Analysis of Carleman Representation of Analytical Recursions

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 224, ARTICLE NO. AY Analysis of Carleman Representation of Analytical Recursions G. Berolaio Department of Physics, Bar-Ilan Uniersity, Ramat-Gan, Israel and Department of Mathematics, Uniersity of Strathclyde, G1 1XH Glasgow, United Kingdom and S. Rabinovich and S. Havlin Department of Physics, Bar-Ilan Uniersity, Ramat-Gan, Israel Submitted by Rai P. Agarwal Received September 15, 1997 We study a general method to map a nonlinear analytical recursion onto a linear one. The solution of the recursion is represented as a product of matrices whose elements depend only on the form of the recursion and not on initial conditions. First we consider the method for polynomial recursions of arbitrary degree and then the method is generalized to analytical recursions. Some properties of these matrices, such as the existence of an inverse matrix and diagonalization, are also studied Academic Press 1. INTRODUCTION In our recent wor 6, 7 we found a correspondence between the logistic map and an infinite-dimensional linear recursion. This correspondence was exploited to obtain a rather general method for the mapping of a polynomial recursion to a linear Ž but infinite-dimensional. one, which was reported in 7. However, studying the literature, we found that our wor is a rediscovery of a nown approach called Carleman linearization 2, 5, 9, 10. * gregory@gauss.maths. strath.ac.u X98 $25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.

2 82 BERKOLAIKO, RABINOVICH, AND HAVLIN Some questions of Carleman linearization, such as the applicability of the method for analytical recursions, are still open. In this paper we thoroughly study this topic, prove convergence of series used in the method, and reveal some useful properties of the linearization, which allows one to understand the correspondence between the original and linearized recursions. The basis of the method is the construction of a transfer matrix, T T 4. This allows one to represent the solution in the form i i, 0 yn et n y, where e is the row Ž transpose. vector 0, 1, 0, 0,... 4 T, y is the column vector n of initial values 1, y, y, y,...,andt is the nth power of the infinite-dimensional matrix T. For introductory purposes we explain the method first for the polynomial recursions Žsee also. 7 and then prove its extension to the general case of analytical recursions. 2. POLYNOMIAL RECURSIONS Here we consider a first-order recursion equation y PŽ y., Ž 1. n1 where P x is a polynomial of degree m: n m P x Ý a x, a m Let y y be an initial value for the recursion Ž We denote by y the column vector of powers of y, y y 4 0, and set the vector e to be the T row vector e It should be emphasized that runs from 0, since in the general case a0 0. In this notation ey is a scalar product that yields ey y. Ž 3. THEOREM 1. For any recursion of type Eq. Ž. 1 there exists a matrix TT 4, defined by such that, 0 ž / m m i Ý i Ý i0 0 PŽ y. ay T y, 0,1,..., Ž 4. y et n y. Ž 5. n The matrix power T n exists for all n and all the operations in the right-hand side of Eq. Ž. 5 are associatie.

3 ANALYSIS OF ANALYTICAL RECURSIONS 83 Proof. For n 0 the statement of the theorem is valid Žsee Eq. Ž 3... def 4 We introduce the column vector y y, where y PŽ y Equation Ž. 4 implies that y1 Ty. Ž 6. Now, by analogy to Eq. Ž. 3, we have y1 ey1 ety. Therefore, the statement of the theorem is true for n 1 as well. Assume that Eq. Ž. 5 is valid for n l for any initial value y. Then y l1 l can be represented as y et y, where y PŽ y. l1 1 1 is considered as a new initial value of the recursion. Then, using Eq. Ž. 6, one obtains yl1 et l y1 et l Ty et l1 y. Note that for and satisfying m we have T 0. Therefore, each row is finite Ži.e., there is only a finite number of nonzero matrix elements in each row.. this proves the existence of powers of T and associativity in Eq. Ž. 5. Thus, the proof is complete. EXAMPLE 1. As one can see, in the general case elements of the matrix T have a form of rather complicated sums. However, they reduce to a fairly simple expression, when the polynomial Ž. 2 has only two terms. To demonstrate this, let us consider the recursion Žfor a good reference on this subect, see. 1 yn1 ynž 1 yn. with y0 y. Ž 7. In this case one has 2 i i Ž i.2 i i0 i Ý ž/ P y yy y. Denoting i, we have and the matrix elements T 2 Ý ž / PŽ y. Ž 1. y are / T Ž 1.. Ž 8 ž. Thus, we recover a nown result 6, 5 for the logistic mapping. Note that, as in Riordan 8, we use the definition of binomial coefficients such that Ž l.0 for the integers l and m, whenever m 0orml. m

