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(1)Optimization, Stability and Cylindrical Decomposition Krister Forsman Department of Electrical Engineering Linkoping University S-581 83 Linkoping Sweden email: krister@isy.liu.se. 1993-04-14. ERTEKNIK REGL. AU. OL TOM R T N ATIC CO. LINKÖPING. Technical reports from the automatic control group in Linkoping are available by anonymous ftp at the address 130.236.24.1 (joakim.isy.liu.se). This report is contained in the compressed Postscript le named /pub/reports/LiTH-ISY-R-1472.ps.Z. 1.

(2) Optimization, Stability and Cylindrical Decomposition Krister Forsman. Department of Electrical Engineering Linkoping University S-581 83 Linkoping Sweden email: krister@isy.liu.se 1993-04-14 Abstract. Some connections between constructive real algebraic geometry and constrained optimization are exploited. We show how the problem of determining the projection of a real-algebraic variety on a certain axis is equivalent to a problem in nonlinear programming. As an application, Grobner bases are used to deal with an optimization problem arising in the theory of local Lyapunov functions. The problems addressed are: determining critical levels of local Lyapunov functions and investigating robustness using Lyapunov functions. Since the tools used come from commutative algebra and algebraic geometry the dierential equations considered are of polynomial type and the Lyapunov functions used are polynomial. Keywords: constrained optimization, Lyapunov theory, stability, polynomial dierential equations, robustness, Grobner bases, elimination theory, nonlinear equation solving, real algebraic geometry, quantier elimination, commutative algebra. 1 Introduction In this paper we discuss the connections between cylindrical algebraic decomposition and constrained optimization: the problem of determining the projection of a real-algebraic variety on a certain axis is equivalent to an optimization problem. As an application we show how Grobner bases can be used to deal with some optimization problems arising in Lyapunov theory. Some problems studied are: nding critical levels of local Lyapunov functions investigating robustness using Lyapunov functions Since the basic framework is commutative algebra and (real) algebraic geometry we consider dierential equations in which all nonlinearities are of polynomial type and Lyapunov functions that are polynomial. We will suppose that the reader is familiar with some basic concepts from commutative algebra, real algebraic geometry and the theory of Grobner bases. The following notation is used:. A B means that A is a proper subset of B , while A B allows A = B . the ideal generated by the set f1 : : : fm is written h f1 : : : fm i. 2.

(3) Z (I ) is the set of real zeroes of the ideal I in R n. The sets Bd W W are introduced in de nition 4.2, the ideal L in formula (14), LR in equation (17), the set D in equation (12) and H in de nition 5.1. Lf V is the Lie-derivative of V w.r.t. f . GB stands for Grobner base, and BGK for the Boege-Gebauer-Kredel algorithm (section 2.2). plex is an abbreviation for purely lexicographic term-ordering and revlex for reverse graded lexicographic term-ordering. g.p. stands for generic position (de nition 2.1). Some of the ideas presented here have been published earlier 12, 13, 14, 16].. 2 Some Facts about Grobner Bases We devote this section to recalling some useful results from the theory of Grobner bases even though the basic de nitions are supposed known. For a thorough introduction to GB see e.g. 6, 17, 19, 22].. 2.1 Some Structure Theorems. We say that p 2 kX1 : : : Xn ] is regular w.r.t. Xi if lm p 2 kXi ] using a plex termordering that ranks Xi the highest. (Here lm denotes \leading monomial".) The following well-known theorem explains how GB can be used to solve systems of polynomial equations. Theorem 2.1 Let a be an ideal in kX1 : : : Xn]. dim a = 0 i for all i there is a p 2 GB (a) such that p is regular w.r.t. Xi . Proof. See 19] or 6, page 209]. 2 Theorem 2.1 does not state that there are n elements in the GB: the set fX12 X1 X2 X22 g is a GB of a zero-dimensional ideal in kX1 X2 ]. For prime ideals the situation is simpler, though: a plex GB for a prime zero-dimensional ideal in kX1 : : : Xn ] has n elements 19]. In 19] it is also proved that generically the n ; 1 rst elements of the GB are linear in the leading variable. Thus the generic look of a plex-GB for a zero-dimensional ideal w.r.t. Xn < : : : < X1 is fX1 ; p1 . X2 ; p2 : : : Xn;1 ; pn;1 pn g. (1). where pi 2 kXn ] for all i, and deg pn > deg pi for i = 1 : : : n ; 1. (Note that the way this theorem is stated in 19] the GB is not reduced.). Denition 2.1 An (arbitrary) ideal that has a GB of the type (1) w.r.t. the plex term2. ordering is said to be in generic position.. Thus a zero-dimensional ideal is in generic position (g.p.) if for two dierent zeroes (a1 : : : an ) and (b1 : : : bn ) we have an 6= bn . Yet another formulation is: a prime zerodimensional ideal is in g.p. i the GB provides a primitive element of the algebraic eld 3.

