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Hilbert space description on nonlinear discrete-time dynamical systems
Physica X 195
1993)137-148
North-Holland
Hilbert space description of nonlinear discrete-time dynamical systems Krzysztof Kowalski Department of Biophysics, Institute of Physiology and Biochemistry, Medical School of Lodz, 3 Lindley St., 90-131 Lodz, Poland Received 17 February 1992 Revised manuscript received 18 August 1992
It is shown that the nonlinear recurrences of the form x n~ 1 = f(xn), where f is analytic in xn, can be brought down to a linear recurrence in Hilbert space with the boson displacement operator.
1. Introduction
In recent years a lot of interest has been devoted to the study of nonlinear dynamical systems. Such systems arise in many branches of physics, engineering and other mathematically based sciences. Examples range from the simplest mechanical systems to the mathematical modelling of the pattern formation (morphogenesis). In a series of papers [1-5] (see also the book [6]) a method has been introduced for the reduction of nonlinear ordinary differential equations and partial differential equations of the evolution type to the abstract Schr6dingerlike equation in Hilbert space. The purpose of the present work is to generalize the treatment to the case of the nonlinear difference equations of the form xn+ 1 =f(xn), where f is analytic in x~ (they are also called recurrences, one-dimensional mappings or discrete-time dynamical systems). Following the scheme of the Hilbert space approach developed by the author, it is demonstrated that such nonlinear recurrences can be cast into a linear, abstract recurrence in Hilbert space with the displacement operator expressed in terms of Bose creation and annihilation operators. The introduced formalism is applied to the study of the linearization transformations for nonlinear difference equations and Feigenbaum-Cvitanovic renormalization equations. 0378-4371/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
138
K. Kowalski / Nonlinear discrete-time systems
2. The linear recurrence in Hilbert space
We first introduce the linear recurrence in Hilbert space corresponding to the following nonlinear recurrence: x,,+~ = f(x,,) ,
(1)
where n E Z+ (the set of non-negative integers) and f: R---~ R is analytic in x~. Proceeding analogously as in the case of ordinary differential equations [1] and partial differential equations [2], we introduce vectors of the form [x, n) = exp[ 1 (x°2 - X o2) ] l x . ) ,
(2)
where x, satisfies (1) and [x~) is the normalized coherent state (see appendix B).
Suppose we are given a boson operator such that
M = k=o ~ ~. a * k [ f ( a ) - a l k '
(3)
where a*, a are the standard Bose creation and annihilation operators. On using (1), (2) and (B.2) we find that the vectors (2) obey the following linear abstract difference equation in Hilbert space: Ix, n + 1) =
Mix, n) .
(4)
Now let xn(Xo) designate the solution of (1) and let Ix0, n) be the solution of (4). Taking into account (2) we find that the following eigenvalue equation holds true: alXo, n ) = x,(Xo)lXo, n ) .
(5)
It thus appears that the integration of (1) is equivalent to the solution of the linear abstract difference equation in Hilbert space (4). Following the scheme presented above, one finds easily that the formalism can be immediately generalized to include non-autonomous, complex multidimensional discrete dynamical systems. Notice that the formal solution to (4) is Ix0, n) = Mnlx0) .
(6)
On calculating the symbol ( x o [ M n l x o ) of the "evolution operator" M n and
K. Kowalski / Nonlinear discrete-time systems
139
using (B.3) and (B.10), one can easily derive the classical formula (7)
x~(Xo) = f f (Xo) .
We now discuss the connection of the actual treatment with the Carleman embedding approach to the nonlinear difference equations [6,7]. Consider eq. (4). On writing (4) in the occupation number representation (see appendix A) we arrive at the following equation: xm,+l =
~'~ M,~,~,xm,,,
(8)
m'EZ+ m X0
Xm 0-- ~
1 2
exp(-~Xo),
where Xm~ = ( m i x , n ) and Mmm, = ( m I g l m ' ) . Now, taking into account (2) and (B.5) we find that Xm~ is given by m Xmn =
~
Xn
1
2
exp(-- ~Xo).
(9)
It follows immediately from (9) that the nonlinear recurrence (1) is embedded into the linear recurrence (8). The Carleman linearization ansatz coincides up to the multiplicative factor ( 1 / ~ ) e x p ( - ½ x 0 2 ) with (9). Thus, it turns out that the Carleman embedding technique corresponds to the particular occupation number representation in the presented canonical Hilbert space approach. E x a m p l e 1. Consider the logistic equation
(10)
xn+ 1 = / z x n ( 1 - x , ) .
The operator M corresponding to (10) is of the form (see (A.4) and (A.5)) M = /.N~0= \( k)N /(-a)
k,
(11)
where N = a+a is the number operator. Taking into account (A.3) we find M n _-/z ~N
i 1. . . . .
n>~2.
