equation (6.47) can be written as
λ(z0, v0)=
 
lim
n → ∞
1
n
n
∑
i=1
ln(fi).     (50)
Hence we can confine ourselves to the calculation of
fi=
||vi'||
||vi||
=
||vi'||
ε
, for i=1...n     (51)
to determine the largest LCE.

Numerically we compute the largest LCE as follows. Given an initial perturbation vector v0, the approximation in (6.48) is used to determine vi+1' for i ≥ 0, that is
Φ(zi + vi)-Φ(zi) ≈ vi+1'. Next we compute the perturbation vector for the i+1-th iterate by using (6.49). The factor fi+1 is computed by using (6.51). Finally, for large n, the largest LCE is computed by using (6.50).

Attractors may be characterized by their Lyapunov spectrum. For a stable steady state or a stable cycle all LCEs are negative. For a quasi-periodic attractor the largest LCE is equal to zero, while all other LCEs are negative. An attractor is called a strange or a chaotic attractor if the corresponding largest LCE is positive, implying sensitive dependence on initial conditions.

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