(4) extension de ned by the ideal. In 19] it is proved that almost all variable changes of the type nX ;1 Xi 7! Xi i < n Xn 7! Xn + ciXi (2) 1. where all ci 2 k, puts a zero-dimensional ideal in generic position. In fact, 18] shows that any zero-dimensional radical ideal is put in generic position by a generic morphism of the type (2).. 2.2 Computational Complexity and the BGK Algorithm. The main disadvantage with Grobner bases is that their computational complexity is in general very high. It has been showed that the complexity for computing a GB for total degree orderings, e.g. revlex, is in general lower than in the case of a plex ordering. There is an approach to elimination theory that is also based on Grobner bases, but does not use the plex term ordering. This approach is primarily due to Boege, Gebauer and Kredel 5] it is also discussed in e.g. 6, 18]. The algorithm, let us call it the BGK algorithm, gives the univariate polynomial that generates the contraction I \ kXi ] for some Xi . Thus, in particular, it works for all zero-dimensional ideals. We call the polynomial obtained the minimal polynomial of Xi , since we think of the eld of fractions of kX1 : : : Xn ]=I as an algebraic extension of k, even though I does not have to be prime, or even zero-dimensional. The idea is rst to compute a Grobner base G w.r.t. a revlex ordering and then compute remainders Rd = rem(Xid G) for successive d:s. Each Rd is considered as an element in the k-vector space of forms of degree d in kX1 : : : Xn ]. For d large enough the expressions for R0 : : : Rd are linearly dependent over k. The linear dependency relation gives us the minimal polynomial for Xi . The reason for using the approach described above is thus that it is typically more ecient than computing a plex GB see 5]. This seems to be the case in the application considered here as well (see section 6). The BGK algorithm is part of the Grobner base package in Maple, available under the name finduni. If the ideal considered is in generic position then the minimal polynomial obtained from BGK tells us how many zeroes there are and whether they are real or not. These questions is of practical importance since we are dealing with a problem in real algebraic geometry (section 4.2). To the knowledge of the author the importance of g.p. ideals in real geometry has not been stressed before. Luckily, it is possible to determine if an ideal is g.p. or not without computing a plex GB:. Denition 2.2 Let I be a zero-dimensional ideal in kX1 : : : Xn ]. The degree of I is the dimension of the k-vector space kX1 : : : Xn ]=I .. 2. The degree of I is equal to the Hilbert series of I evaluated at 1. It can be computed from any GB of I . It is clear that I is in g.p. i this number is equal to the degree of the minimal polynomial for Xn . The experience of the author is that the computation of the degree (and even the Hilbert series) is very fast once a GB, not necessarily plex, is known. Again we stress that the BGK algorithm might terminate even if the ideal in question is not zero-dimensional. The condition kXi ] \ I 6= 0 is necessary and sucient for termination we will see examples of this in e.g. section 4. 4.