N i1
~ in
n-1 N --
+ Ep= 1 ip )]
g'/zl(n-r);, (__ a ) g'/=~i~
lr+ 1
(12)
140
K. Kowalski / Nonlinear discrete-time systems
Hence, by virtue of (6), (9) and (B.5) the formal solution xn(Xo) of (10) can be written as (13a)
Xl(Xo) = (llMlxo) e x p ( ½ X o ) = ~ X o ( 1 - X o ) ,
x.(Xo) = (llM"lx0) exp(½x0)
n
(l)[n i
= /x x o
~.
il
i 1 , • . . ,i n
--
)]
r
1 + Ep= 1 . lp lr-
n 1>2.
/j,E~_ ~ (n_r)ir(__Xo)S~=l is
1
(13b)
3. Linearization transformations
Before proceeding with the linearization transformations we first discuss the case of the general functional equation of the form q b ( f ( x ) ) - - ~b(x),
(14)
where the real valued functions ~b, f and ~ are analytic in x E R. Consider the following identity, which is an immediate consequence of (5): aM = Mf(a) .
(15)
Taking into account (15) we find 6(a)M = M6(f(a)),
(16)
where ~b is an arbitrary analytic function. By virtue of (16) and (14) we have ¢k(a)M = M~b(a) .
(17)
On taking the Hermitian conjugation of (17) and using (B.7) we finally arrive at the following equation:
M*I4,) = 1¢'),
(18)
where I~b)= ~b(a*)10) and I~b)= ~(a*)10 ). It thus appears that the functional equation (14) can be brought down to the
K. Kowalski I Nonlinear discrete-time systems
141
Hilbert space equation (18). One finds easily that (14) corresponds to the particular Bargmann representation (see appendix B) for the abstract equation (18). On the other hand, writing (18) in the occupation number representation we get ~b(fo) = q*o,
(19a)
k
Fik[ f ~ . . . . .
f~ok-i+')]d~ (i)( fo) = q/ok) ,
(19b)
i=1
where k = 1, 2 . . . . . are given by
0% f~0r) =f{')(0), q/0k) = q/k)(0) and F i k [ f ' , . . . ,
F1k = f(I,) , Fi~,+, = f'F~_ak + F~k, Fek = f ' ~ .
f(k-i+l)] (20a)
1 < i < k + 1,
(20b) (20C)
The relations (19) and (20) can be derived by taking the nth derivative of both sides of (14). We have thus shown that the particular occupation number representation for the abstract equation (18) corresponds to the method of the power series (Maclaurin series) expansions. We now discuss the linearization transformations. Suppose that ~b: R--+ R is an analytic linearization transformation for eq. (1), that is rb(f(x)) = a~b(x),
(21)
where ~b is analytic in x and A-~ 1 is a constant. On putting in (14) q,(x)=A~b(x) and using (18) we find that the following equation holds: M*I4~ ) = AI~ ) ,
(22)
where I~b) = 4,(a*)10). Evidently, the linearization transformation is linked to the solution of (22) by 4~(x) = (xl~b> exp(½x2) ,
(23)
where Ix) is a normalized coherent state. It thus appears that the problem of determining analytic linearization transformations for the nonlinear recurrence (1) reduces to the solution of the Hilbert space eigenvalue equation (22).
142
K. Kowalski / Nonlinear discrete-time systems
Example 2. Consider the linearization transformations for the logistic equation
(10). We make the ansatz
[g(N)a +
h ( N ) ] l 4,) = 10),
(24)
where g and h are analytic in N. The ansatz (24) follows from the assumption that 4,n = ( n 14,) satisfies the linear first order difference equation. On using (01 4,) = 0 and putting (114,) = 1 we obtain g(0) = 1. Notice that such ansatz is equivalent to the following one: [~(N)a* +/~(N)]I 4,> = 11),
(25)
where/~(1) = 1. Setting g ( N ) ~ 1 and h ( N ) = v N and using (A.2) and (B.7) we find easily the linearization transformation for (10), where ~ = 2 such that 6(x) = - ½In(1 - 2 x ) .
(26)
The solution of (10), where ~ = 2 generated by (26) is x.(Xo) = ½[1 - (1 - 2Xo)2"].
(27)
As it pointed out by a referee the solution (27) can be also obtained via the linear transformation ~0(x) = 1 - 2 x ,
(28)
reducing (10), where /.~ = 2 to the solution of the equation In+ 1 = X 2. The well-known solution of (10) for ~ = 4 can be derived easily with the help of the actual treatment by involving in (24) the terms quadratic in N (we set g ( N ) = a N + 1 and h ( N ) = ~ N 2 ) . Finally, we note that we could apply (19), where 0~0*) = A4,~0~) in the study of linearization transformations for eq. (1). However, an experience with the logistic equation (10) shows that we then obtain from (19) the relations of the form "~0 ,,~(n+,) = cn(/~ ) 4,(o,) with unknown general form of c n for arbitrary n, and the approach based on the ansatz (24) is more effective.
