Math.comput.sci. 2 (2009), 493–515 c 2009 Birkh¨
auser Verlag Basel/Switzerland 1661-8270/030493-23, published online March 5, 2009 DOI 10.1007/s11786-008-0067-1
Mathematics in Computer Science
Investigating Algebraic and Logical Algorithms to Solve Hopf Bifurcation Problems in Algebraic Biology Thomas Sturm, Andreas Weber, Essam O. Abdel-Rahman and M’hammed El Kahoui Abstract. Symbolic methods to investigate Hopf bifurcation problems of vector fields arising in the context of algebraic biology have recently obtained renewed attention. However, the symbolic investigations have not been fully algorithmic but required a sequence of symbolic computations intervened with ad hoc insights and decisions made by a human. In this paper we discuss the use of algebraic and logical methods to reduce questions on the existence of Hopf bifurcations in parameterized polynomial vector fields to quantifier elimination problems over the reals combined with the use of the quantifier elimination over the reals and simplification techniques available in REDLOG. We can reconstruct most of the results given in the literature within a few seconds of computation time and extend the investigations on these systems to previously not analyzed related systems. Especially we discuss cases in which one suspects that no Hopf bifurcation fixed point exists for biologically relevant values of parameters and system variables. Here we focus on logical and algebraic techniques of finding subconditions being inconsistent with the hypothesis of the existence of Hopf bifurcation fixed points. Mathematics Subject Classification (2000). 37G10; 92C45; 03C10. Keywords. Hopf bifurcations; oscillations; dynamical systems; quantifier elimination over the reals; logical inconsistency checking.
1. Introduction Symbolic methods to investigate Hopf bifurcation problems of vector fields arising in the context of algebraic biology have recently obtained renewed attention [1, 2, 11, 13]. A major reason for this attention is the relationship between
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Hopf bifurcation fixed points and the occurrence of oscillations. Although this relationship is subtle – we refer to [1, 13] for nice introductions containing further references – the topic of investigating “oscillations” in systems for biologically or chemically relevant values of parameters is of such great interest that considerable work has been done to investigate various biological and chemical systems with respect to Hopf bifurcation fixed points, see e.g. [10, 12, 14, 15, 19] to mention only a few. There exist software packages such as AUTO1 or XPPAUT2 , which locate Hopf bifurcations by means of numerical calculations. They allow one to evidence the existence of Hopf bifurcations. However, if one wants to prove that no Hopf bifurcation fixed point exists at all, then numeric methods fail in principle. Furthermore, if there are ranges of parameters for which there are Hopf bifurcations but these ranges are rather small, then numeric methods might not detect them – a possibility that is not only of theoretical interest in the context of algebraic biology and chemical reaction systems. Whereas theoretically the problem is known to be decidable [9, 11, 13] the symbolic investigations carried out for specific parameterized polynomial vector fields arising from larger examples, e.g. the ones investigated in [1, 11], have not been fully algorithmic up to now but required a sequence of symbolic computations intervened with ad hoc insights and decisions made by a human, and sometimes of sophisticated coordinate transforms. The aim of this paper is to demonstrate how to proceed fully algorithmically for the examples discussed there. For this we proceed as follows: We use the method described by El Kahoui and Weber [9, 20] to generate from the symbolic description of the respective ordinary differential equation a first-order formula in the language of ordered rings, where our domain is the real numbers. This is always possible if the vector field consists of polynomials in the variables and parameters. If one suspects that there is no Hopf bifurcation fixed point or one just wants to assert that there is one, then one can apply quantifier elimination to the existential closure, i.e. all parameters are existentially quantified, of our generated formula. More generally, applying quantifier elimination to the original formula yields in principle a quantifier-free semi-algebraic description of the parameters for which Hopf bifurcation fixed points exist. In practice this latter variant does not finish within reasonable time at present. There is, however, an alternative, which provides at least one sample solution in the positive case; viz. extended quantifier elimination [22]. Moreover, if no Hopf bifurcation fixed point exists it is often feasible to construct a subconjunction of the exact condition described in [9, 20]; if quantifier elimination of the existential closure of such a subconjunction yields false one knows a fortiori that the exact Hopf condition from [9, 20] must be false, too. 1 indy.cs.concordai.ca/auto/ 2 www.math.pitt.edu/
~bard/xpp/xpp.html
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For the quantifier elimination we use REDLOG [6, 16], which provides an automatic combination of virtual substitution methods [21] with partial CAD [5]. After noticing quite soon that most of our generated formulae were intractable by the regular elimination method, we developed a modified quantifier elimination procedure called positive quantifier elimination. This is based on the observation that in the considered examples, the variables as well as almost all of the parameters can be restricted to positive values. It has turned out that this knowledge greatly supports the elimination process. A current version of REDLOG including positive quantifier elimination is available for free download on the REDLOG website3 . All computational examples discussed throughout this paper are contained in the online database REMIS there [16]. We can reconstruct the results of [1, 2] in some seconds of computation time. Also the main example given in [11] could be handled in principle – without using the sophisticated coordinate transform used there. Moreover, we can analyze [2, System (1)] and [2, System (2)] with respect to Hopf bifurcation fixed points for n up to 12 quite easily, a task that has not been undertaken in [2]. The main results of Section 2, 3.1, and 3.3 have been published in [17], all of Section 3.2 is new in this journal version of [17].