(5) 2.3 Other Methods. As mentioned there are some advantages of using symbolic instead of numerical methods for equation solving. But there are other methods than GB for solving systems of polynomial equations. The most obvious alternative is probably characteristic sets 8, 24]. These provide a method for triangulating systems of polynomial equations. It has been proved that for prime ideals characteristic sets can be viewed as a special case of GB via localization techniques 8, 15]. It is sometimes claimed that characteristic sets are more ecient than GB from a computational point of view, but it is questionable if this is still true when the \tricks" described in e.g. 15] are used before applying Buchberger's algorithm. Furthermore, it seems hard or even impossible to implement algorithms that compute characteristic sets for arbitrary (not necessarily prime) ideals. Another alternative could be resultants. Resultants are sometimes more convenient than GB for solving a system of two bivariate equations. If the number of equations is larger than two, they often turn out to be less ecient than GB. Another disadvantage with resultants are the parasitic solutions that may occur.. 2.4 Inequations. It is possible to include inequations in the framework of commutative algebra: if we wish to consider the system f1 = : : : = fm = 0 q 6= 0 (3) where fi q 2 kX1 : : : Xn ] we may study the ideal h f1 : : : fm uq ; 1 i k X1 : : : Xn u] (4) where u is an auxiliary variable see e.g. 11]. This is known as the Rabinovich trick after 23]. We will make use of it in section 4.. 3 Constrained Optimization and CAD We will now see how problems in constrained optimization can be viewed as special cases of cylindrical algebraic decomposition in case the goal function and the constraints are polynomial functions. Though rather natural, this connection does not seem to have been exploited earlier: A thorough search in the databases INSPEC and Mathematics (encompassing e.g. AMS Mathematical Reviews), covering material published before the fall of 1992, for the intersection between f nonlinear programming or constrained optimization g and f cylindrical (algebraic) decomposition or real (algebraic) geometry or quanti er elimination g gave zero hits. Consider the problem of maximizing a function subject to equality constraints:. CO Given m + 1 (polynomial) functions. V c1 : : : cm : R n ! R. determine the maximum value of V on the real variety f(x1 : : : xn ) 2 R n : c1 = : : : = cm = 0g. 5. (5).

(6) If we introduce an auxiliary \value variable" d we immediately see that (CO) is equivalent to the following problem in real algebraic geometry:. CAD Determine the projection of the real variety f(x1 : : : xn d) 2 R n+1. :V. ;d. = c1 = : : : = cm = 0g. (6). on the (real) d-axis.. The \standard" solution to (CAD) is to apply the cylindrical algebraic decomposition, see e.g. 2, 10]. The standard solution to (CO) in nonlinear programming is to introduce Lagrange multipliers 1 : : : m . A condition (see e.g. 20, p. 314]) on a solution x to the optimization problem is then that c1 = : : : = cm = 0 rV ; T rc = 0 (7) As we want to determine the a semi-algebraic set it seems likely that the nonlinear programming approach is somehow related to the basic principles of CAD, even if the author has not been able to verify this formally. In this paper we will solve the problem (CO) using Grobner bases on the system (7).. 4 Local Lyapunov Functions Lyapunov theory is a commonly used mathematical tool in dierent areas engineering: e.g. automatic control 27], power systems analysis 21] and mechanics 1]. It is probably the only general tool for determining stability for a system of nonlinear ode:s. Let us now see how constrained optimization and elimination theory enter into the study of local Lyapunov functions.. 4.1 Background. The main purpose of this subsection (4.1) is to recall some standard results in Lyapunov theory and establish notation and terminology. The results and theorems are well known and therefore stated without proofs. Consult e.g. 25, 27] for details.. Denition 4.1 A polynomial p 2 R X1 : : : Xn] is positive if p(0) = 0 and 8x 2 R n n f0g. : p(x) > 0. p is non-negative if p(0) = 0 and 8x 2 R n : p(x) 0. We say that p is locally positive if p(0) = 0 and there is a neighborhood C of the origin such that 8x 2 C n f0g : p(x) > 0. p is locally non-negative if p(0) = 0 and there is a neighborhood C of the origin such that 8x 2 C : p(x) 0. 2 6.

