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3 Lyapunov Exponents and Methods of Their Analysis
where V, C are the matrices of the dimension n × n, y
k
i stands for the k-th component
of vector y i , and y
k
i+m is the k-th component of the vector y i+m . If A is a solution to
the mentioned equations, then the LEs can be found in the following way
λ i = lim
n→∞
1
nτ
n
j=1
ln A j e
j
i ,
where {e j } is a set of basic vectors in tangent space ζ j .
The algorithm can be realized in a way similar to the computation of LEs of the
ODEs given analytically.
Let us choose an arbitrary basis {e
s
} and then follow the changes in the length of the
vector A j e
s . As the vectors A j e
s grow and their orientations change, it is necessary
to perform their orthogonalization and normalization by using, for example, the
Gramm-Schmidt procedure. Then, the procedure is repeated for the new basis.
The mentioned method allows one to estimate a spectrum of nonnegative LEs.
However, the method has a serious disadvantage—it is highly sensitive to noise and
errors.
3.9 Modification of the Neural Network Method [14, 15]
We proposed a novel and counterpart method to compute LEs based on a modification
of the neural network method (see Fig. 3.2)
To realize the neural network algorithm, the following criteria were taken into
account:
(i) the network is sensitive to the input information (information is given in the
form of real numbers);
(ii) the network is self-organizing, i.e. it yields the output space of solutions only
based on the inputs;
Fig. 3.2 One-layer neutral
network