3.4 Benettin’s Method
83
˜
λ 1 =
1
T
ln
x 1
ε
.
(3.8)
The time intervals T is chosen in the way to keep the perturbation amplitude
less than linear dimension of the phase portrait non-homogeneity and less of the
being investigated attractor dimension. Let us consider the normalized vector of
perturbation x
1 = εεx 1 / x 1 , and the associated new perturbation point ˜
x
1 =
x 1 + x
1 . We prolong the so far described procedure by taking into account the
points x 0 and ˜
x 0 instead of x 1 and ˜
x 1 , respectively.
Repeating the mentioned procedure M times, one may estimate λ as the averaged
arithmetic value of the quantities ˜
λ l estimated on each computational step, i.e. we
have
λ ∼ =
1
M
M
i=1
˜
λ l =
1
M
M
i=1
1
T
ln
x i
ε
=
1
MT
M
i=1
ln
x i
ε
.
(3.9)
In order to achieve more accurate estimation, one may arbitrarily choose larger
values of M and carry out the computations for a different initial point x 0 .
The so far presented method can be employed only if the evolutionary equations
governing system dynamics are known (in the case of experimental data the governing
equations are typically not known).
In order to obtain numerically the Lyapunov spectrum one may use the modified
algorithm generalizing the classical Benettin’s approach. Instead of using one trajectory, a few of them are considered based on perturbations of a few initial points. A
number of the perturbed trajectories is equal to the dimension of the phase space. In
order for smooth realizations of the latter approach, a numerical method based on the
governing equations derivation based on a variation procedure is employed [4]. Since
LLE plays a crucial influence on the evolution of all perturbed trajectories, then each
step of the algorithm requires not only normalization of the perturbation vectors but
also their orthogonalization. We consider the procedure of numerical estimation of
the Lyapunov spectrum of a given dynamical system. We assume for simplicity that
the phase space is three-dimensional (3D).
Let r 0 be a certain point belonging to the attractor. We fix a certain relatively
small (in comparison to the attractor dimension) positive number ε and we choose
the perturbation points x 0 , y 0 and z 0 to keep the length of perturbation vectors
x 0 = x 0 − r 0 , ,y 0 = y 0 − r 0 , ,z 0 = z 0 − r 0 equal to ε, and we keep them mutually orthogonal. Now, the points r 0 , x 0 , y 0 and z 0 evaluate into points r 1 , x 1 , y 1 and
z 1 , respectively, within the monitored time interval T . We consider the new perturbation vectors x 1 = x 1 − r 1 , ,y 1 = y 1 − r 1 , ,z 1 = z 1 − r 1 , and again we carry
out their orthogonalization based on the known method of Gramm-Schmidt [5].
After the orthogonalization process, the obtained perturbation vectors x
1 ,
y
1 , z
1 are orthonormalized, i.e. they are mutually orthogonalized and they
have unit length. Now, we again employ the normalization procedure of the orthogonalized perturbation vectors to make their length equal to ε in the following way
x
1 = x
1 · ε, ,y
1 = y
1 · ε, ,z
1 = z
1 · ε.
(3.10)
83
˜
λ 1 =
1
T
ln
x 1
ε
.
(3.8)
The time intervals T is chosen in the way to keep the perturbation amplitude
less than linear dimension of the phase portrait non-homogeneity and less of the
being investigated attractor dimension. Let us consider the normalized vector of
perturbation x
1 = εεx 1 / x 1 , and the associated new perturbation point ˜
x
1 =
x 1 + x
1 . We prolong the so far described procedure by taking into account the
points x 0 and ˜
x 0 instead of x 1 and ˜
x 1 , respectively.
Repeating the mentioned procedure M times, one may estimate λ as the averaged
arithmetic value of the quantities ˜
λ l estimated on each computational step, i.e. we
have
λ ∼ =
1
M
M
i=1
˜
λ l =
1
M
M
i=1
1
T
ln
x i
ε
=
1
MT
M
i=1
ln
x i
ε
.
(3.9)
In order to achieve more accurate estimation, one may arbitrarily choose larger
values of M and carry out the computations for a different initial point x 0 .
The so far presented method can be employed only if the evolutionary equations
governing system dynamics are known (in the case of experimental data the governing
equations are typically not known).
In order to obtain numerically the Lyapunov spectrum one may use the modified
algorithm generalizing the classical Benettin’s approach. Instead of using one trajectory, a few of them are considered based on perturbations of a few initial points. A
number of the perturbed trajectories is equal to the dimension of the phase space. In
order for smooth realizations of the latter approach, a numerical method based on the
governing equations derivation based on a variation procedure is employed [4]. Since
LLE plays a crucial influence on the evolution of all perturbed trajectories, then each
step of the algorithm requires not only normalization of the perturbation vectors but
also their orthogonalization. We consider the procedure of numerical estimation of
the Lyapunov spectrum of a given dynamical system. We assume for simplicity that
the phase space is three-dimensional (3D).
Let r 0 be a certain point belonging to the attractor. We fix a certain relatively
small (in comparison to the attractor dimension) positive number ε and we choose
the perturbation points x 0 , y 0 and z 0 to keep the length of perturbation vectors
x 0 = x 0 − r 0 , ,y 0 = y 0 − r 0 , ,z 0 = z 0 − r 0 equal to ε, and we keep them mutually orthogonal. Now, the points r 0 , x 0 , y 0 and z 0 evaluate into points r 1 , x 1 , y 1 and
z 1 , respectively, within the monitored time interval T . We consider the new perturbation vectors x 1 = x 1 − r 1 , ,y 1 = y 1 − r 1 , ,z 1 = z 1 − r 1 , and again we carry
out their orthogonalization based on the known method of Gramm-Schmidt [5].
After the orthogonalization process, the obtained perturbation vectors x
1 ,
y
1 , z
1 are orthonormalized, i.e. they are mutually orthogonalized and they
have unit length. Now, we again employ the normalization procedure of the orthogonalized perturbation vectors to make their length equal to ε in the following way
x
1 = x
1 · ε, ,y
1 = y
1 · ε, ,z
1 = z
1 · ε.
(3.10)