(7) We de ne negative etc. analogously. Now consider a dynamical system. f : R n ! R n x = (x1 : : : xn ). x_ = f (x). (8). We suppose that f (0) = 0 and that the components of f are polynomial in x: 8i :. fi 2 R X1 : : : Xn ]. (9). Since f is polynomial, the solution of (8) is uniquely determined by an initial value 2 R n 9]. We write (t ) for the solution starting at , so is a function R

(8) R n ! R n .. Denition 4.2 Given a V. 2 R X1 : : : Xn ]. the regions W W and Bd are de ned by W = fx 2 R n (Lf V )(x) < 0g f0g W = fx 2 R n (Lf V )(x) 0g Bd = f x 2 R n V (x) d g. where Lf V is the Lie-derivative of V P w.r.t. f , i.e. the time derivative of V along a trajectory @V = V_ . 2 of the system (8): Lf V = rxV f = fi @x i. This means that the boundary @Bd is a level surface of V for each d. It is immediately obvious from the de nition that. ab. Ba Bb. ). (10). for a b 2 R .. Denition 4.3 A function V. 2 R X1 : : : Xn ]. system (8) if Lf V is non-positive.. is a (polynomial) Lyapunov function for the. 2. Thus V is a Lyapunov function if W = R n .. Denition 4.4 A function V (8) if Lf V is locally. 2 R X1 : : : Xn ] is a local non-positive and 9d 2 R : Bd W .. Lyapunov function for the system. If V is a local Lyapunov function for (8) and d is such that Bd. 2 Bd. ). 8t 0 :. 2. . W then. (t ) 2 Bd. In other words the set Bd is invariant if V is a local Lyapunov function and Bd W . In particular, if V is a Lyapunov function for (8) Bd is invariant for all d. Bd is not necessarily connected make a partition Bd = Bdi (11) i. of Bd into disjoint connected sets. Since is continuous in t we have the following: Theorem 4.1 If V is a local Lyapunov function for (8) and Bdi W then. 2 Bdi. ). 8t 0 :. 7. (t ) 2 Bdi.

(9) For a polynomial V the partition (11) consists of a nite number of components: see theorem 2.4.5 in 4]. The number of connected components of Bd depends on the value of d. The most important application of Lyapunov theory is stability theory. In order to establish stability using Lyapunov functions we need one more concept.. Denition 4.5 A function p : R n ! R is radially unbounded if p(x) ! 1 as kxk ! 1. 2 Thus Bd is bounded for all d i V is radially unbounded. The most important theorem of this subsection is then the following:. Theorem 4.2 If V is a positive, radially unbounded, local Lyapunov function for (8) and. Bd0 W then where Bd0 tem (8).. 3. 2 Bd0. ). lim (t ) = 0. t!1. 0. Thus the origin is a (locally) asymptotically stable equilibrium of the sys-. 4.2 Determining Critical Levels. Having de ned local Lyapunov functions and explained their connection to stability theory above, the following important problem immediately arises: How do we choose d in order to get Bd as large as possible, while still inside of W ?. If Bd W we can guarantee stability in Bd , according to theorem 4.2. In e.g. 7] an ecient method for solving this problem is wanted. This section, which is the gist of the paper, describes an algorithm for solving this problem via Grobner bases, in the case of polynomial nonlinearities. To study the problem we introduce the set D R :. D = fd 2 R Bd W g. (12). This set is obviously the projection of Z (h V ; d Lf V i) on the d-axis thus it is semi-algebraic. It is a simple consequence of property (10) that D is connected, i.e. consists of a single interval. In view of the property (10) we may reformulate the problem as: which is the smallest d > 0 such that @Bd \ @W 6= (13) If we view the problem as one of optimization we can formulate it as follows: we wish to minimize V subject to Lf V = 0. This observation is also made in e.g Shields and Storey 26]. Summing up we get n new equations while introducing one new variable, the Lagrange multiplier . So we get a system of n + 2 equations in n + 2 variables:. V ; d = 0 Lf V = 0. rx (Lf V ) ; rx V. =0. i.e. we should study the ideal L = hV. ; d. @V ; @ L V : : : @V ; @ L V i Lf V @x @x1 f @xn @xn f 1 8. (14).

