ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA. Alois Klc Pavel Pokorny Jan Rehacek. (Communicated by Milan Medved')

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1 Math. Slovaca 5 () No. 1 nn{nn aoc Mathematical Institute Slovak Academy of Sciences ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA Alois Klc Pavel Pokorny Jan Rehacek (Communicated by Milan Medved') ABSTRACT. Possibilities of using the Baker-Campbell-Hausdor (BCH) formula to describe the!-limit behavior of dynamical systems generated by two alternating vector elds (zig-zag dynamical systems) are studied. It is shown that in the case when the two vector elds generating the zig-zag dynamical system are linear the usage of the BCH formula is useful. Limitation for nonlinear case are discussed. 1. Introduction In this paper we are going to study a particular class of dynamical systems that we chose to call \zig-zag dynamical systems" for its dynamics is determined by two smooth vector elds alternately operating on the phase space. Motivation for the study of such systems can be found in [5] and partially also in [6]. Both papers deal primarily with the study of periodic points of the zig-zag system mostly in the case where the two vector elds are F -related by some involutive dieomorphism F. In this article we will consider arbitrary smooth vector elds.. Preliminaries Let M be a smooth manifold of dimension m with two smooth vector elds u v: M! TM Mathematics Subject Classification: Primary 58F5 58F8 58D5 58F99 Secondary 17B66. K e y w o r d s: dynamical system period map Lie group Lie bracket exponential map. This work has been supported by the grant MSM 347 of the Czech Ministry of Education. Numerical computation has been done at the Supercomputer Center of the Masaryk University Brno and at the IBM-VSCHT- CVUT Joint Supercomputer Center Prague. 1

2 ALOIS KLIC PAVEL POKORNY JAN REHACEK dened on it. The ows of these vector elds will be denoted by ' t t where t R respectively. Next we consider a p periodic function r p (t) dened on [ p) by for t [p) r p (t)= (1) 1 for t [p p) : The zig-zag dynamical system is now described by the following non-autonomous dierential equation with a p-periodic piece-wise continuous right hand side _x = f(x t) =u(x)+r p (t) v(x)u(x) : () x _x=u(x) _x=v(x) P (x )= p p (x ) p (x ) P (x ) P 3 (x ) P 4 (x ) Figure 1. The trajectory of the zig-zag dynamical system. Starting from the initial condition x the motion in the phase space is governed by the equation _x = u(x) for the rst half of the period and by the equation _x = v(x) for the second half of the period. The vector eld u has the ow ' t and the vector eld v has the ow t.

3 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA Let us denote by (tx ) the solution of () satisfying the initial condition ( x )=x. Using the phase ows ' t and t we can express this solution for t [ p) in the form ' t (x (tx )= ) for t [p) tp (' p (3) (x )) for t [p p) : Let us dene P (x) = (px): (4) Then the mapping P is a period map (also called a stroboscopic map) for the equation () and the! -limit behavior of the solution of () can be described by the! -limit behavior of the mapping P i.e. by the! -limit behavior of the orbit x P(x )P (x )::: = P n (x ) 1 n=. From (3) and (4) it follows that The situation is depicted on Fig. 1. P = p ' p : (5) 3. Averaging method and the BCH formula In [5] we used the averaging method for nding periodic solutions of () with which we have associated an autonomous averaged system _x = u(x)+v(x) (6) whose trajectories as is well known approximate solutions of () for small p on some nite interval. The right hand side of (6) is the rst term of the Baker- Campbell-Hausdor (BCH) formula which for the reader's convenience we now briey review hoping that in the process we illustrate its importance in the study of the zig-zag systems. The BCH formula for matrices. From the classical theory of Lie groups and Lie algebras it follows that the matrix exponential satises e A e B =e (A+B) if and only if AB = BA i.e. if the two matrices commute. In case the matrices A B do not commute then e A e B =e C where the matrix C is given by the BCH formula (see [1] [3]) A [A B] B [A B] + (7) C = A + B + 1 [A B]+ 1 1 where [A B] =AB BA is the commutator of the matrices A and B. We will use this fact in Sections 5 and 6. 3

