Rev. Acad. Canar. Cienc., XII (Núms. 1-2), 225-235 (2000) - (Publicado en julio de 2001)
Prigogine dynamical systems based on Bernoulli shifts of real functions
FRANZ PICHLER
Institute of Systems Science
Johannes Kepler University Linz
A-4040 Linz, Austria
ABSTRACT
The baker-transform and the related dynamical system is usually studied
with the unit-square as its state-space. In this paper we define the bakertransform
and related discrete-time dynarnical system with the non-negative
real numbers as state-space. Furthermore we consider the extension of the so
defined baker transform to real-valued functions. As a mathematical
instrument we use the generalized Walsh-Functions as originally introduced
by Fine (1950). The A-transform of Prigogine is proven to be identical to
dyadic convolution. It is shown that the method of Prigogine to construct
irreversible processes can also in this case be applied successfully.
O. lntroduction
The baker-transform is extensively studied in mathematics. lts representation as a Bemoulli
shift operation is wellknown [l], [2]. l. Prigogine and his coworkers have shown how the
baker-transform and the related discrete time dynamical system (baker-Prigogine dynamical
system) can be used to construct irreversible processes. The essential mathematical too! for
this is provided by the so-called A-transform [3], [4]. In a former report it was shown by the
author that by use of Walsh Fourier analysis, the A-transform can be considered as a discrete
dyadic convolution [5]. A similar approach has been suggested in [6]. In this paper we study
the baker transform on the non-negative real numbers R+. lt is shown that in this case the
generalized Walsh functions as originally defined by N.J. Fine [7] can be used to study
irreversibility in the line of Prigogine's work. The A-transform is equivalent in this case to the
dyadic convolution defined on the non-negative real numbers [O,oo).
1t is shown that dyadic differentiation as introduced by Butzer in analysis provides means to
construct specific examples of the A-transform.
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1. Preliminary Consideration
Prigogine (1977 Nobel price for chemistry) and his group explore complex phenomena by
bottom up methods starting at deterministic microscopic models with conservative dynamics
to be lifted up to macroscopic models with dissipative dynamics and collective variables for
state definition. The hope is, to explore possible emerging properties of attracting chaos which
can be observed in macroscopic pattems of behavior. Besides of physics and chemistry the
field of biology and socio-economics is considered to be promising for the application of this
approach in modeling.
In the work of Prigogine ([3], [4]) several examples serve for the demonstration of the
method. Besides of the horse-shoe map of Smale the baker transform, a specific case of a Ksystem
(K stands for Kolmogoroff) is used there to define conservative discrete-time
dynamical systems with chaotic behaviors. For the case of the state space Li (0,1 )2 , consisting
of the hilbert space of square-integrable functions on the unit-square [0,1) x [0,1), Prigogine
is able to construct a unitary transformation A which allows to transform a conservative
(reversible) dynamic process into a dissipative (and therefor irreversible) dynamic process
and vice versa.
2. Representation of the baker transform on [0,oo)
By D[O,oo) we denote the subgroup of the dyadic group D which consists of ali sequences u
which are 0-periodic to the left. Between [O,oo) and D[O,oo) we define the following (natural)
map e: (O,oo)~ D[O,oo). For sE[O,oo) with s = I,s;Z-; we define e(s):=u with u(i)=s¡ for
i=-N
ali i with -N$;i<oo and u(i):=O for ali i with i<-N. The map e is bijective and its inverse
e- 1:D[O,oo) ~ [O,oo) therefor exists. lt is easy to confirm that the Bemoulli-shiftoperation ~in
D[O,oo) corrresponds by e in [0,oo) to a multiplication by Z-1 s.
e-1 (u~ l) = Z-1 s (l)
From this result it is evident, that the baker-transform B - if restricted to the domain
Q2[0,l)x[O,l)- results in [O,oo) asan ordinary multiplication by 2-1• In detail this can be seen
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as follows: For se [O,oo) and the baker-transform B we define the following map
B-:[O,oo)-?[0,oo): For se [O,oo) the value B-(s) is given by
B-(s):=r(B{r"1(s))), (2)
where r is the bijective map r:Q2[0,l)x[0,1)-? [O,oo) given by r(x,y)=s with s = :Ls;T; and
i=-N
s¡=X-;+1 fori=0,-1 ,-2,-.. ,-N and s;=y¡ for i=l,2,3,. ... , where x= :f,x;T; and y= y= f,y;2-;.
i=l i=I
Since the baker transform B defined on Q2[0, 1) x [O, 1) is represented in D[O, oo) by a 1-step
shift to the "right" (Bemoulli shift) it follows from (1) that
B- (s)=2-1s. (3)
We call B- the real baker-transform. Although B- looks rather trivial it can be considered as
the representation of the (not so trivial) dyadic baker-transform on [O,oo). Next we study the
effect of the real baker-transform for function sfe L2[0,oo ). To begin with, we compute first
the real baker-transform of the generalized Walshfunctions 'lf(J),roe [0,00), as defined by
Fine[7]. Here we repeat this definition.