(10) in R x1 : : : xn d]. As mentioned we presuppose that we do not have rV 6= 0 = r(Lf V ) at the relevant point. It is thus sucient that the hypersurface V ; d is not singular for the critical d. A plex GB for L w.r.t. a term-ordering of the type fx1 : : : xn g. >d. (15). gives us the contraction L \ R d]. If L \ R d] 6= 0 we thus obtain a polynomial p in d only, derived from the original equations. Suppose that p has m distinct positive real roots:. r1 < r2 < : : : < rm. (16). (we do not care about negative or non-real roots, or about counting the multiplicity of the roots). Put r0 = ;1 rm+1 = 1 and recall the de nition of D (equation (12)). It is well known that if d0 2 D and i is such that d0 2 (ri ri+1 ) then (ri ri+1 ) D. It is possible to have D 6= (ri ri+1 ), namely if one of ri ri+1 renders a solution that is non-real in some of the other variables (x1 : : : xn ): see example 6.1. It is clear that in order to have D = (ri ri+1 ) it is sucient, but not necessary, that the ideal L be in generic position (de nition 2.1). In many cases there is a symmetry in the problem that prevents L from being in g.p. E.g. if V is homogeneous and Z (Lf V ) is symmetric then a solution (x d) occurs together with (;x d) or something similar. This is the case in examples 4.1 and 6.1 below. Now, L in de nition 14 is not zero-dimensional. Since Lf V has a local maximum at the origin r(Lf V )(0) = 0 (see 20, section 6.1]), so for x = 0 d = 0 any value of satis es the equations. Thus we have a one-dimensional component of the solution manifold, given by d = 0 x = 0 and arbitrary. This is not a big problem in symbolic computation for two reasons: there are still only nitely many d:s solving the system and the BGK algorithm works as long as L \ R d] 6= 0. we can use the Rabinovich trick described in section 2.4 to exclude the case d = 0. However, the one-dimensional component of Z (L) does cause trouble when numerical methods are applied. And the experience of the author is that it is not convenient to use the Rabinovich trick in numerics, unfortunately. We introduce the ideal LR = L + h u d ; 1 i. . kx1 : : : xn d u]. (17). where u is the Rabinovich variable. In the \normal" case thus dim LR = 0. We demonstrate on a simple example how to apply the methods described above:. Example 4.1 The equation 2 M dtd 2 + K dtd = P ; Pe sin . (18). is a simple model of a single machine electrical power system. Here is a phase dierence between two voltages, P is proportional to the power delivered and Pe is the mechanical 9.

