Unlimited dimension increase of chaotic attractors
under linear filtering
A.A. Kipchatov and E.L. Kozlenko
Saratov State University, Saratov, Russia
It is known that there are two types of the attractor complication:
(i) for finite impulse response
filters trajectories of the attractor, reconstructed from output
oscillations, stretch and fold up many times, but its fine structure
is preserved and the dimension increase takes place only on finite scales
and for 
the dimension remains unchanged
(observational dimension increase);
(ii) for infinite impulse response filters fractal splitting of the trajectories
of the reconstructed attractor takes place and dimension undergoes real
increase [1]. The value of assumed dimension increase for 1st-order
filters is equal to 1 according to Badii and Politi hypothesis [2].
But for high-order the degree
of the dimension increase is not clear. The present work is focused on
the analysis of maximum complication of chaotic oscillations in high-order
filters.
We consider the simplest chaotic oscillations, emerging from logistic map
,
and the simplest ith-order recursive filter realised by sequential
connecting of i 1st-order recursive filters
where 
-- output signal, 
-- input signal,
i -- ordinal number of filter in the chain, 
-- coefficients,
determining filter cutoff frequency.
The analysis of attractors, reconstructed by means of time delays method from output oscillations of the filter can be done qualitatively -- through the set of Poincare sections.
As a result the qualitative similarity of attractors topological structures of the input unfiltered chaotic oscillations and the kth-Poincare sections of the reconstructed attractors of kth-order filtered chaotic oscillations are obtained. The repeating character of the complexity increase of the chaotic oscillations are: the 1st-order filter converts line (input attractor is the square parabola) into a surface via superfractalization; the 2st-order filter leads to fractal splitting of the surface and transforming attractor in 3D-object; nth-order filter gives the fractal splitting of n-dimentional object and transformation of it in n+1-dimentional object.
The same results have been obtained in the case of Henon map taken as
generator of more complicated chaotic signal: typical chaotic attractor
of the Henon map ( 
) has been revealed in the sections
of the attractors of filtered oscillations.
So, we can state, that the dimension of chaotic oscillations, put through the high-order filter, is
where i -- order of the filter. That meanes the possibility of the infinite increase of the dimension of chaotic oscillations, allowing to generate the test chaotic oscillations with a given dimension.
This work was financial supported by Russian fund of fundamental researches (grant 95-02-06262).