2. Positive quantifier elimination In the context of algebraic biology it is often known that all (or at least most) variables and parameters are positive. Let us assume first that all variables and parameters are positive. Such a global information greatly supports the quantifier elimination process by virtual substitution. We call the resulting procedure positive quantifier elimination. We do not go into technical details on this here but indicate a few major points, where positivity can be exploited. To start with, recall that the practical applicability of substitution methods depends crucially on efficient and powerful simplification of intermediate and final results [7]. These simplification methods can be considerably improved by positivity assumptions on all variables: • There is an established simplification strategy which checks for terms that are (strict) trivial squaresums [7]. These are by definition sums of monomials with positive coefficients and even degree in all variables (and a positive absolute summand). Obviously, trivial squaresums are positive semi-definite, and strict trivial squaresums are positive definite. Thus corresponding atomic formulae can be evaluated to true or false or at least be simplified to equations or negated equations. For instance, we generally know that 3x2 y 4 + 7x2 ≥ 0 . On the assumption that all variables are positive, we need not even check for even degrees. For instance, we may equivalently replace by true the atomic 3 www.redlog.eu
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formula 3xy + 7x > 0 . Notice that technically we need only check for the literal occurrence of a minus sign in some canonical recursive or distributive representation of left hand side polynomials. • Exploiting (generalized) trivial squaresums can be extended to testing multiplicative factors of the occurring polynomials. For instance, with positive quantifier elimination we can go ax2 + axy 2 − bx2 − by 2 ≤ 0 ←→ (ax − b)(x + y 2 ) ≤ 0 ←→ ax − b ≤ 0 . Since our positivity assumptions greatly increase our chance to discover such squaresums, it pays off in many cases to factorize more systematically. In order to optionally do so we have introduced into REDLOG a switch rlsifaco (“simplifier factorization with ordering relations”), which is going to play a role in the discussion of our computation examples later on. Next, we illustrate two situations, where we can exploit positivity within the virtual substitution process itself: • When virtually substituting into atomic formulae quotients with parametric denominators one must in general multiply with the square of the denominator in order to preserve signs. As an example consider c ≡ acy + acz 2 − by 2 − 2byz 2 − bz 4 > 0 . (ax − b > 0) x// y + z2 With positive quantifier elimination one heuristically discovers non-negative denominators or at least non-negative factors based on strategies as sketched for simplification above. In our example, this allows us to simply drop the positive principal denominator after formal substitution: c (ax − b > 0) x// ≡ ac − by − bz 2 > 0 . y + z2 At this point one possibly avoids subsequent degree violations for quantified variables in the denominator; in our example, z might be a quantified variable. • There is a degree decreasing shift operation which essentially divides all exponents of quantified variables by the GCD of these exponents [8]. Doing so, one obviously has to take care of positivity, e.g., when switching between even and odd degrees. This operation can be simplified and slightly generalized on our positivity assumption. As a simple example consider the following transformation, where with positive quantifier elimination we need not add a condition z ≥ 0: ∃z(az 2 − bz 4 > 0 ∧ z 4 − 3z 2 < 0) ←→ ∃z(az − bz 2 > 0 ∧ z 2 − 3z < 0) . For our set of input formulae discussed throughout this paper, it turns out that all optimizations discussed above become relevant at some point. For input
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problems from other fields of applications, positive quantifier elimination has not been systematically evaluated yet. Notice that positive quantifier elimination as a concept is not at all restricted to virtual substitution methods. For instance, during the projection phase of partial CAD one could drop polynomials that are definite on positivity assumptions. It is theoretically quite straightforward to equip positive quantifier elimination with an optional argument to pass a list of parameters and variables that are not known to be positive. This is, however, not yet implemented at present. We are now going to discuss how to alternatively deal with a parameter that is not known to be positive, say λ1 . In applications, it might happen that the parameter occurs in a linear equation. In this special case, one can eliminate it by solving for λ1 in one of the linear equations involving this parameter. Algorithmically, this strategy can be realized in a preprocessing step, e.g. the one realized by Brown and Groß [4]. This preprocessing is in fact a bit more general but still restricted to special cases. One generally applicable procedure is to perform a case distinction on λ1 > 0, λ1 < 0, or λ1 = 0. This results in three positive quantifier elimination runs, where in the two latter cases we substitute λ1 ← −λ1 and λ1 ← 0, respectively. Iterating this procedure for other general parameters λ2 , . . . , λn yields 3n many case distinctions, so that the case distinction is only feasible for small n, i.e. there are only few parameters not known to be positive. There is another theoretically interesting option: For quantified non-positive variables we observe that every real number is a difference of two positive real numbers and go λ1 ← λ1 − λ01 , where λ01 is quantified in the same way as λ1 . Notice that both our substitution techniques sketched above are completely algorithmic. Moreover they can easily be implemented on top of any implementation of positive quantifier elimination.