(11) power input. For details, consult 21, chapter 4]. If we put M = K = Pe = 1 and write x1 = x2 = _ we get x_ 1 = x2 x_ 2 = ;x2 + P ; sin x1 The equilibria are thus given by x2 = 0 P = sin x1 . If P = 0 then the origin is an equilibrium which is seen to be stable using linearization. It is not dicult to see that the origin is not globally stable. A local Lyapunov function helps us estimate the domain of attraction (DOA) of the origin. In order for this example to t into the real algebraic framework we have to approximate the sine-function occurring with a polynomial (other approaches than approximation are also possible). If we do the approximation sin x1 x1 ; 16 x31 we will not overestimate the DOA. Thus we get a system determined by the vector eld f = (;x2 ;x2 ; x1 + 61 x31 ) (19) A local Lyapunov function for this f is (20) V = 23 x21 + x1x2 + x22 A computation of the plex GB for LR w.r.t. the term-ordering u > x2 > x1 > > d gives us the minimal polynomial for d which is. p = 36 d4 + 376 d3 ; 4770 d2 ; 1575 d ; 3375 having. (21). ;17:776 : : : . 7:7019 : : : for real roots. LR is not in g.p. so, disregarding the background of the problem, we can't be sure a priori that d = 7:7 is a solution to the optimization problem. Since we have access to the entire GB in this case it easy to check that this d-value corresponds to a real zero of the ideal. Thus we have D = (;1 7:7) here. For some computational aspects of this example, see section 6. 2. 5 Robustness against Structured Uncertainty In engineering it is very important to ensure that the solutions and designs made for a particular problem do not rely too heavily on the underlying models. This is known as the robustness problem in control theory. In this section we will use Lyapunov theory to deal with a special case of the robustness problem. An instance of the nonlinear robustness problem is that of a system with structured uncertainty: x_ = f (x

(12) ) x = (x1 : : : xn ) (22) where

(13) = (

(14) 1 : : :

(15) q ) are real, time-invariant parameters. The components of f are assumed to be polynomial in both x and

(16) :. fi 2 R x1 : : : xn

(17) 1 : : :

(18) q ] i = 1 : : : n If a given function V is a Lyapunov function for f or not will now depend on

(19) :. (23). Denition 5.1 Given V , de ne the region H R q to be the set of all

(20) such that V is a. 2. local Lyapunov function for the system (22).. 10.

(21) We suppose that H is non-empty, and that we have access to a nominal value

(22) 0 2 H of the uncertain parameters. The regions W and W in de nition 4.2 now depend on

(23) : W (

(24) ) = f x 2 R n (Lf V )(x

(25) ) < 0 g f0g (24) W (

(26) ) = f x 2 R n (Lf V )(x

(27) ) 0 g and we want to know if Bd W (

(28) ) or not. In the previous section the problem of nding the largest d such that Bd W holds for

(29) 0 was considered. Here, we keep d xed and vary

(30) . First we consider an instance of the (local) stability robustness problem: Given d, for what values of

(31) is Bd W (

(32) )?. In words: for what values of the parameters

(33) can we guarantee stability within Bd using the local Lyapunov function V ? If we call this set Md we have. Md = f

(34) 2 H Bd W (

(35) ) g (25) where H is the set of parameters allowed for (de nition 5.1) and W (

(36) ) is de ned in equation (24). This problem can be approached with Grobner bases, in the following manner. The ideals L LR now lie in R x1 : : : xn

(37) 1 : : :

(38) q ]. We are interested in nding out for what

(39) 2 R q that L has real zeroes. This means that we are looking for

(40) such that @Bd \ @W (

(41) ) 6= (26) Having done this we obtain a hypersurface in parameter space, giving us information on @Md . We need to know in which component of the region bounded by @Md that (26) is satis ed. This is easy, since we know that

(42) 0 belongs to this component. Also, we need to know if @Md intersects @H or not, i.e. if all the parameters de ned by our hypersurface are in the allowed region H or not. Furthermore, we would like to know if the region bounded by the hypersurface is connected or not which is certainly not a simple problem. The ideal L is generated by n + 2 polynomials in n + q + 1 variables (the x, the

(43) and ): it typically contains an element in

(44) only. The Grobner base for L w.r.t. a termordering of the type fx1 : : : xn g > f

(45) 1 : : :

(46) q g (27) gives us a generator for the wanted subideal. The following example indicates how these ideas work in practice.. Example 5.1 Let f (x) = (;x1 + x21 x2 ;x2 ) V = x21 + x22 and d = 1. It is easy to see that V is a local Lyapunov function for the system i > 0. We have

(47) = ( ) and

(48) 0 = (2 1). The Grobner base for the ideal L with respect to x2 > x1 > > > contains the polynomial p = 4 3 2 + 27 4 ; 16 4 ; 132 2 2 + 64 3 ; 132 2 (28) ;96 2 + 4 2 + 64 ; 16 11.