4 ALOIS KLIC PAVEL POKORNY JAN REHACEK The BCH formula for vector elds. Here we would like to provide some heuristic and motivational reasoning. Following [1] we denote by Di 1 c (M) the group of C1 dieomorphisms with compact support. The set Di 1 c (M) is an innite-dimensional Lie group whose Lie algebra is the algebra X c (M) of smooth vector elds on M.For u v X c (M)wedenote by [uv] the usual Lie bracket of the two vector elds (see e.g. []). It is well known that a vector eld u X c (M) generates a phase ow ' t Di 1 c (M) t R. The dieomorphism '1 is called the time one map of the ow ' t. The correspondence is a well dened mapping u 7! ' 1 Exp: X c (M)! Di 1 c (M) (8) that is called the exponential mapping. This mapping is an analogy of the exponential mapping for nite dimensional Lie groups. Unfortunately for the vector elds this exponential mapping does not have the nice properties of the nite dimensional case (see [1] [4] [7] [11]) particularly it is neither one-to-one nor surjective near the identity (cf. [1] [7]) so there are dieomorphisms from Di 1 c (M) that are arbitrarily close to the identity and yet do not have any pre-image in X c (M) under the exponential mapping. The set Exp X c (M) = E (9) will be called the set of embeddable dieomorphisms i.e. the dieomorphisms that can be embedded intoaow' t of some vector eld u X c (M). Typically the set E is a rather irregular rst category like subset of Di 1 c (M) (see [9]). Let us now return to our zig-zag dynamical system and consider the vector elds u v from (). Then and and thus and Exp(u) =' 1 Exp(v) = 1 ' p = Exp(pu) p = Exp(pv) : Let us assume for this moment the validity of the BCH formula even in this innite-dimensional case i.e. let us assume that the following holds Exp(v) Exp(u) = Exp w(v : u) : (1) 4

5 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA One might expect that then the vector eld w(v : u) is given by the BCH formula w(v : u) =(v+u)+ 1 [vu]+ 1 1 v [v u] u [v u] + : (11) In Paragraph 7 we show that this is not true and we derive a correct formula. With respect to relation (1) the period map P for the zig-zag system would be given by where P = p ' p = Exp(pv) Exp(pu) = Exp w(pv : pu) (1) w(pv : pu) =p(v+u)+ p p3 [vu]+ v [v u] u [v u] + : (13) 1 Then we could embed the period map P into the phase ow of the vector eld w(pv : pu). The orbits of the mapping P would lie on trajectories of the vector eld w and the! -limit behavior of these orbits will be determined by the! -limit behavior of the trajectories of w. At the same time we see that using the averaging method we obtain the rst term in the BCH formula. In Paragraph 4 the case is investigated when the right-hand side of (11) is reduced to its rst term and in Paragraphs 5 and 6 we will consider two cases in which the BCH formula for matrices can be applied for the study of qualitative properties of the linear zig-zag systems. Finally in Paragraph 7 the nonlinear case of zig-zag system is treated. In the last Paragraph 8 the numerical study is provided as a demonstration of using the BCH formula for an example of zig-zag system. 4. The case [u v] = In this paragraph we will assume that the Lie bracket of the vector elds u v from () is equal to i.e. [u v] =: (14) It is well known (see [ p. 155 Theorem 7.1]) that the relation (14) is equivalent to ' t s = s ' t for all t s R (15) where ' t and t are the phase ows of the vector elds u v respectively. In this case the following theorem holds. 5

6 ALOIS KLIC PAVEL POKORNY JAN REHACEK Theorem 1. Let u v be two vector elds in X c (M) and ' t t their phase ows. If [u v] = then is the phase ow of the vector eld u + v. t = t ' t t R (16) P r o o f. The rst two properties of the ow are evident. (i) = ' =idj M (ii) t s = t ' t s ' s = t+s ' t+s = t+s. For (iii) we need to show that We have or more briey d d t (x) dt dt (t (x)) = d dt = u t (x) + v t (x) : t ' t (x) t t (x) t ' d' t (x) dt = v t (x) + t ' t (x) u ' t (x) _ t = _t + t _'t = v + t u (17) where we use the notation from [1 p. 3]. According to [ p. 141 Theorem 5.7] we obtain that t u = u so that _ t = u + v which was to be proved. Remark 4.1. In this case the BCH formula takes the form Exp(v) Exp(u) = Exp(u + v) (18) and the period map for the equation () is simply P (x) = p (x): The orbits of the mapping P lie on the trajectories of the vector eld u + v. Remark 4.. For each u X c (M) and for each F Di 1 c (M) Exp(F u)=fexp(u) F 1 (19) as follows from considerations in [ p. 137 Example 4]. The relationship (19) implies that for all F Di 1 c (M) 6 F Exp X c (M) F 1 Exp X c (M) : () Let us now recall the Thurston's theorem (see [1 p. 4]):

