- Open Access
Higher difference structure of some discrete processes
© Shahverdian et al.; licensee Springer 2012
Received: 20 October 2012
Accepted: 8 November 2012
Published: 26 November 2012
A method for analyzing discrete processes based on consideration of their higher difference structure is presented. Two kinds of processes, deterministic systems (discrete dynamical systems) and the sequences of random binary independent variables, are considered. Two main statements, computation of higher absolute differences and restoring the original by a given higher order difference image, are studied. Some applications to dynamical systems are given.
MSC:39B12, 39A50, 37E05.
This paper presents some results on difference analysis, which is a method for studying irregular time series and orbits of discrete dynamical systems. Such an analysis, suggested in [1–3], is based on consideration of higher absolute differences taken from successive terms of a given time series or orbits of a given system. This is motivated by an observation that some natural systems (e.g., the visual cortex, ) process the information contained in signals difference structure. The difference analysis reveals some new aspects of nonlinearity; for instance, a difference analog for Lyapunov exponent , used for chaos discrimination, is introduced .
The content of this paper is as follows. In Section 2 deterministic processes are considered, Section 3 deals with random independent binary processes. In both cases, we are interested in computation of higher order absolute differences, recovering the original by a given difference image as well as in the limiting behavior of difference processes. Section 4 contains the proofs of results presented.
2 Deterministic processes
This section considers higher absolute differences taken from successive terms of discrete deterministic systems. Two main problems, computation of higher differences and restoring the original, are studied. Some applications to discrete dynamical systems are given.
2.1 Higher absolute differences of deterministic processes
In difference analysis, a given time series or an orbit X is decomposed into two constituents: the sign component S and the magnitude (or height) component H. The S-component reflects the alternation in monotony (increase/decrease) of higher order absolute differences taken from successive terms of X. The component S does not depend on the very exact values of these differences, while the component H consists of these absolute differences and does not depend on their sign distribution. At every step (associated with the order of differences) of the decomposition process, the orbit X can be completely restored by first terms of X and the two components. Since the behavior of X at infinity is determined by the limiting behavior of its S- and H-components, such an analysis can also be useful in dynamical systems (see, e.g., ).
In this way, a given time series X is decomposed into two other, the sign and magnitude components and . We call and the m th S- and H-components of X, respectively, and denote , , and .
exists, then for every (for every ) the limit () also exists.
The following theorem asserts that every absolute difference is a linear form of the original X and presents a formula for its computation. The rule (5) below is a generalization of the well-known additive scheme for constructing the Pascal triangle of binomial coefficients. This theorem is a consequence of relations (2); the proof (conducted by induction) is straightforward and is omitted.
2.2 Restoring the original by difference image
At every m th step (associated with the order m of differences) of the decomposition process, the original time series can be completely restored by its first m entries and the two components S and H. Indeed, the following theorem provides us with analytical expressions for computation of the original X by the components , and first m terms (below, it is denoted ):
We note that Eq. (7) represents a countable infinite system of higher order difference equations (which are nonlinear since we deal with absolute differences) and its solution by Eq. (6) is given. The proof of this theorem is straightforward (by induction) and is omitted.
2.3 Application to dynamical systems
We consider the orbits of the maps f defined on the unit segment and the limiting (as the order m of differences tends to +∞) behavior of their difference series . We claim that this behavior is mainly due to some function ℋ () which can be determined as a solution to some difference-functional equation.
The next statement presents a difference equation for computation of the limiting function ℋ and establishes the special property of such functions: once vanished, ℋ remains zero on all the further orbit of the map f. In the following two statements, denotes the entire part of number a (maximal integer which does not exceed a) and is its fractional part.
- (1)The function ℋ satisfies the following functional equation:(10)
If for some we have , then for all .
- (3)If for some we have , then there is a number N, such that
The proof of this theorem is straightforward: the point (1) of theorem is a consequence of Eq. (2) and points (2) and (3) follow directly from the point (1).