4. Feigenbaum-Cvitanovic renormalization equations We now discuss the Feigenbaum-Cvitanovic renormalization equations (universal equations) [8] within the actual Hilbert space approach. Consider the
K. Kowalski / Nonlinear discrete-time systems
143
universal equation of the form
f(/3x) =/3f(f(x)),
(29)
where f is analytic in x and /3 is a scaling parameter. On setting in (14) qb(x)=-/3f(x), ~b(x)-f(/3x) and using (18) together with (A.2), (A.4) and (A.5) we arrive at the following equation:
=/3N-1lf),
M*lf)
(30)
where I f ) = f ( a + ) 1 0 ) • The solution to (30) and the solution to (29) are related by
f(x) = exp( lix),z
(31)
where Ix) is a normalized coherent state. Thus, it turns out that the universal equation (29) can be brought down to the abstract equation in Hilbert space (30).
Example 3. On putting in the ansatz (25) th = f and g(N) -= v, /~(N) - 1 we immediately get a t
f(a*)
-
(32)
. 1 + yam
-
-
Hence, using (29), we obtain the solution to the universalequation (29), where /3 = 2 such that X
f(x) = - -
1 - 3'x
,
(33)
where 3' is a constant. The function (33) was originally found in ref. [9]. We now examine the case of the occupation number representation for the abstract equation (30), that is, we take into consideration eq. (19). By virtue of the identities ~b(x)-=/3f(x) and O(x)=-f(/3x), these equations take the form
f(fo) =/3-if0 ,
(34a)
k
i=l
Fik[f; . . . . . f(ok-i+l)lf(')(fo) = /3k-lf(0k) ,
where k = 1, 2 . . . . .
oo and Fik are given by (20).
(34b)
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K. Kowalski / Nonlinear discrete-time systems
E x a m p l e 4. Consider first the case f0 ¢ 0. On calculating (34b) for k = 1, 2, putting f ~ ¢ 0 and making use of (34a) we arrive at the following equation:
_
f0 f(fo)
f'(Yo)
(35)
Solving this trivial equation we get f ( x ) = V ' x 5 + 3",
(36)
where 3" is an integration constant. Now, taking into account (34a) we find that (36) is the solution of the universal equation (29), where /3 = 1 / ~ . It can be checked easily that the function (36) is a member of the family of solutions to (29) of the form
L ( x ) = (x + 3")1,o ,
(37)
where the corresponding scaling parameters are/34 = 2-1/% We now discuss the case f0 = 0. On setting f0 = 0 and using (34b) together with (20) one finds easily the solution to (34b) such that i!
f ; = 1,
/3 = 2 ,
-_
f(oi) = 2i_ 1 f ; '
' ,
i = 2, 3, . . . ,
(38)
where fo ~ 0. Hence, putting f'~/2 = 3' we arrive at the solution of the universal equation (29) of the form c¢
f(x) = ~ 3"i-lxii=1
(39)
x___x___ 1-3"x '
that is, the solution (33) is obtained. Making use of (34b) and (20) the author established the existence of the solution to eq. (34b), satisfying f0 = 0 ,
t
__
f0 - 1
~
tt__
f0 - 0 ,
/3 = X/2,
f (02') = 0 ,
(4o) f(2i+l)
o
ttt
wtti
= cif o ,
i = 2, 3 . . . . .
where f 0 ~ 0. A few first terms in the power series expansion for the corresponding universal function f ( x ) are
K. Kowalski / Nonlinear discrete-time systems
f(x)=
x +
1 x3
+
10 x5
+
350
x
7
+
17150 9 x
+
145
(x11) ,
(41)
where we put f'~' = 1.
5. Conclusions In the present work it is shown that very general nonlinear recurrences (embracing everything that arises in practice) may be cast into the linear recurrence in Hilbert space. As in the case with ordinary differential equations [1] and partial differential equations [2], the actual treatment generalizes the Carleman linearization approach [6,7] which corresponds to the particular occupation number representation. The scheme of linearization in Hilbert space described herein, can be treated as a generalization of the classical method of variation of constants to the case of difference equations. Indeed, eq. (22) has the structure analogous to the equations arising in the linearization of nonlinear ordinary and partial differential equations [5] which have been shown to generalize the method of variation of constants. The introduced formalism has been applied to the study of linearization transformations and Feigenbaum-Cvitanovic renormalization equations. The simplicity and ubiquity of the approach based on the ansatz (24) suggest that it could be a useful tool in the investigation of functional equations (14) arising in the theory of nonlinear recurrences. On the other hand, the examples of applications of the recursive setting (19) and (20) for obtaining coefficients of a composite function presented herein, indicate that it could be of importance in the study of universal equations. As a matter of fact one could apply instead of (19) and (20), the method of the substitution of a series into series or the Faa di Bruno formula. Nevertheless, such methods are much more complicated.