(49) Example no BGK plex GB 4.1 15 291 6.1 4.5 10.1 6.2 43 > 1000 6.3 41 > 1000 6.4 112 > 1000. Table 1: Computation times (CPU-seconds) for dierent methods for nding the critical level of a local Lyapunov function. So, in parameter space, the region for which B1 W ( ) is bounded by the curves f ( ) . p( ) = 0 g f ( 0) g. (29). where p is as in equation (28). (Recall that H is de ned by > 0). The above analysis is somewhat incomplete: we have not showed that M is connected, for instance. If we assume that this is the case we can conclude that. M1 = f ( ) 2 R 2 p( ) < 0 ^ > 0 g. (30). 2. since p(

(50) 0 ) < 0.. The problem of performance robustness is possible to handle in a way similar to the one above. For details we refer the reader to 12] or 14].. 6 Complexity Issues In this section we compare the time for computing a plex GB for LR with that of BGK. A theoretical complexity analysis seems dicult. However, for a comparison the two most obvious methods were applied on some examples in Maple version V release 2 on a Sun Sparc 2 with 32 MBytes of memory. The running times obtained are displayed in table 1. The methods are applied to the ideal LR . The plex GB:s are computed w.r.t. the termordering u > xn > : : : > x1 > > d. The following examples were designed to provide some extra material for the complexity comparison above.. Example 6.1 If. V = 12 (x21 + x22) f = (;x1 + x22 ;2x2 ) 12. (31).

(51) we get that the plex GB for LR w.r.t. d < < x1 < x2 < u has 5 elements and contains the polynomial p = 8d2 ; 71d + 4 (32) having roots 8.82 and 0.057. LR is not in g.p. and, in fact, d = 0:057 does not give a real solution in x2 . 2. Example 6.2 The system x_ 1 = ;x1 + 2x22 x_ 2 = x21 ; x2 + x23 x_ 3 = ;x21 ; x3. (33). has a local Lyapunov function V = 12 (x21 + x22 + x23 ). Computing a plex GB for LR seems to be intractable. However, as we apply the BGK algorithm we nd that LR contains the polynomial. p = 17334272 d7 ; 431947776 d6 + 3748548608 d5 ; 15851676672 d4 +37610535168 d3 ; 50549069952 d2 + 33916643352 d ; 7476843429 having the approximate roots. (34). 0 0:395 1:686 2:077 5:146 12:417 and two non-real roots. A computation of the Hilbert series of LR shows that the degree of LR is 7. This means that LR is in g.p. so the root d = 0:395 corresponds to a real solution in x and . Thus D = (;1 0:395). 2. Example 6.3 The system f = (;x1 ; x1 x2 ;x2 ; x31 ) with the local Lyapunov function. V = x21 +2x1 x2 +2x22 gives a polynomial for the critical d of degree 7 with coecients 1010 . 2. Example 6.4 The system f = (;2x1 + x42 + 17x22 3x1 x52 ; x2 ) with the local Lyapunov. function V = x21 + x22 gives a polynomial for the critical d of degree 10 with coecients 1014 .. 2. Acknowledgement This work was nancially supported by the Swedish Council for Technical Research (TFR).. References 1] V.I. Arnold. Mathematical Methods of Classical Mechanics, volume 60 of GTM. Springer, second edition, 1989. 2] D.S. Arnon, G.E. Collins, and S. McCallum. Cylindrical algebraic decomposition i: The basic algorithm. SIAM J. Comput., 13(4):865{877, November 1984. 13.

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