Particularly, this theorem immediately implies the following.
be infinite. Then for all large enough .
For instance, if f is such that (e.g., this can be the so-called 4-logistic map or Bernoulli shift studied in deterministic chaos [5, 6]) and , then for every the semi-orbit (11) is infinite and hence, by Corollary 2.4, for such f, ℋ is identically zero on every periodic orbit: if , then for every . Particularly, it follows that if is such that and is dense on I, then the null-set is also dense on I.
3 Random independent discrete processes
This section considers the higher absolute differences taken from successive terms of random independent binary sequences (see also ). The same two problems as in Section 2, computation of higher order absolute differences and restoring the original, are studied. When proving Theorem 3.2, we underline some connections of difference analysis with discrete models of self-organized criticality (SOC), a physics-originated theory (e.g., [8, 9]) which studies the large systems of interacting microsystems.
3.1 Higher absolute differences of random binary processes
The following Theorem 3.1 is an analog for Theorem 2.1. To demonstrate the analogy, instead of , we use and denote and ().
where the coefficients are the terms of kth line of the triangle ℙ.
3.2 Restoring the original by difference image
for hold. The solution to this task is given by Theorem 3.2 which is an analog for Theorem 2.2. The proof of Theorem 3.2 is conducted by involving some physics-related notions such as particles, energy, etc. Particularly, this allows us to demonstrate some connections of the difference analysis with a modification of some SOC-related models.
correspond to a given k. Equation (14) means that our task is reduced to solving some countable system of linear algebraic equations; since, for every i which is not contained in the sample s from Eq. (15), we have , the system of equations (14) coincides with the system (16) below.
In the next formulation, for a positive number a, denotes the entire part of a (maximal integer which does not exceed a), and for natural a and b, we denote , where r is residual from dividing a by b (and if ).
- (1)are positive and arbitrary, and for , is equal to(17)
- (2)for , is equal to(18)
Then, for every , the sample satisfies Eq. (16).
It can be noticed from formulation of this theorem that the sought (whose index m is given as ) are such that are positive and arbitrary, while for , we have ; that is, the above-mentioned restored process ξ is not unique but depends on k arbitrary positive numerical parameters.
To prove Theorem 3.2, we use the following physics-related model, which can be treated as a modification of some lattice models (e.g., the BTW-model, ) of SOC. Let us describe the model that we use. Let, for , be a lattice from . At every moment of discrete time at the vertices of , which are found above the diagonal L connecting the vertices and , some material particles with positive or negative mass (or charge) are situated. Each particle performs uniform rectilinear motion with the unit speed on the vertical segments connecting the vertices and () of . At the (initial) moment , all the particles are found on L and possess the unit mass. Each moment t, when a number (=p) of particles reach simultaneously the upper side of the lattice (one can see that such particles should possess the same mass m), these particles disappear, and at the next moment , at every position on L a new particle of the mass emerges (and the motion of particles continues). In SOC terminology, the p particles hit the threshold (the upper side of the square ), and an avalanche of intensity p at the moment t in the system S of the moving particles occurs.
of the system S of particles on for every moment t; here, and denote the y-coordinate and mass of the particle which at the moment t is found at the vertex . The proof of Theorem 3.2 is based on the following lemma.
4 Proof of results
(it is assumed that ) holds.
One can see that (27) coincides with the point (1) of the theorem, and this point of the theorem is proved.
In terms of the lattice model, the inequality is interpreted as the threshold attainment and the substitution of is interpreted as an event of emergence of new particles on each of positions of the diagonal L. One can see that if this variable substitution procedure is sequently applied to all the equations in the system (28), we obtain the above-defined lattice model of moving particles. Then, by applying Lemma 3.3 (where the time variable t coincides with the variable T in Eq. (28)) and assuming that the potential U (Section 3.2) is assigned as , we obtain the sought Eq. (18). Theorem 3.2 is proved. □
Authors thank the editing group for corrections in the linguistics of the manuscript.
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