7 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA Theorem T. For any smooth manifold M the identity component Di 1 c (M) in Di 1 c (M) is a simple group. This theorem and the relation () clearly imply that the set Exp X c (M) cannot be a subgroup of the group Di 1 c (M) for if it were one then by () it would have to be a normal subgroup which contradicts the Theorem T. Remark 4.3. Suppose that Exp(u) Exp(v) = Exp w(u : v) (1) where w X c (M) i.e. the dieomorphism ' 1 1 is embeddable. Then for each F Di 1 c (M) we get Exp(F u) Exp(F v) = Exp F w(u : v) () since on the left hand side we have according to (19) F Exp(u) F 1 F Exp(v) F 1 = F Exp(u) Exp(v) F 1 = F Exp w(u : v) F 1 = Exp F w(u : v) : 5. The linear zig-zag system In this paragraph we set M = R n and consider two linear vector elds u(x) =Ax and v(x) =Bx (3) where A B are arbitrary square matrices of order n x R n. The corresponding phase ows are then so that That implies ' t (x) =e ta x and t (x) =e tb x (4) Exp(Ax) =e A x and Exp(Bx) =e B x: Exp(Ax) Exp(Bx) =e A e B x=e C x= Exp(Cx) (5) where the matrix C is determined by (7). Let us now return to the period map P for the equation () where the vector elds u and v are given by (3). Then considering (4) we obtain P (x) = p ' p (x)=e pb e pa x=e D x 7

8 ALOIS KLIC PAVEL POKORNY JAN REHACEK where the matrix D is according to (7) given by D = p(b + A)+ p [BA]+ p3 1 B [B A] A [B A] + : (6) Thus the mapping P (x) can be embedded into the ow t (x) which is the phase ow of the linear system _x = Dx : The orbits of the period map P are lying on the trajectories of this linear system. Thus we have proved the following theorem. Theorem. Let A B be matrices of order n and r p (t) be a function dened by (1). Let furthermore P : R n! R n be the period (time p) map of the system _x = Ax + r p (t)[bx Ax] : Then for suciently small p > there exists a matrix D given by (6) such that every orbit O P = P n (x ): nn of the mapping P lies on the trajectory of the linear system _x = Dx passing though the initial state x. Remark 5.1. Let us return to the relation (5). It was obtained by using the BCH formula for matrices. We will show how to write this relation using the BCH formula for linear vector elds. For this purpose we must consider that the usual bracket product of linear vector elds satises [Ax Bx] =BAx ABx = [A B]x : When we introduce a more convenient bracket product (see [7 p. 141]) [v u] = [v u] i.e. the appropriate bracket product is just the negative of the usual bracket product of vector elds then [Ax Bx] =[AB]x and the vector eld Cx in (5) can be expressed in the form Cx = Ax +Bx + 1 [Ax Bx] which is just the BCH formula for linear vector elds. 8 Ax [Ax Bx] Bx [Ax Bx] + (7)

9 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA 6. Nonhomogeneous linear zig-zag system In this paragraph we set M = R n again and consider two vector elds u(x) =A(xx ) and v(x) =B(xx 1 ) (8) where x x 1 R n and x 6= x 1. Remark 6.1. The case when x = x 1 can be converted to the linear case using () from the Remark 4.3. If we choose F in the form F (x) =x+x then F (Ax) =A(xx ) and F (Bx) =B(xx ) and we obtain easily a result similar to Theorem. The vector elds (8) determine nonhomogeneous linear dierential equations with the corresponding phase ows _x = A(x x ) and _x = B(x x 1 ) ' t (x) =x +e ta (xx ) and t (x) =x 1 +e tb (xx 1 ): (9) The corresponding period map has the form Let us set P (x) = p ' p (x)=x 1 +e pb x +e pa (xx )x 1 : (3) e pb e pa =e D where the matrix D is given by (6) and let us consider the dierential equation with the phase ow _x = D(x x ) t (x) =x +e td (xx ) (31) into which wewant toembed the mapping P = p ' p so that Substituting (3) and (31) into (3) we get p ' p = 1 : (3) x 1 +e pb x +e pa (xx )x 1 =x +e D (xx ) and after simplication E e pb e pa x = x 1 +e pb (x x 1 )e pb e pa x (33) 9