For OJ= LOJ;T; and s= Ls;T; thegeneralizedWalshfunction 'lf(J):[0,1)-?Risdefined
i=-N i=-M
by
r if LOJ;S¡_¡ e ven
i¡r.,(s)= (4)
-1 if L0J¡S¡_¡ odd
In line with our earlier definition of B- we define now the real baker-transform b_ of a
function feL2 [0,oo) by b-f(s)=f(B-(s)). Using (3) we have in general
b_f(s) = f(T' s) (5)
For the case of a generalized Walshfunctions 'l'(J) this results in
b. ('I'., )(s) = i¡r., {2-1 s) = 'l'r•., (s) since in general we ha ve vr., (2m s) = i¡r2 • ., (s) forme U:.
We see that for ali (roe [oo)) we have
b.(1/1.,) = '1' 2-1., (6)
This result is analogous to the result which was derived in [5] for the dyadic Walshfunctions.
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The theory of the generalized Walshfunctions as developed originally by Fine (7) shows that
each function f E L2 [o, oo) can be represented by an integral in the following forrn
f(s) = J F(m)r¡r.,(s)dw (7)
o
Here the function F denotes the generalized Walsh-Fouriertransforrn which is given by
F(m) = J f (s)r¡r.,(s)ds (8)
o
Using the representation (7) we get for b_ (f) the following Walsh-Fourier representation
- - -
b_ (f)(s) = f (B_ )s)) = J F(m)r¡r .,(B_ (s))dm = J F(m)r¡r.,(2-1 s)dm = J F(OJ)lfFi-•., (s)dw
o o
Substituting 2-1 m H ál gives
b_(f)(s) = J 2F(2m)r¡r.,(s)dm (9)
o
We see that the generalized WF-transforrn of b_(f) is given by
WFT(b_ (f))(ál) = 2F(2m) (10)
The formula (10) allows an effective computation of the real baker-transforrn b_ (f) by
means of the generalized WF-transforrn by "chasing" the following commutative diagram.
The computational efficiency is assured by the use of the existing fast algorithm for
computing the generalized WF-transforrn (WFFT-algorithm).
L2[0,oo)
b_
L2[0,.,.,
wnl IWff'
(1 l)
L2[0,oo) L2[0,.,.,
2F(2(.))
This concludes the mathematical analysis of the real baker-transforrn. The next chapters will
<leal with systemstheoretical problems.
3. Baker-Prigogine Dynamical Systems on L2[0,oo)
We have seen, that the baker-transforrn B, if restricted to dyadic points, can be represented on
[O,oo), using the real baker transforrn B_ by multiplication by 2-1; B-(s)=2-1s for SE [0,00) . The
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in verse baker-transformation B"1 is then represented by B-:_1 on [O,oo) by multiplication by 2;
B-:_1(s) = 2s . For the real baker-transformation of functions from L2[0,oo) we have shown by
(6)that b_l/I., =1/1 2• 1.,. Sirnilarlywehave
(12)
Furthermore we showed in (10) that WFI'(b_f)((J)) = 2F(2m). In a similar way it can be
shown that forfEL2[0,oo) we have
(13)
3.1 Real baker-dynamics
We define now in direct analogy to the definition of the baker-Gibbs dynamics which has
been developed in [5] the appropriate dynamics for the state-space L2[0,oo). The place of the
evolution operator u=b-1 of [5] is then taken by U_ = b-:_1 • A distribution function p:E~R
considered in the book of Nicolis-Prigogine [4] corresponds in our dynamics to a function
pe L2[0,oo ).
Let the elements of L2[0,oo) be considered as states of a discrete-time autonomous dynamical
system. For the state p1 of this dynamics at time t, the state Pt+l at time t+ 1 is computed by
P1+1 =U_p, (14)
The global state-transition function cp(U_) is then given by
P, =u:-•• Po (15)
We cal! the so defined dynamical system rb := ( qi(U. ), L2 [o, oo) [o, oo)) the real baker-dynamics.