Acknowledgements I would like to thank a referee for helpful comments. This work was supported by KBN grant 2 0903 91 01.
Appendix A. Occupation number representation We first recall the basic properties of Bose creation and annihilation operators a t, a. These operators obey the Heisenberg algebra
146
K. K o w a l s k i / N o n l i n e a r discrete-time systems
[a, a*] = I ,
[a, I1 = [a*, I] = 0 ,
(A.1)
where I is the identity operator. Let f : R ~ R be an analytic function. The commutation relations (A. 1) imply [a, f(a+)] = df(a+)/da + .
(A.2)
Suppose now we are given the number operator N = a*a. It can be easily checked that the following formulae hold true:
akf(N) = f ( N + k)a k ,
1 k~
k-1
a+kak = ~1 1--[ ( N - j ) =
(A.3)
(k)
,
(A.4)
]=0
k N k=0
Let us assume that there exists in the Hilbert space ~ , in which the Bose operators act, a unique, normalized vector [0) (vacuum vector) such that al 0) = 0.
(A.6)
We also assume that there is no nontrivial closed subspace of ~ which is invariant under the action of operators a, a +. The state vectors In), n E Z+, defined as follows: a tn
In) = ~
10),
(A.7)
are the eigenvectors of the number operator, that is
Uln) = nln).
(A.8)
These vectors form an orthonormal and complete set, i.e. we have
(nln') = a,,,,
In)
(A.9) I.
(A.IO)
n•Z+
The action of the Bose operators on the vectors
In )
is of the form
K. Kowalski / Nonlinear discrete-time systems
aln)=x/-~ln-1),
147
a*ln)=~ln+ l).
(A.11)
It follows that n!
~1/2
akln)=((n-k)!/ In-k)' a'kln)=((n+k)')n!
1/2
In + k ) .
(A.12)
Appendix B. Coherent states
We now briefly sketch the basic properties of coherent states. The coherent states ]z), where z E C are usually defined as eigenvectors of the annihilation operators, i.e.
alz)
= zlz) •
(B.1)
The normalized coherent states can be also defined by Iz ) = exp(- ½Izl =) exp(za*)lO ) .
(B.2)
These states form the overcomplete set. We have (z I w) = exp[- ½(Izl 2 + I wl ~ - 2 z ' w ) ] ,
(B.3)
f d/z(z)Iz>(zl
(B.4)
= I,
R2
where the asterisk designates the complex conjugation and d/x(z)= (1/'rr) d(Rez) d(Im z). The passage from the occupation number representation to the coherent states representation is given by z
(nlz) = ~
n
exp(- ½lz]:).
(B.5)
Now let ]~b) be an arbitrary state. Using (A.10) and (B.5), we find that the function ~b(z*)= (z[ ~b) can be written in the following form:
~(z*) -- 8(~,) exp(- ½Iz12),
(B.6)
where q~(z*) is an analytic (entire) function. Taking into account (B.1) and (B.5) we arrive at the following abstract form of (B.6),
148
K. Kowalski / Nonlinear discrete-time systems
(B.7)
= ~(a*)lo). Making use of (B.4) and (B.6) we obtain the following relation: (4~10) = f d / z ( z ) e x p ( - l z l 2) ~ * ( z * ) ~ ( z * ) .
(B.8)
R2
The representation (B.8) is called the Bargmann representation. The action of the Bose operators in this representation is of the form d
TF*
atqb(z *) = z * ~ b ( z * ) .
(B.9)
T h e counterpart of the Dirac delta function in the Bargmann representation is the reproducing kernel ~ ( w * , z) defined by
q~(w*) = f d/x(z) exp(-Izl 2) :~(w*, z) ,~(z*),
(B.IO)
R2
where ~ ( w * , z ) = exp(w*z).
References [1] [2] [3] [4] [5] [6]
K. Kowalski, Physica A 145 (1987) 408. K. Kowalski, Physica A 152 (1988) 98. K. Kowalski and W.-H. Steeb, Progr. Theor. Phys. 85 (1991) 713. K. Kowalski and W.-H. Steeb, Progr. Theor. Phys. 85 (1991) 975. K. Kowalski, Physica A 180 (1992) 156. K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization (World Scientific, Singapore, 1991). [7] W.-H. Steeb, in: Nonlinear Phenomena in Chemical Dynamics (Springer, Berlin, 1981). [8] P. Cvitanovic, ed., Universality in Chaos (Adam Hilger, Bristol, 1984). [9] J.E. Hirsch, M. Nauenberg and D.J. Scalapino, Phys. Lett. A 87 (1982) 391.
