Institute of Catalysis, Bulgarian Academy of Science
1113 Sofia, BULGARIA

Coarse-grained structure of a physical (strange) attractor. Analytical solution

The notion of a physical attractor is inherently consistent with the notion of the boundedness of fluctuations. The conjecture behind the boundedness is that a stable long-term evolution of any natural system is possible if and only if the fluctuations that the system exerts are bounded so that the system permanently stays within the thresholds of stability. Thus, the thresholds of stability match the boundaries of a finite volume in the phase space so that every phase trajectory is confined in that volume called physical attractor [1]. Further, we proved [1] that any bounded irregular sequence (BIS) has certain universal properties that are insensitive to the increment statistics. These properties are: (a) the power spectrum comprises a continuous band that uniformly fits the shape 1/fa(f), where a(f) monotonically increases starting from 1 at f = 1/T (T is the length of the sequence) up to p (p is arbitrary but p > 2 as f approaches infinity; (b) the physical (strange) attractor is non-homogeneous; (iii) the Kolmogorov entropy is finite. It has been proven also that the strange attractors that are related to the simulated dynamical systems has the above properties.

The present task is to reveal the influence of the incremental statistics onto the properties of the bounded irregular sequences after  coarse-graining. The problem appears rather controversial  because the boundedness imposes restrictions on the range of the correlations among successive increments. Indeed, an arbitrary  increase of the correlation length certainly makes the amplitude of certain fluctuations to exceed the thresholds of stability.  So, the boundedness imposes a finite radius of the increment correlations. Then, it is to be anticipated that the coarse-grained structure of  any BIS is universal.  To the most surprise, it turns out that there are 3 levels of coarse-graining: (I) the low level whose major property is that the structure of the BIS matches the white noise behavior; (ii) the meso-level  where the sequence of excursions matches telegraphic noise behavior. At the same time, the excursions retain certain specific property related to the incremental statistics; (iii)  macro-level where the properties of any BIS are universal ( insensitive to any detail of the increment statistics). It should be stressed that the universal properties on that level of coarse-grained are due to the  proof that  under coarse-graining the symmetric random walk with constant step appear as the global attractor for any fractal  Brownian motion. On the other hand, it should be stressed that the chaotic properties listed above remain unaffected by coarse-graining. Thus, both white noise and telegraphic noise behavior are rather alias effects: they are typical for sequences of certain finite length.  The asymptotic behavior of every BIS, regardless to the level of coarse-graining, is covered always by the chaotic properties established in [1].

The above separation is made on the grounds of certain Poissonian like behavior of the excursion appearance. The particularity of that behavior is imposed by the boundedness and the lack of long-range correlations among increments. More precisely, the lack of long-range correlations among increments renders a uniform convergence of the average to the mean of any BIS. The boundedness provides that every excursion is loaded  in a specific “embedding  time interval”. The length of that interval is exerted as a random choice from certain range of almost equiprobable values. In turn, these properties ensure that: (i) the successive excursions are independent from one another events; (ii) the probability for an excursion is insensitive to its position in the sequence; (iii) no more than one excursion can develop at any point of the sequence.  It should be stressed that the “pulse” like behavior (the separation of the successive excursions from one another) appear as a robust to coarse-graining behavior. Recently it is proven that such behavior is typical on quantum level [2]. The importance of the “pulse” like behavior is that it provides that the fluctuations remain bounded at any scale of averaging, respectively at any level of coarse-graining.


  1. M. K. Koleva and V. C. Covachev , “Common and Different Features Between  the  Behavior of the Chaotic Dynamical Systems and the $1/f^\alpha$-type Noise”, FNL, 1,  (2001) R131-149
  2. M. K. Koleva,  “Size-Independent Non-Equilibrium Fluctuations. Alias $1/f^alpha$-type Noise”, cond-mat/0212357