By means of the Walsh-Fourierintegral representation of a state p1 at time t, which
is given by
p,(s) = J R1 (m)1¡t.,(s)dm (16)
o
we compute the next state Pi+• by
- -1 1
Pi+i (s) =(U p , )(s) = f R, (m)1¡t2.,(s)dm = f-R, (-m)l¡t .,(s)dw
o 02 2
This means, that (as expected by (13))
l 1
Rt+I (m) = 2 R, <2m) (17)
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Consequently, the WF-transform R, of p, can be computed from the initial state Po by
R, (m) = 2-<1-ro J R0 (TCt-ro >m) (18)
Next we want to investigate the "energy-function" H _ : L2 [o, oo) ~ [O, oo )of a real bak:er-dynamics
rb.
We have in general
H_(p,) = f p,2 (s)ds .
o
For the difference H _ (p r+t) - H _ (p,) we compute using ( 16), (17) and the parseval' s
equation
f- - R,2(m)dm-f R,2(m)dm =o
o o
This result shows that the real bak:er-dynarnics rb constitutes, as expected, a conservative
dynamical system.
3.2 A-Transformation for the real baker-dynamics
Following the methods, as developed by Prigogine et al. [3], [4], now we want to assign to a
real bak:er-dynarnics rb an irreversible dynamical system. Let the real bak:er-dynarnics rb as
before be given by rb=( <p(U _ ), L2 [o, oo) [o, oo )).
Let A denote a dyadic convolution operator A : L2 [o, oo) ~ L2 [O, oo) which is generated by a
weighting function l, that means that A = l ®. Now we require from A the following
properties:
For the WF-transform L of l we want that for
L(O)=l and O~m1 <üJ2 <oo itshouldfollowthat 1:2:L(üJ1)>L(m2)>0 (19)
In other words, by (19) we require that L starts with value 1 and is a positive monotonically
decreasing function in the hilbert space L2 [O, oo).
In analogy to our approach in [5) we use A as a state assignment map to establish a
dynamorphic image of the given bak:er-dynarnics rb.
p, =Ap, (te (0,oo) (20)
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The condition (19) as required in our case for a dyadic convolution operator A assures the
existence of the inverse operator A-1• For the evolution-operator W of the assigned
dynamorphic system we have therefor
W_ =AU_A-1 (21)
Similarly as above we define for the state space L2 [O, oo) an associated discrete-time
autonomous dynamical system with local state transition function w _. cp(W ) denotes the
associated global state transition function derived by iterating w _.
We call the dynamical system rbP := ( ¡p(W_ ), L2 [O, oo} [O, oo)) defined in this way, the real
bak:er-Prigogine dynamics.
Since U_ = A-1w_A and p, = A-1 p, we have a one to one correspondence between the real
bak:er-dynamics rb and the associated real bak:er-Prigogine dynamics rbP.
3.3 Irreversibility of the real baker-Prigogine dynamics
As in the case of the bak:er-Prigogine dynamics which is investigated in [5] there is also in the
case of a real bak:er-Prigogine dynamics rbP the proof of its irreversibility of most importance.
We approach this problem by investigating its "energy-function" H.. which is given by
H_(p,) := J p,2 (s)ds (22)
o
Let t>t0 ;:::o. Wewanttocompute H_(p,)-H_(p0 ). Wehave
- - - - H _ (p,) = J p12(s)ds = f .R,2(w)dw = f L2 (w)R,2(w)dw = f L2 (w)210-' Rg (2'0-' w)dw
o o o o
Substitution of 210 - 1 m ~ m gives finally H _ (p,) = f L2 (21-1º w)Rg (m)dm. With this result
o
we have then
H _ (p, )-H _ (p,.) = f (L2 (21-1° W)- L2 (W))Rg (m)dw <o
o
since for t>to follows by (19) L2(21-1° w)- L2 (w) <O.
This shows that the "energy-function" H.. of a real bak:er-Prigogine dynamics rbP is a
Ljapunov-functional and the transfonnation A2 is a Ljapunov-variable. Therefor the
dynamical system rbP = (q¡(W_ ),L2 [0,oo }[O,oo)) is irreversible.
As a result we see, that the A-transfonn of Prigogine, which is in our case of real function
analysis given by the dyadic convolution of real functions defined on [O,oo) provides a state
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isomorphisms to associate to a conservative reversible real baker-dynamics rb the
corresponding dissipative irreversible real baker-Prigogine dynamics rbP.
