3.5 Wolf’s Method
85
less than the dimension of the reconstructed attractor (both points x 0 and
x 0 should
be chosen for different time instants). Then we monitor evolution of both points on
the reconstructed attractor unless a distance between them is not larger than the given
quantity ε max . The obtained points x 1 and
x 1 refer to the time evolution T 1 .
Then we consider again the series (3.14) and we look for a point
x
1 , being close to
x 1 , where the following formula is satisfied
x
1 − x 1
= ε 1 < ε. The vectors
x 1 − x 1
and
x
1 − x 1 should possess (possibly) the same direction. The mentioned procedure
is repeated for the points x 1 and
x 1 .
Repeating the so far described procedure M times, the LLE is estimated based on
the following formula
λ ˜
=
M−1
k=0
ln(ε
k /ε k )/
M
k=1
T k .
(3.15)
The given method has been employed by the authors of this monograph to the
systems with known spectra of LEs such as the Hénon map, Rössler and Lorenz
systems and Mackey-Glass equations [7]. Besides, the method is applicable to study
the Belousov-Zhabotinsky reaction [8] and the Couette-Taylor flow [9].
Wolf et al. [6] introduced a few constraints on the choice of the embedding dimension and time delay τ while carrying out the attractor reconstruction for getting the
most precise estimation of LEs.
Based on investigation of the Rössler system [10] and Belousov-Zhabotinsky
reactions [11], the influence on the time delay required for attractor reconstruction,
the time of system development between steps of changes, length of the exchange
vector and the minimum possible vector length on the numerically obtained results
for the LLE computation have been illustrated and discussed.
In addition, it has been illustrated how variation of time of the systems development between the mentioned changes from 0.5 to 1.5 of the orbit almost always
implied stable estimation of the LLEs for the Lorenz system, Rössler attractor [7]
and the Belousov-Zhabotinsky reaction [12].
The so far described algorithms can be used to detect chaos and to estimate the
corresponding parameters based on the available experimental data while estimating
the first positive LEs. Besides, the carried out numerical experiments showed that the
deterministic chaos can differ from the external noise (Belousov-Zhabotinsky attractor) and topological complexity (Lorenz attractor). However, it requires a special
choice of the used data, and the studied attractor should have a large dimension.
3.6 Rosenstein’s Method [13]
The method proposed by Rosenstein [13] is simple in realization and allows to
get results relatively fast but in fact it yields a special function [13] instead of the
numerical value of λ 1 of the following form
85
less than the dimension of the reconstructed attractor (both points x 0 and
x 0 should
be chosen for different time instants). Then we monitor evolution of both points on
the reconstructed attractor unless a distance between them is not larger than the given
quantity ε max . The obtained points x 1 and
x 1 refer to the time evolution T 1 .
Then we consider again the series (3.14) and we look for a point
x
1 , being close to
x 1 , where the following formula is satisfied
x
1 − x 1
= ε 1 < ε. The vectors
x 1 − x 1
and
x
1 − x 1 should possess (possibly) the same direction. The mentioned procedure
is repeated for the points x 1 and
x 1 .
Repeating the so far described procedure M times, the LLE is estimated based on
the following formula
λ ˜
=
M−1
k=0
ln(ε
k /ε k )/
M
k=1
T k .
(3.15)
The given method has been employed by the authors of this monograph to the
systems with known spectra of LEs such as the Hénon map, Rössler and Lorenz
systems and Mackey-Glass equations [7]. Besides, the method is applicable to study
the Belousov-Zhabotinsky reaction [8] and the Couette-Taylor flow [9].
Wolf et al. [6] introduced a few constraints on the choice of the embedding dimension and time delay τ while carrying out the attractor reconstruction for getting the
most precise estimation of LEs.
Based on investigation of the Rössler system [10] and Belousov-Zhabotinsky
reactions [11], the influence on the time delay required for attractor reconstruction,
the time of system development between steps of changes, length of the exchange
vector and the minimum possible vector length on the numerically obtained results
for the LLE computation have been illustrated and discussed.
In addition, it has been illustrated how variation of time of the systems development between the mentioned changes from 0.5 to 1.5 of the orbit almost always
implied stable estimation of the LLEs for the Lorenz system, Rössler attractor [7]
and the Belousov-Zhabotinsky reaction [12].
The so far described algorithms can be used to detect chaos and to estimate the
corresponding parameters based on the available experimental data while estimating
the first positive LEs. Besides, the carried out numerical experiments showed that the
deterministic chaos can differ from the external noise (Belousov-Zhabotinsky attractor) and topological complexity (Lorenz attractor). However, it requires a special
choice of the used data, and the studied attractor should have a large dimension.
3.6 Rosenstein’s Method [13]
The method proposed by Rosenstein [13] is simple in realization and allows to
get results relatively fast but in fact it yields a special function [13] instead of the
numerical value of λ 1 of the following form