1993)137-148
North-Holland
Hilbert space description of nonlinear discrete-time dynamical systems Krzysztof Kowalski Department of Biophysics, Institute of Physiology and Biochemistry, Medical School of Lodz, 3 Lindley St., 90-131 Lodz, Poland Received 17 February 1992 Revised manuscript received 18 August 1992
It is shown that the nonlinear recurrences of the form x n~ 1 = f(xn), where f is analytic in xn, can be brought down to a linear recurrence in Hilbert space with the boson displacement operator.
1. Introduction
In recent years a lot of interest has been devoted to the study of nonlinear dynamical systems. Such systems arise in many branches of physics, engineering and other mathematically based sciences. Examples range from the simplest mechanical systems to the mathematical modelling of the pattern formation (morphogenesis). In a series of papers [1-5] (see also the book [6]) a method has been introduced for the reduction of nonlinear ordinary differential equations and partial differential equations of the evolution type to the abstract Schr6dingerlike equation in Hilbert space. The purpose of the present work is to generalize the treatment to the case of the nonlinear difference equations of the form xn+ 1 =f(xn), where f is analytic in x~ (they are also called recurrences, one-dimensional mappings or discrete-time dynamical systems). Following the scheme of the Hilbert space approach developed by the author, it is demonstrated that such nonlinear recurrences can be cast into a linear, abstract recurrence in Hilbert space with the displacement operator expressed in terms of Bose creation and annihilation operators. The introduced formalism is applied to the study of the linearization transformations for nonlinear difference equations and Feigenbaum-Cvitanovic renormalization equations. 0378-4371/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
138
K. Kowalski / Nonlinear discrete-time systems
2. The linear recurrence in Hilbert space
We first introduce the linear recurrence in Hilbert space corresponding to the following nonlinear recurrence: x,,+~ = f(x,,) ,
(1)
where n E Z+ (the set of non-negative integers) and f: R---~ R is analytic in x~. Proceeding analogously as in the case of ordinary differential equations [1] and partial differential equations [2], we introduce vectors of the form [x, n) = exp[ 1 (x°2 - X o2) ] l x . ) ,
(2)
where x, satisfies (1) and [x~) is the normalized coherent state (see appendix B).
Suppose we are given a boson operator such that
M = k=o ~ ~. a * k [ f ( a ) - a l k '
(3)
where a*, a are the standard Bose creation and annihilation operators. On using (1), (2) and (B.2) we find that the vectors (2) obey the following linear abstract difference equation in Hilbert space: Ix, n + 1) =
Mix, n) .
(4)
Now let xn(Xo) designate the solution of (1) and let Ix0, n) be the solution of (4). Taking into account (2) we find that the following eigenvalue equation holds true: alXo, n ) = x,(Xo)lXo, n ) .
(5)
It thus appears that the integration of (1) is equivalent to the solution of the linear abstract difference equation in Hilbert space (4). Following the scheme presented above, one finds easily that the formalism can be immediately generalized to include non-autonomous, complex multidimensional discrete dynamical systems. Notice that the formal solution to (4) is Ix0, n) = Mnlx0) .
(6)
On calculating the symbol ( x o [ M n l x o ) of the "evolution operator" M n and
K. Kowalski / Nonlinear discrete-time systems
139
using (B.3) and (B.10), one can easily derive the classical formula (7)
x~(Xo) = f f (Xo) .
We now discuss the connection of the actual treatment with the Carleman embedding approach to the nonlinear difference equations [6,7]. Consider eq. (4). On writing (4) in the occupation number representation (see appendix A) we arrive at the following equation: xm,+l =
~'~ M,~,~,xm,,,
(8)
m'EZ+ m X0
Xm 0-- ~
1 2
exp(-~Xo),
where Xm~ = ( m i x , n ) and Mmm, = ( m I g l m ' ) . Now, taking into account (2) and (B.5) we find that Xm~ is given by m Xmn =
~
Xn
1
2
exp(-- ~Xo).
(9)
It follows immediately from (9) that the nonlinear recurrence (1) is embedded into the linear recurrence (8). The Carleman linearization ansatz coincides up to the multiplicative factor ( 1 / ~ ) e x p ( - ½ x 0 2 ) with (9). Thus, it turns out that the Carleman embedding technique corresponds to the particular occupation number representation in the presented canonical Hilbert space approach. E x a m p l e 1. Consider the logistic equation
(10)
xn+ 1 = / z x n ( 1 - x , ) .
The operator M corresponding to (10) is of the form (see (A.4) and (A.5)) M = /.N~0= \( k)N /(-a)
k,
(11)
where N = a+a is the number operator. Taking into account (A.3) we find M n _-/z ~N
i 1. . . . .
n>~2.