4 84 BERKOLAIKO, RABINOVICH, AND HAVLIN 3. RECURSION SOLUTION FOR ARBITRARY ANALYTIC FUNCTIONS IN THE RHS In this section we consider complex maps analytic in a dis B r z: zr4 centered at the origin. Real analytic maps are considered as a special case of complex maps. Let the series Ý 0 a z f z converge absolutely in the dis B z: z r 4 r. We construct the follow- ing series absolutely convergent in the dis B 4: ž / ž / ž / Ý a z a aa z a aa z Ý T z. 0 DEFINITION 1. The matrix T T 4, 0 is said to be the transfer matrix of the analytic function f z if Ý T z fž z.. THEOREM 2. Let gž z. be analytic in the dis B R, f: Br BR be analytic in the dis B r; let the matrices T and S be the transfer matrices of the functions fž z. and gž z.. Then the matrix ST is the transfer matrix of the composition g fž z.. Proof. First we prove the existence of the matrix product ST. For arbitrary, 0r, one has fž z.r in the closed dis z, for some,0r. Thus, the power fž z. satisfies fž z. Ž R. and by Cauchy s inequality for the coefficients of the power series repre- sentation we have T Ž R.. Therefore, the series 0 ST i Ý SiT Ý Si R 0 0 converges absolutely, since, by definition, the series Ý 0 Siz is absolutely convergent in B R. Thus, the matrix product ST exists and, furthermore, we have S T z S z R Ý Ý i Ý Ý i z S Ž R. r Ý Ý i 0 0

5 ANALYSIS OF ANALYTICAL RECURSIONS 85 and the series Ý0 ST i z converges absolutely in the dis B. Changing the order of summation 4, one has Ý Ý i Ý i Ý i 0 i0 0 i0 Ž ST. z S T z S fž z. gž f z. gf z. 9 i Considering the limit r, we infer that Eq. Ž. 9 holds in the dis B r. This observation completes the proof. In this way we can write down the solution of the recursion zn1 fž zn. with z0 z. Ž 10. THEOREM 3. Let f: Br Br be analytic with transfer matrix T. Then, for any initial point z B r, T where, e 4 and z z i 4 i,1 i0 i0. zn et n z, Since real analytic functions can be analytically continued to the complex plane, the above analysis can be employed in the real functions case as well. For practical use instead of the condition gž z.: BRBr the stronger condition, Ý i i0 b x r for x R, can be used, where the bi are coefficients of the series decomposition of the function f. The following example presents a family of functions whose transfer matrices form is invariant under multiplication. EXAMPLE 2. We consider the function fž x. axž b x.. The components of the transfer matrix T are ž / ab for 0, T 1 Ž otherwise.

6 86 BERKOLAIKO, RABINOVICH, AND HAVLIN If g x cx d x and S is the transfer matrix of g x, then the components of the matrix ST are ž / a l 1 1 ž / ž / ž / l0 d l 1 ac bd ž / ž / ž / ab ab i 1 ž i 1 / i i 0 0 i 1 i1 i Ý Ý ST ST cd ab i0 cb Ý. One can see that the matrix elements Ž ST. Ž 11.. have the same form as in Eq. 4. OPERATIONS WITH TRANSFER MATRICES An interesting question to as now is: What is the inverse of the transfer matrix, for example, of matrix Ž. 8? The answer to this question and a possible method to calculate a general form of elements of the inverse matrix gives THEOREM 4. Let T be the transfer matrix of the analytic function fž x., Ž. Ž. 1 f00, f 0 0. Then T exists and is the transfer matrix of the inerse 1 function f Ž x.. 1 Note that by the inverse function f Ž x. we understand the formal 1 1 Taylor series which, under substitution, produces f f x ff Ž x. x. The series converges if and only if the function defined by the implicit function theorem as a solution of fž x. y for initial value fž 0. 0is analytic at the point y 0. Thus, only local information about the function fž x. is used in the theorem. Proof. The statement of the theorem implies that T is an upper triangular matrix. Indeed, fž x. can be represented as fž x. xž x. and therefore fž x. x Ž x. and T 0 for. Existence and unique- ness of the inverse matrix, which is also an upper triangular matrix, is proved in 3. To prove the theorem, we consider a formal Taylor series 1 g x T x. Ý 1 0 Now we can form a transfer matrix S of the function gž x.. Then Ž ST. 1 1, since the first row of the matrix S is the same as that of the matrix

