86
3 Lyapunov Exponents and Methods of Their Analysis
y(i, t) =
1
t
ln d j (i), d j (i)
= min
x j
x j − x
j
,
(3.16)
where x j stands for the considered point, and x
j refers to its neighbour point.
The carried out algorithm relies on the coupling of d j and the Lyapunov exponents
d j (i) ≈ e
λ 1 (it)
. LLE is defined by an angle of inclination of the most linear part of
the computed function.
3.7 Kantz Method [10]
The Kantz algorithm [10] allows for computation of LLE by looking for all neighbours in the vicinity of the studied trajectory and yields the average distance between
neighbours and the studied trajectory as a function of time (or relative time). It is
based on the following equation
S(τ ) =
1
T
T
t=1
ln(
1
|U t |
i∈U t
|x t+τ − x i+τ |),
(3.17)
where x t —an arbitrary signal point; U t —neighbourhood of x t ; x i —neighbour of x t ;
τ —relative time multiplied by the frequency of the choice; T —dimension of the
choice; S(τ )—elongation factor in domain of the linear increase of the curve whose
slope defines the Lyapunov exponent, i.e. e
λτ
∞e
S(τ ) . However, the requirement of
the linear increase of the curve yields occurrence of new errors. Though the method
is useful and enough accurated for the case of systems with known values of LLEs,
the choice of parameters and regions when the mentioned linear increase occurs is
realized in an arbitrary way.
3.8 Method Based on Jacobian Estimation [11, 12]
This method has been proposed in references [11, 12]. Its main idea is to use an
algorithm, the scheme of which is illustrated in Fig. 3.1. A sphere of small radius
ε is taken. After a few iterations m, a certain operator T
m transforms this sphere
into an ellipsoid having a 1 , . . . , a p half-axes. The sphere is stretched along the axes
a 1 , . . . , a s > ε, where s is the number of positive LEs. For sufficiently small ε, the
operator T
m is close to the sum of the shear operator and the linear operator A. The
LLEs are computed as averaged eigenvalues of the operator A on the whole attractor.
A vector ζ j is chosen, and a set {ζ k i }(i = 1, . . . , N ) of i-th neighbourhood vectors
is found. The following set of vectors y i ≡ ζ k i − ζ j , where ||y i || ≤ ε, is taken. After
m successive iterations, the operator T
m transforms the vector ζ j into ζ j+m , and the
3 Lyapunov Exponents and Methods of Their Analysis
y(i, t) =
1
t
ln d j (i), d j (i)
= min
x j
x j − x
j
,
(3.16)
where x j stands for the considered point, and x
j refers to its neighbour point.
The carried out algorithm relies on the coupling of d j and the Lyapunov exponents
d j (i) ≈ e
λ 1 (it)
. LLE is defined by an angle of inclination of the most linear part of
the computed function.
3.7 Kantz Method [10]
The Kantz algorithm [10] allows for computation of LLE by looking for all neighbours in the vicinity of the studied trajectory and yields the average distance between
neighbours and the studied trajectory as a function of time (or relative time). It is
based on the following equation
S(τ ) =
1
T
T
t=1
ln(
1
|U t |
i∈U t
|x t+τ − x i+τ |),
(3.17)
where x t —an arbitrary signal point; U t —neighbourhood of x t ; x i —neighbour of x t ;
τ —relative time multiplied by the frequency of the choice; T —dimension of the
choice; S(τ )—elongation factor in domain of the linear increase of the curve whose
slope defines the Lyapunov exponent, i.e. e
λτ
∞e
S(τ ) . However, the requirement of
the linear increase of the curve yields occurrence of new errors. Though the method
is useful and enough accurated for the case of systems with known values of LLEs,
the choice of parameters and regions when the mentioned linear increase occurs is
realized in an arbitrary way.
3.8 Method Based on Jacobian Estimation [11, 12]
This method has been proposed in references [11, 12]. Its main idea is to use an
algorithm, the scheme of which is illustrated in Fig. 3.1. A sphere of small radius
ε is taken. After a few iterations m, a certain operator T
m transforms this sphere
into an ellipsoid having a 1 , . . . , a p half-axes. The sphere is stretched along the axes
a 1 , . . . , a s > ε, where s is the number of positive LEs. For sufficiently small ε, the
operator T
m is close to the sum of the shear operator and the linear operator A. The
LLEs are computed as averaged eigenvalues of the operator A on the whole attractor.
A vector ζ j is chosen, and a set {ζ k i }(i = 1, . . . , N ) of i-th neighbourhood vectors
is found. The following set of vectors y i ≡ ζ k i − ζ j , where ||y i || ≤ ε, is taken. After
m successive iterations, the operator T
m transforms the vector ζ j into ζ j+m , and the