N i1
~ in
n-1 N --
+ Ep= 1 ip )]
g'/zl(n-r);, (__ a ) g'/=~i~
lr+ 1
(12)
140
K. Kowalski / Nonlinear discrete-time systems
Hence, by virtue of (6), (9) and (B.5) the formal solution xn(Xo) of (10) can be written as (13a)
Xl(Xo) = (llMlxo) e x p ( ½ X o ) = ~ X o ( 1 - X o ) ,
x.(Xo) = (llM"lx0) exp(½x0)
n
(l)[n i
= /x x o
~.
il
i 1 , • . . ,i n
--
)]
r
1 + Ep= 1 . lp lr-
n 1>2.
/j,E~_ ~ (n_r)ir(__Xo)S~=l is
1
(13b)
3. Linearization transformations
Before proceeding with the linearization transformations we first discuss the case of the general functional equation of the form q b ( f ( x ) ) - - ~b(x),
(14)
where the real valued functions ~b, f and ~ are analytic in x E R. Consider the following identity, which is an immediate consequence of (5): aM = Mf(a) .
(15)
Taking into account (15) we find 6(a)M = M6(f(a)),
(16)
where ~b is an arbitrary analytic function. By virtue of (16) and (14) we have ¢k(a)M = M~b(a) .
(17)
On taking the Hermitian conjugation of (17) and using (B.7) we finally arrive at the following equation:
M*I4,) = 1¢'),
(18)
where I~b)= ~b(a*)10) and I~b)= ~(a*)10 ). It thus appears that the functional equation (14) can be brought down to the
K. Kowalski I Nonlinear discrete-time systems
141
Hilbert space equation (18). One finds easily that (14) corresponds to the particular Bargmann representation (see appendix B) for the abstract equation (18). On the other hand, writing (18) in the occupation number representation we get ~b(fo) = q*o,
(19a)
k
Fik[ f ~ . . . . .
f~ok-i+')]d~ (i)( fo) = q/ok) ,
(19b)
i=1
where k = 1, 2 . . . . . are given by
0% f~0r) =f{')(0), q/0k) = q/k)(0) and F i k [ f ' , . . . ,
F1k = f(I,) , Fi~,+, = f'F~_ak + F~k, Fek = f ' ~ .
f(k-i+l)] (20a)
1 < i < k + 1,
(20b) (20C)
The relations (19) and (20) can be derived by taking the nth derivative of both sides of (14). We have thus shown that the particular occupation number representation for the abstract equation (18) corresponds to the method of the power series (Maclaurin series) expansions. We now discuss the linearization transformations. Suppose that ~b: R--+ R is an analytic linearization transformation for eq. (1), that is rb(f(x)) = a~b(x),
(21)
where ~b is analytic in x and A-~ 1 is a constant. On putting in (14) q,(x)=A~b(x) and using (18) we find that the following equation holds: M*I4~ ) = AI~ ) ,
(22)
where I~b) = 4,(a*)10). Evidently, the linearization transformation is linked to the solution of (22) by 4~(x) = (xl~b> exp(½x2) ,
(23)
where Ix) is a normalized coherent state. It thus appears that the problem of determining analytic linearization transformations for the nonlinear recurrence (1) reduces to the solution of the Hilbert space eigenvalue equation (22).
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K. Kowalski / Nonlinear discrete-time systems
Example 2. Consider the linearization transformations for the logistic equation
(10). We make the ansatz
[g(N)a +
h ( N ) ] l 4,) = 10),
(24)
where g and h are analytic in N. The ansatz (24) follows from the assumption that 4,n = ( n 14,) satisfies the linear first order difference equation. On using (01 4,) = 0 and putting (114,) = 1 we obtain g(0) = 1. Notice that such ansatz is equivalent to the following one: [~(N)a* +/~(N)]I 4,> = 11),
(25)
where/~(1) = 1. Setting g ( N ) ~ 1 and h ( N ) = v N and using (A.2) and (B.7) we find easily the linearization transformation for (10), where ~ = 2 such that 6(x) = - ½In(1 - 2 x ) .
(26)
The solution of (10), where ~ = 2 generated by (26) is x.(Xo) = ½[1 - (1 - 2Xo)2"].
(27)
As it pointed out by a referee the solution (27) can be also obtained via the linear transformation ~0(x) = 1 - 2 x ,
(28)
reducing (10), where /.~ = 2 to the solution of the equation In+ 1 = X 2. The well-known solution of (10) for ~ = 4 can be derived easily with the help of the actual treatment by involving in (24) the terms quadratic in N (we set g ( N ) = a N + 1 and h ( N ) = ~ N 2 ) . Finally, we note that we could apply (19), where 0~0*) = A4,~0~) in the study of linearization transformations for eq. (1). However, an experience with the logistic equation (10) shows that we then obtain from (19) the relations of the form "~0 ,,~(n+,) = cn(/~ ) 4,(o,) with unknown general form of c n for arbitrary n, and the approach based on the ansatz (24) is more effective.