10 ALOIS KLIC PAVEL POKORNY JAN REHACEK where E is the identity matrix. Assuming that E e D is regular we can solve this equation for x : x = E e D 1 x 1 +e pb (x x 1 )e D x : (34) This fully determines the phase ow (31). Remark 6.. The previous argument shows that the embedding of the dieomorphism (3) into the ow (31) may not be always possible but if the matrix E e D is not regular then the dieomorphism (3) can still be embedded into aow of a more general vector eld namely where x D is a constant vector. _x = Dx + x D Example 6.1. As an illustration we will discuss a simple example from [5] which is called \blinking nodes" there. We will consider two two-dimensional vector elds u(x y) =(x+1y) v(x y) =(x1y) (35) and the corresponding dierential equations and _x = x +1 _y=y _x = x 1 _y = y whose phase ows are described explicitly as ' t (x y) = 1+(x1) e t ye t t (x y) = 1+(x+1)e t ye t : In this case considering the notation (8) we obtain A = B = E x =(1) = x 1 pa = pb = pe : The equation (34) has then the form 1 x = = he e pe i 1 hx +x e pe x e pe i hee pei 1 h E+e pe e pei x :

11 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA and Considering the matrix identities we obtain e e pe p = e p i 1 he e pe 1 = 1ep 1 1ep e pe e pe E = (e p 1) 1 y x = " (e p 1) 1ep (e p 1) 1ep # (e p 1) 1 = e p 1 ep +1 : x Figure. The solid line is the trajectory of the zig-zag system \blinking nodes". Our goal is to nd a vector eld whose trajectory (the dashed line) contains the orbit of the period map of the \zig-zag" system. The point x is the attractor of the period map P. The situation is depicted in Fig. where the solution curve of the zig-zag system () is marked by a solid line while the trajectory of the phase ow (31) is marked by a dashed line. We note that this result agrees with the result from [5] which was obtained by a dierent way. 11

12 ALOIS KLIC PAVEL POKORNY JAN REHACEK 7. Nonlinear zig-zag system From the reasons discussed in [7] it follows that the generalization of the BCH formula from nite dimensional Lie groups to innite dimensional Lie groups is possible only in the case when this innite dimensional Lie group can be provided with a real analytic structure. This is not possible for Di 1 c (M) (cf. [7]). Thus the BCH formula in general setting for the group Di 1 c (M) does not even make sense. Another diculty with the BCH formula for the group Di 1 c (M) follows from the Remark 4. namely even when the dieomorphisms ' 1 = Exp(u) and 1 = Exp(v) are embeddable then their composition 1 ' 1 is not necessarily embeddable because the set Exp X c (M) is not a group. Despite of this it seems to us reasonable to consider the following problem. Problem Formulation. Let us consider two vector elds u v X c (M) (suciently \small") with the phase ows ' t t i.e. ' 1 = Exp(u) and 1 = Exp(v) : Let us suppose that 1 ' 1 Exp X c (M) (36) i.e. the composition of corresponding 1-ows is an embeddable dieomorphism. That means there exists a vector eld w X c (M) with the phase ow t satisfying 1 = 1 ' 1 (37) or Exp(w) = 1 ' 1 : (38) Let us try to nd the vector eld w satisfying (38). There may exist more than one such vector elds because the mapping Exp is not one-to-one. It is obvious that the vector eld w depends on the two vector elds u and v. We denote this dependence w(v : u). We want to express the vector eld w in terms of the two vector elds v and u. The method used below is based on the expression of the phase ow ' t of a vector eld u X c (M) using the relation [8 (1.19)] i.e. for t R x M we have the Lie series ' t (x) = For details see [8]. 1 1X k= t k k! uk (x) =x+tu(x)+ t u (x)+ : (39)

13 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA We will work with local coordinates and let us set M = R n for simplicity and the vector elds will be given using the coordinate functions in the form u(x) = v(x)= nx nx i=1 i=1 u i v i where x =(x 1 x x n ) R n. Let us recall that the Lie bracket of these vector elds is given by nx [u v] = u j j : i We will use the short form where Further we will write briey Let us denote for t = p and i=1 j=1 [u v] =u(v)v(u)=v uu v = j ij=1:::n u = u(u) u 3 = u(u ) etc. y = ' p (x) =x+pu(x)+ p u (x)+ (4) z = p (y) =y+pv(y)+ p v (y)+ : (41) Now wewant tondavector eld w with the phase ow t such that i.e. 1 = p ' p (4) 1 (x) =z= p (y): (43) We want to nd the vector eld w = w(v : u) as a formal series w(x) =w 1 (x)+w (x)+w 3 (x)+= 1X k=1 w k (x) (44) 13