3.4 Realization by dyadic dift'erentiation
In this final section we investigate how the A-transformation of a real baker-dynamics can be
realized by dyadic differentiation. For a function f E L2 [O, oo) we denote by f C•l the (strong)
Gibbs-derivative, which is given by (see for example [9], [10])
It is known that for the generalized W alshfunctions rp.,, ( m E [o, oo)) , we have
V'~ l = líJl/f.,
(23)
(24)
(24) shows that the generalized Walshfunctions 'l'm are the eigenfunctions of the Gibbsdifferential
operator DC1l which is defined by D[ll¡ = ¡(11_
However, by (24) it is also evident that DC•l is a dyadic convolution operator, that means, that
there exists a function d E I! [o, oo ), such that DC•l = d ® and furthermore that ¡[ll can
consequently be represented by the integral
f 111(s) = f d(t ® s)f (t)dt (25)
o
Lct D denote the WF-transform of d. Then by the dyadic convolution theorem and
considering (24) we have
D( m) = m (26)
By mcans ofD we define the transformation L0 : [O,oo )~R by
Lo :=e-o
This menas, that we have
LD(m) =e_.,
We see, that LD (0) = 1 and furthermore that for O!!> m1 < m2 < oo we have
1;;:; LD(m1) > LD(m2 ) >O.
This shows that Lv meets the requirements (19). Consequently AD =ID® where ID
is given by
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(27)
(28)
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10 (s) = J L0 (m)r¡t., (s)dm
o
is a suitable A-transformation in the sense of Prigogine.
This means that (tp(W_(D)),L2 [0,oo1[0,oo), where
W_(D) = ADU Ai,
is a specific real baker-Prigogine dynamics.
(29)
(30)
The following generalization of this specific example of a real baker-Prigogine dynamics is
straightforward. For non-negative real numbers c1,c2, ••• ,cn we define by vl1l a polynomial
operator by
p(D[1J) := cpl1J + c2D[2J + ... + c.vl·l (31)
Application of p(D(1J) to 'I'., gives
p(Dl1l)l/f., = (c1Dl1l +c2Dl2l + ... +c.vl•l)l/f., = (c1m+c2m2 + ... +c.m")r¡t., (32)
Together with vl1l also p(Dl1l) is a dyadic convolution operator. Therefor there exists a
function p(d)e L2[0,oo) such that p(Dl1l) = p(d) ®. The WF-transform WFT(p(d)) of p(d)
is by (8) given here by the polynomial
p(D) = c1D+c2D 2 + ... +c.D" with p(D)(m) = c1m+c2m2 + ... +c.m"
Let now Lp<D> be defined by
L -e-p<D>
p(D) .-
Then wehave
L (m) ·- e -p(c,OJ+c2ai, + ... +c,01' )
p(D) .-
(33)
(34)
(35)
The function Lp<D> meets in analogy to L0 the requirements (19) and we have with the
dynamical system (tp(W_p(D)),L2 [0,oo)[O,oo) where
W_ (p(D)) =A p(Dp _A~<D>
a further example for a real baker-Prigogine dynamics.
lts A-transformation A p<D> is given by
Ap(D) = lp(D)@
(36)
(37)
The weighting function l p<D> can be computed again by the in verse WF-transformation of
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lp<D>(s) = J Lp<D>(m)l¡F.,(s)dm
o
4. Summary and Conclusion
(38)
Starting from the example of the wellknown 2-dimensional baker transform B as considered
in Prigogine works [3], [4] which has been investigated further by the author in [5], the
restriction of B on the set of dyadic rational numbers on the horizontal line is interpreted as
the real baker transform b~ for the set [O,oo) of nonnegative real numbers. B~ is trivially given
by the division of a real number s E [o, oo) by 2; B~ (s) = r' s. For real functions f the real
baker-transform b~ is defined by (b~f)(s) := f (T' s). In consequence it follows that the real-valued
W alshfunctions 1fJ., of Fine are with respect to b~ a closed set of functions since
With respect to the spectral domain of Walsh-Fourier analysis it follows, that the real baker
transform of functions results in an inverse real baker transform of the associated WalshFourier
Spectrum. This property allows to prove that the energy function of the discrete time
system rb with the real baker transform a state transition is conservative. However, if we use a
dyadic invertible convolution operator l with non-decreasing spectral representation A to
assign the states from rb to rbP (real baker Prigogine dynamics) we find that rbP is dissipative
and irreversible.
The paper shows how the well known example of Prigogine for the A-transform can be
demonstrated by the most simplest case with the interval [O,oo) as state space and where A is
defined by dyadic convolution. Furthermore our paper demonstrates the construction of
different possible A-transforms using the concept of dyadic differentiation. From our
viewpoint it might be possible to find models for practica! examples taken from physics,
chemistry or biology which use the approach of Prigogine to associate a (rnicroscopic)
dynamical system of conservative and reversible kind to a (mesoscopic) dynamical system
with collective variables which is dissipative and irreversible.
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The University of Chicago Press, Chicago 1973
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