4. Feigenbaum-Cvitanovic renormalization equations We now discuss the Feigenbaum-Cvitanovic renormalization equations (universal equations) [8] within the actual Hilbert space approach. Consider the
K. Kowalski / Nonlinear discrete-time systems
143
universal equation of the form
f(/3x) =/3f(f(x)),
(29)
where f is analytic in x and /3 is a scaling parameter. On setting in (14) qb(x)=-/3f(x), ~b(x)-f(/3x) and using (18) together with (A.2), (A.4) and (A.5) we arrive at the following equation:
=/3N-1lf),
M*lf)
(30)
where I f ) = f ( a + ) 1 0 ) • The solution to (30) and the solution to (29) are related by
f(x) =
(31)
where Ix) is a normalized coherent state. Thus, it turns out that the universal equation (29) can be brought down to the abstract equation in Hilbert space (30).
Example 3. On putting in the ansatz (25) th = f and g(N) -= v, /~(N) - 1 we immediately get a t
f(a*)
-
(32)
. 1 + yam
-
-
Hence, using (29), we obtain the solution to the universalequation (29), where /3 = 2 such that X
f(x) = - -
1 - 3'x
,
(33)
where 3' is a constant. The function (33) was originally found in ref. [9]. We now examine the case of the occupation number representation for the abstract equation (30), that is, we take into consideration eq. (19). By virtue of the identities ~b(x)-=/3f(x) and O(x)=-f(/3x), these equations take the form
f(fo) =/3-if0 ,
(34a)
k
i=l
Fik[f; . . . . . f(ok-i+l)lf(')(fo) = /3k-lf(0k) ,
where k = 1, 2 . . . . .
oo and Fik are given by (20).
(34b)
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K. Kowalski / Nonlinear discrete-time systems
E x a m p l e 4. Consider first the case f0 ¢ 0. On calculating (34b) for k = 1, 2, putting f ~ ¢ 0 and making use of (34a) we arrive at the following equation:
_
f0 f(fo)
f'(Yo)
(35)
Solving this trivial equation we get f ( x ) = V ' x 5 + 3",
(36)
where 3" is an integration constant. Now, taking into account (34a) we find that (36) is the solution of the universal equation (29), where /3 = 1 / ~ . It can be checked easily that the function (36) is a member of the family of solutions to (29) of the form
L ( x ) = (x + 3")1,o ,
(37)
where the corresponding scaling parameters are/34 = 2-1/% We now discuss the case f0 = 0. On setting f0 = 0 and using (34b) together with (20) one finds easily the solution to (34b) such that i!
f ; = 1,
/3 = 2 ,
-_
f(oi) = 2i_ 1 f ; '
' ,
i = 2, 3, . . . ,
(38)
where fo ~ 0. Hence, putting f'~/2 = 3' we arrive at the solution of the universal equation (29) of the form c¢
f(x) = ~ 3"i-lxii=1
(39)
x___x___ 1-3"x '
that is, the solution (33) is obtained. Making use of (34b) and (20) the author established the existence of the solution to eq. (34b), satisfying f0 = 0 ,
t
__
f0 - 1
~
tt__
f0 - 0 ,
/3 = X/2,
f (02') = 0 ,
(4o) f(2i+l)
o
ttt
wtti
= cif o ,
i = 2, 3 . . . . .
where f 0 ~ 0. A few first terms in the power series expansion for the corresponding universal function f ( x ) are
K. Kowalski / Nonlinear discrete-time systems
f(x)=
x +
1 x3
+
10 x5
+
350
x
7
+
17150 9 x
+
145
(x11) ,
(41)
where we put f'~' = 1.
5. Conclusions In the present work it is shown that very general nonlinear recurrences (embracing everything that arises in practice) may be cast into the linear recurrence in Hilbert space. As in the case with ordinary differential equations [1] and partial differential equations [2], the actual treatment generalizes the Carleman linearization approach [6,7] which corresponds to the particular occupation number representation. The scheme of linearization in Hilbert space described herein, can be treated as a generalization of the classical method of variation of constants to the case of difference equations. Indeed, eq. (22) has the structure analogous to the equations arising in the linearization of nonlinear ordinary and partial differential equations [5] which have been shown to generalize the method of variation of constants. The introduced formalism has been applied to the study of linearization transformations and Feigenbaum-Cvitanovic renormalization equations. The simplicity and ubiquity of the approach based on the ansatz (24) suggest that it could be a useful tool in the investigation of functional equations (14) arising in the theory of nonlinear recurrences. On the other hand, the examples of applications of the recursive setting (19) and (20) for obtaining coefficients of a composite function presented herein, indicate that it could be of importance in the study of universal equations. As a matter of fact one could apply instead of (19) and (20), the method of the substitution of a series into series or the Faa di Bruno formula. Nevertheless, such methods are much more complicated.