14 ALOIS KLIC PAVEL POKORNY JAN REHACEK where w k (x) X c (M) and w k (pv : pu) =p k w k (v:u): (45) By substituting from (4) into (41) we get after simplication so that z = x + pu(x)+ p u (x)+ p3 3! u3 (x)+o(p 4 ) n +p v x+pu(x)+ p u (x)+o(p 3 ) + p v x + pu(x)+o(p ) + p3 3! o v 3 x + O(p) = x + pu(x)+ p u (x)+ p3 3! u3 (x)+o(p 4 ) n +p v(x)+v (x) hpu(x)+ p u (x)+o(p 3 ) + p v (x)+ v (x) pu(x)+o(p ) + p3 3! =x+pu(x)+ p u (x)+ p3 3! u3 (x)+o(p 4 ) +p + p io nv(x)+pu v(x) + p u o v(x) +O(p 3 ) v (x)+pu v (x) +O(p ) + p3 3! v 3 (x) +O(p) v 3 (x)+o(p) z = x + p v(x)+u(x) + p3 3! Now we rewrite (43) in the form + p v (x)+u v(x) +u (x) + v 3 (x) +u 3 (x)+3u v(x) +3u v (x) + O(p 4 ) : (46) z = 1 (x) =x+w(x)+ 1 w (x)+ 1 3! w3 (x)+ (47) and we use (44) for w(x). Then (when omitting x ) 14 w =(w 1 +w +w 3 +)(w 1 + w + w 3 + ) =w +w (w )+w (w )++w (w )+w++w (w )+ 3 1 (48) w 3 = w(w )=(w 1 +w +) w +w (w )+w (w ) =w 3 1 +w(w )+w 1 (w)+ (49) 1 :

15 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA From (48) and (49) we put into (47) and we get z = x +(w 1 +w +w 3 +)+ 1 w 1 +w 1 (w )+w (w 1 ) ! [w3 1 + ]+O(p4 ): By comparing (46) and (5) and using (45) we get (5) p 1 : w 1 = v + u (51) p : w + 1 w 1 = 1 v +u(v)+u (5) p 3 : w w 1 (w )+1 w (w 1 )+ 1 3! w3 1 = 1 3! v 3 +3u (v)+3u(v )+u 3 : (53) Now we put from (51) into (5) using and we get w 1 =(v+u)(v + u) =v +v(u)+u(v)+u (54) w = 1 u(v) v(u) = 1 [v u] = 1 [vu] (55) where we use the modied Lie bracket [ ] from Remark 5.1. Now we put into (53) using w 3 1 = w 1 (w 1 )=(v+u) v +v(u)+u(v)+u (56) =v 3 +v (u)+v u(v) +v(u )+u(v )+u v(u) +u (v)+u 3 1 u(v)v(u) = 1 v (u)v u(v) +u v(u) u (v) w 1 (w )=(v+u) (57) w (w 1 )= 1 v(u)u(v) (v+u)= 1 v u(v) +v(u )u(v )u v(u) : (58) After putting (56) (57) and (58) into (53) and simplifying we get w 3 = 1 v [v u] u [v u] = This result suggests the following hypothesis: v [v u] u [v u] : (59) Hypothesis. One of the vector elds satisfying (38) i.e. Exp(w) = Exp(v) Exp(u) is the vector eld w(v : u) given by the following BCH formula w(v : u) =v+u+ 1 [vu] v [v u] u [v u] + and the convergence of the right-hand side implies that the dieomorphism 1 ' 1 is embeddable. Remark 7.1. It is useful to note that our hypothesis is also supported by the result from the Paragraph 5 because the result (7) from Remark 5.1 obtained by a fully rigorous way agrees with our hypothesis for linear vector elds. 15

16 ALOIS KLIC PAVEL POKORNY JAN REHACEK 8. Numerical experiment As mentioned in the Remark 7.1 the full proof of our Hypothesis is not complete in the sense that the problems with the convergence in the BCH formula are not yet solved. Because of this we have made the following numerical experiment to shed more light on the relation between the! -limit behavior of the orbits of period mapping P and the! -limit behavior of the trajectories of the vector eld w(v : u) Invariant curve of the period map P for the zig-zag dynamical system \blinking cycles". Figure 3. Let us have two planar vector elds ( x 1) y ( x 1) ( x 1) + y u(x y) = (x 1) + y y (x 1) + y ( x + 1) y ( x + 1) ( x + 1) + y v(x y) = (x + 1) + y y (x + 1) + y (6) (61) each of which has a stable limit cycle with the radius p and the center (1 ) (1 ) respectively. Consider now the zig-zag dynamical system () with these vector elds for the switching period p = : where = 3:5. 16