7 ANALYSIS OF ANALYTICAL RECURSIONS 87 1 T. On the other hand, Theorem 4 implies g f x Ý0 ST 1x x. Therefore, the powers Žg fž x.. are equal to x Ý 0 x Ý ST x, which implies Ž ST. 0. Thus, ST is the unity matrix, 1 1 ST and g x f Ž x.. The theorem is proved. 1 EXAMPLE 3. For the logistic mapping f x x 1 x we have f Ž y. Ž 1 ' 1 4y. 2 Žwe eliminate the second branch Ž1 ' 1 4 y. 2 of the inverse function because it violates the condition 1 Ž.. 1 f f00. To get an explicit form of elements T, one can notice 1 that f Ž x. is simply related to the generating function cž x. Ž 2x. 1 Ž1 ' 14x. of Catalan numbers 8. Respectively, the powers Ž c x. are decomposed with the aid of ballot numbers as follows: cž x. Ýa 1, x, where l m a l m l1, m lmž m / are ballot numbers. This relation immediately gives T 1 a 1,, i.e., / 2 1 T. 2ž Writing down the orthogonality relation T 1 T E in matrix components, one finds the combinatorial identity 2l ÝŽ 1. ž / ž /, l 2l l Žsee also. 8. The additional condition f Ž. 0 1 maes possible the diagonalization of a transfer matrix. THEOREM 5. Let T be the transfer matrix of the analytic function fž x., Ž. Ž. 1 f00, f 0 0, 1. Then transfer matrices D and D exist such that T D 1 D, Ž 12. where is a diagonal matrix with elements.

8 88 Proof. BERKOLAIKO, RABINOVICH, AND HAVLIN We rewrite Eq. Ž 12. as follows: DT D and note that the structure of matrices T and is rather simple and the equation gives a straightforward algorithm for calculating the matrix D. Indeed, 1 1 Ý i i i1 D T T D. This relation defines off-diagonal elements of the matrix D while the diagonal ones remain arbitrary. Thus, we have a family of suitable upper triangular matrices. To prove that the family contains a transfer matrix, we fix the undetermined element D11 to be, say, a 0 and, as before, consider a formal series h x D x. Ý 1 1 This allows one to construct a transfer matrix S for the function hž x. and, since is also the transfer matrix of the function Ž x. x, we can apply the reasoning from the previous proof to confirm that the equation ST S holds. Indeed, the first rows of the matrices on the left- and right-hand sides are equal due to the fact that S1 D 1. The rest of the equation is implied by Theorem 4. The theorem is proved. COROLLARY 1. Let f: Br Br be an analytic function. Then for any initial point z Br we hae the following representation: zn ed 1 n Dz. 1 1 EXAMPLE 4. The correspondences T f x, Dh x, D h Ž x., and Ž x. allow us to rewrite Eq. Ž 12. in the functional form fž x. h 1 Ž hž x.., where h x Ý D x. The elements D obey the linear recursion i Ý i 1i i1 D T D. For some special cases this recursion admits an explicit solution for D 1 which gives us information about h x. For example, using this result, it is

9 ANALYSIS OF ANALYTICAL RECURSIONS 89 possible to recover the well-nown representation of the logistic map for 4: ž ' / Ž '. Ž ' x 1x sin 2 arcsin x sin 4 arcsin x, 2 which corresponds to f x 4x 1x and h x arcsin x and im- plies the easy-to-analyze solution f x sin 2 arcsin x, n n ' n where f x is nth iteration of the function f x. 2 ' 2 5. SUMMARY In this wor we prove convergence of the series used in Carleman linearization of nonlinear analytic recursions. We establish a correspondence between analytic functions and infinite matrices of a certain type. This correspondence is proven to extend to operations with matrices. Namely, for an analytic function, satisfying certain conditions, and the corresponding matrix the following is true: the inverse matrix corresponds to the formal inverse series of the function. A similar result is obtained for the diagonalization of the matrix which corresponds to the formal representation fž x. h 1 Ž hž x.. of the given function fž x., where f Ž. 0 and the functions hž x. and 1 h Ž x. are given in the form of series. Convergence of this series is a possible subect for future research. ACKNOWLEDGMENTS We than Dany Ben-Avraham, Iain Stewart, and Michael Grinfeld for a critical reading of this manuscript. REFERENCES 1. R. P. Agarwal, Difference Equations and Inequalities, Deer, New Yor, T. Carleman, Application de la theorie des equations integrates lineaires aux systemes ` d equations differentielles nonlineaires, Acta Math. 59 Ž 1932., R. G. Cooe, Infinite Matrices and Sequence Spaces, Macmillan, London, 1950.

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