Acknowledgements I would like to thank a referee for helpful comments. This work was supported by KBN grant 2 0903 91 01.
Appendix A. Occupation number representation We first recall the basic properties of Bose creation and annihilation operators a t, a. These operators obey the Heisenberg algebra
146
K. K o w a l s k i / N o n l i n e a r discrete-time systems
[a, a*] = I ,
[a, I1 = [a*, I] = 0 ,
(A.1)
where I is the identity operator. Let f : R ~ R be an analytic function. The commutation relations (A. 1) imply [a, f(a+)] = df(a+)/da + .
(A.2)
Suppose now we are given the number operator N = a*a. It can be easily checked that the following formulae hold true:
akf(N) = f ( N + k)a k ,
1 k~
k-1
a+kak = ~1 1--[ ( N - j ) =
(A.3)
(k)
,
(A.4)
]=0
k N k=0
Let us assume that there exists in the Hilbert space ~ , in which the Bose operators act, a unique, normalized vector [0) (vacuum vector) such that al 0) = 0.
(A.6)
We also assume that there is no nontrivial closed subspace of ~ which is invariant under the action of operators a, a +. The state vectors In), n E Z+, defined as follows: a tn
In) = ~
10),
(A.7)
are the eigenvectors of the number operator, that is
Uln) = nln).
(A.8)
These vectors form an orthonormal and complete set, i.e. we have
(nln') = a,,,,
In)
(A.9) I.
(A.IO)
n•Z+
The action of the Bose operators on the vectors
In )
is of the form
K. Kowalski / Nonlinear discrete-time systems
aln)=x/-~ln-1),
147
a*ln)=~ln+ l).
(A.11)
It follows that n!
~1/2
akln)=((n-k)!/ In-k)' a'kln)=((n+k)')n!
1/2
In + k ) .
(A.12)
Appendix B. Coherent states
We now briefly sketch the basic properties of coherent states. The coherent states ]z), where z E C are usually defined as eigenvectors of the annihilation operators, i.e.
alz)
= zlz) •
(B.1)
The normalized coherent states can be also defined by Iz ) = exp(- ½Izl =) exp(za*)lO ) .
(B.2)
These states form the overcomplete set. We have (z I w) = exp[- ½(Izl 2 + I wl ~ - 2 z ' w ) ] ,
(B.3)
f d/z(z)Iz>(zl
(B.4)
= I,
R2
where the asterisk designates the complex conjugation and d/x(z)= (1/'rr) d(Rez) d(Im z). The passage from the occupation number representation to the coherent states representation is given by z
(nlz) = ~
n
exp(- ½lz]:).
(B.5)
Now let ]~b) be an arbitrary state. Using (A.10) and (B.5), we find that the function ~b(z*)= (z[ ~b) can be written in the following form:
~(z*) -- 8(~,) exp(- ½Iz12),
(B.6)
where q~(z*) is an analytic (entire) function. Taking into account (B.1) and (B.5) we arrive at the following abstract form of (B.6),
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K. Kowalski / Nonlinear discrete-time systems
(B.7)
= ~(a*)lo). Making use of (B.4) and (B.6) we obtain the following relation: (4~10) = f d / z ( z ) e x p ( - l z l 2) ~ * ( z * ) ~ ( z * ) .
(B.8)
R2
The representation (B.8) is called the Bargmann representation. The action of the Bose operators in this representation is of the form d
TF*
atqb(z *) = z * ~ b ( z * ) .
(B.9)
T h e counterpart of the Dirac delta function in the Bargmann representation is the reproducing kernel ~ ( w * , z) defined by
q~(w*) = f d/x(z) exp(-Izl 2) :~(w*, z) ,~(z*),
(B.IO)
R2
where ~ ( w * , z ) = exp(w*z).
References [1] [2] [3] [4] [5] [6]
K. Kowalski, Physica A 145 (1987) 408. K. Kowalski, Physica A 152 (1988) 98. K. Kowalski and W.-H. Steeb, Progr. Theor. Phys. 85 (1991) 713. K. Kowalski and W.-H. Steeb, Progr. Theor. Phys. 85 (1991) 975. K. Kowalski, Physica A 180 (1992) 156. K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization (World Scientific, Singapore, 1991). [7] W.-H. Steeb, in: Nonlinear Phenomena in Chemical Dynamics (Springer, Berlin, 1981). [8] P. Cvitanovic, ed., Universality in Chaos (Adam Hilger, Bristol, 1984). [9] J.E. Hirsch, M. Nauenberg and D.J. Scalapino, Phys. Lett. A 87 (1982) 391.