17 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA Then the corresponding period map P has the invariant closed curve depicted in Fig Figure 4. To approximate the dynamics of the zig-zag system \blinking cycles" shown in Fig. 3 we use the rst 1 to 4 terms of the BCH formula. When taking the rst 1 or terms the resulting dynamical system has a qualitatively dierent attractor (a stable stationary point) not shown here. When taking the rst 3 or 4 terms the resulting dynamical system has a stable limit cycle that is almost indistinguishable from the invariant curve of the original zig-zag system. Cf. Fig. 3. Now we construct using (6) and (61) the vector elds w 1 = v + u w = 1 [v u] w 3 = 1 1 w 4 = 1 4 v u [v u] v u [v u] i.e. these vector elds are successive terms in the BCH formula. The computation of w k was made using the computer algebra system Mathematica. Further we construct the vector elds 17

18 ALOIS KLIC PAVEL POKORNY JAN REHACEK w 11 = w 1 w 1 = w 1 + w w 13 = w 1 + w + w 3 w 14 = w 1 + w + w 3 + w 4 and we solve numerically the dierential equations _x = w 1i (x) i =134: (6) The rst two systems have only stable steady states while the last two systems have an almost indistinguishable stable limit cycle depicted in Fig. 4. The reader can convince himself that the coincidence of the invariant curve from Fig. 3 with the closed trajectory in Fig. 4 is surprisingly good. This demonstrates clearly how the BCH formula can be used to approximate the dynamics of a zig-zag dynamical system by an autonomous system. The BCH formula gives an essential renement of the averaging method studied in [5]. REFERENCES [1] BANYAGA A. : The Structure of Classical Dieomorphism Groups. Math. Appl. 4 Kluwer Academic Publishers London [] BOOTHBY W. M. : An Introduction to Dierentiable Manifolds and Riemannian Geometry Academic Press New York [3] BOURBAKI N. : Elements de mathematique. Fasc. 6: Groupes et algebres de Lie. Chap. I: Algebres de Lie. Actualites Sci. Indust. 185 (nd ed.) Hermann Paris (French) [4] HAMILTON R. S. : The inverse function theorem of Nash and Moser Bull. Amer. Math. Soc. (N.S.) 7 (198) 65{. [5] KLI C A. POKORN Y P. : On dynamical systems generated by two alternating vector elds Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996) 15{3. [6] KLIC A. REHACEK J. : On systems governed by two alternating vector elds Appl. Math. 39 (1994) 57{64. [7] MILNOR J. : Remarks on innite dimensional Lie groups. In: Relativity Groups and Topology II Les Houches (1983) North Holland Amsterdam-New Yourk 1984 pp. 17{157. [8] OLVER P. J. : Applications of Lie Groups to Dierential Equations Springer-Verlag New York [9] PALIS J. : Vector elds generate few dieomorphisms Bull. Amer. Math. Soc. 8 (1974) 53{55. [1] VARADARJAN V. S. : Lie Groups Lie Algebras and Their Representations Prentice-Hall Inc. New Yersey

19 ZIG-ZAG DYNAMICAL SYSTEMS AND THE BAKER-CAMPBELL-HAUSDORFF FORMULA [11] WOJTYNSKI W. : One-parameter subgroups and the B-C-H formula Studia Math. 111 (1994) 163{185. Received September Department of Mathematics Prague Institute of Chemical Technology Technicka 5 CZ{166 8 Praha 6 CZECH REPUBLIC Pavel.Pokorny@vscht.cz Department of Mathematics Georgian Court College Lakewood NJ 871 U.S.A. 19

Departement Elektrotechniek ESAT-SISTA/TR Dynamical System Prediction: a Lie algebraic approach for a novel. neural architecture 1

Katholieke Universiteit Leuven Departement Elektrotechniek ESAT-SISTA/TR 1995-47 Dynamical System Prediction: a Lie algebraic approach for a novel neural architecture 1 Yves Moreau and Joos Vandewalle