90
3 Lyapunov Exponents and Methods of Their Analysis
e =
c
k=d+1
( ˆ
x k − x k )
2
c − d
.
(3.19)
When the network is being trained, sensitivity of the output is defined in each
time step by computing partial derivatives of all averaged points of the time series
in each time step x k− j :
ˆ
S( j) =
1
c − j
c
k= j+1
∂ ˆ
x k
∂ x k− j
.
(3.20)
In the case of the network given by (3.18), the partial derivatives have the following
form
∂ ˆ
x k
∂ x k− j
=
n
i=1
a i j b i sec h
2
a i0 +
d
m=1
a im x k−m
.
(3.21)
The largest value j is the optimal embedding dimension, and the key role is played
by ˆ
S( j) as in the false nearest neighbours method. The individual values of ˆ
S( j) yield
a quantitative estimate of the importance of each time step using the associated terms
of the autocorrelation function or coefficients of the associated linear model.
The weight coefficients of the trained neural network are substituted to the matrix
of solutions, and the input data are used to define the initial state. The computation of
the spectrum is realized by employment of the generalized Benettin algorithm based
on the obtained system of equations.
References
1. Awrejcewicz, J., Krysko, A.V., Erofeev, N.P., Dobriyan, V., Barulina, M.A., Krysko, V.A.:
Quantyfying chaos by various computational methods. Part 1: Simple systems. Entropy 20(3)
175 (2018)
2. Awrejcewicz, J., Krysko, A.V., Erofeev, N.P., Dobriyan, V., Barulina, M.A., Krysko, V.A.:
Quantyfying chaos by various computational methods. Part 2: Vibrations of the BernoulliEuler beam subjected to periodic and colored noise. Entropy 20(3) 170 (2018)
3. Kolmogorov A.N.: General theory of dynamical systems and classical mechanics. In: International Mathematical Congress in Amsterdam, pp. 185–208. Fizmatgiz, Moscow (1961). (in
Russian)
4. Benettin, G., Galgani, L., Strelcyn, J.M.: Kolmogorov entropy and numerical experiments.
Phys. Rev. A 14, 2338–2345 (1976)
5. Beklemishev, D.V.: Course of Analytical Geometry and Linear Algebra. Nauka, Moscow
6. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a
time series. Phys. D 16, 285–317 (1985)
7. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197,
287–289 (1977)
8. Hudson, J.L., Mankin, J.C.: Chaos in the Belousov-Zhabotinskii reaction. J. Chem. Phys. 74,
6171 (1981)
9. Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans.
Roy. Soc. A 223(605–615), 289–343 (1923)
3 Lyapunov Exponents and Methods of Their Analysis
e =
c
k=d+1
( ˆ
x k − x k )
2
c − d
.
(3.19)
When the network is being trained, sensitivity of the output is defined in each
time step by computing partial derivatives of all averaged points of the time series
in each time step x k− j :
ˆ
S( j) =
1
c − j
c
k= j+1
∂ ˆ
x k
∂ x k− j
.
(3.20)
In the case of the network given by (3.18), the partial derivatives have the following
form
∂ ˆ
x k
∂ x k− j
=
n
i=1
a i j b i sec h
2
a i0 +
d
m=1
a im x k−m
.
(3.21)
The largest value j is the optimal embedding dimension, and the key role is played
by ˆ
S( j) as in the false nearest neighbours method. The individual values of ˆ
S( j) yield
a quantitative estimate of the importance of each time step using the associated terms
of the autocorrelation function or coefficients of the associated linear model.
The weight coefficients of the trained neural network are substituted to the matrix
of solutions, and the input data are used to define the initial state. The computation of
the spectrum is realized by employment of the generalized Benettin algorithm based
on the obtained system of equations.
References
1. Awrejcewicz, J., Krysko, A.V., Erofeev, N.P., Dobriyan, V., Barulina, M.A., Krysko, V.A.:
Quantyfying chaos by various computational methods. Part 1: Simple systems. Entropy 20(3)
175 (2018)
2. Awrejcewicz, J., Krysko, A.V., Erofeev, N.P., Dobriyan, V., Barulina, M.A., Krysko, V.A.:
Quantyfying chaos by various computational methods. Part 2: Vibrations of the BernoulliEuler beam subjected to periodic and colored noise. Entropy 20(3) 170 (2018)
3. Kolmogorov A.N.: General theory of dynamical systems and classical mechanics. In: International Mathematical Congress in Amsterdam, pp. 185–208. Fizmatgiz, Moscow (1961). (in
Russian)
4. Benettin, G., Galgani, L., Strelcyn, J.M.: Kolmogorov entropy and numerical experiments.
Phys. Rev. A 14, 2338–2345 (1976)
5. Beklemishev, D.V.: Course of Analytical Geometry and Linear Algebra. Nauka, Moscow
6. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a
time series. Phys. D 16, 285–317 (1985)
7. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197,
287–289 (1977)
8. Hudson, J.L., Mankin, J.C.: Chaos in the Belousov-Zhabotinskii reaction. J. Chem. Phys. 74,
6171 (1981)
9. Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans.
Roy. Soc. A 223(605–615), 289–343 (1923)