3. Computation examples 3.1. Models of genetic circuits The examples investigated in [1] consist of a family of ordinary differential equations of the following form: d G(t) = ϑ γ0 − G(t) − G(t)P (t)n , dt d P (t) = nα γ0 − G(t) − G(t)P (t)n + δ M (t) − P (t) , dt d M (t) = λ1 G(t) + γ0 µ − M (t) , ([1, Sys. (1)]) dt where n is a natural number. We have renamed λ to λ1 because lambda is obviously an inconvenient choice for a variable name in Lisp systems. All variables and parameters except λ1 are known to be positive. For a description of the model from which this family of ordinary differential equations arise, we refer to [1].
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eq1n := diff(G(t),t)=theta*(gamma0-G(t)-G(t)*P(t)^n); eq2n := diff(P(t),t)= n*alpha*(gamma0-G(t)-G(t)*P(t)^n)+delta*(M(t)-P(t)); eq3n := diff(M(t),t)=lambda1*G(t)+gamma0*mu-M(t); fcns1 := {G(t),P(t),M(t)}; params1 := [theta,alpha,gamma0,mu,lambda1]; paramcondlist := [theta>0, gamma0>0, mu>0, delta>0, alpha>0]; funccondlist := [G(t)>0, P(t)>0, M(t)>0]; lown:= 0; upn :=10; for nn from lown to upn do eq11 := eval(subs(n=nn,eq1n)); eq21 := eval(subs(n=nn,eq2n)); eq31 := eval(subs(n=nn,eq3n)); DEHopfexistence({eq11,eq21,eq31}, fcns1,params1,funccondlist,paramcondlist); od;
Figure 1. A Maple script for generating the first-order input formulae ϕ2 , . . . , ϕ10 for Section 3.1. Using a Maple library4 implementing the methods described in [9] we can generate first-order formulae stating the question on the existence of Hopf bifurcation fixed points (taking into account the known positivity conditions) by the Maple script in Figure 1. Here we generate formulae up to n = 10 in a slight extension of the cases considered in [1]. There it is proved that no Hopf bifurcations exist for n ≤ 8 but for n = 9. In Figure 2 we present the generated first-order input formulae for n = 2 and n = 9. The system variables are renamed to vv1, vv2, vv3 by the Maple library; as they occur in quantified form only, this renaming is of little concern. For better readability they are typeset here as v1 , v2 , v3 . Apart from this, we literally give the formulae as they are output by Maple. They happen to contain some redundant parentheses. As indicated in the introduction, we consider for now exclusively the existential closures ∃ϕ2 , . . . , ∃ϕ10 of our formulae. Then quantifier elimination yields either true or false depending on whether or not Hopf bifurcations exist. We are going to algorithmically treat the positive cases in more detail later on in Section 3.1.3. 3.1.1. Applying regular quantifier elimination. Using the regular quantifier elimination of REDLOG or QEPCAD [3] with their default settings we could not decide 4 The current version of the library can be obtained at cg.cs.uni-bonn.de/project-pages/ symbolicanalysis/.
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∃v1 ∃v2 ∃v3 (((0 < v1 ∧ 0 < v3 ) ∧ 0 < v2 ) ∧ (((((((ϑ(γ0 − v1 − v1 v32 ) = 0 ∧ λ1 v1 + γ0 µ − v2 = 0) ∧ 2α(γ0 − v1 − v1 v32 ) + δ(v2 − v3 ) = 0) ∧ (0 < ϑδ + ϑv32 δ + 2λ1 ϑv1 v3 δ ∧ ϑv32 + 2ϑδ + 8ϑαv1 v3 + 4αv1 v3 ϑδ + 4αv1 v33 ϑδ + 8αv1 v3 δ + δ 2 + ϑδ 2 + 16α2 v12 v32 + ϑv32 δ 2 + 2ϑ2 v32 δ + ϑ2 v34 δ + δ + ϑ2 δ + ϑ2 + 2ϑ2 v32 + ϑ2 v34 + 4αv1 v3 + ϑ + 2ϑv32 δ + 8ϑv33 αv1 − 2λ1 ϑv1 v3 δ = 0)) ∧ 0 < ϑ) ∧ 0 < γ0 ) ∧ 0 < µ) ∧ 0 < δ ∧ 0 < α)),
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∃v2 ∃v1 ∃v3 (((0 < v1 ∧ 0 < v3 ) ∧ 0 < v2 ) ∧ (((((((ϑ(γ0 − v1 − v1 v39 ) = 0 ∧ λ1 v1 + γ0 µ − v2 = 0) ∧ 9α(γ0 − v1 − v1 v39 ) + δ(v2 − v3 ) = 0) ∧ (0 < ϑδ + ϑv39 δ + 9λ1 ϑv1 v38 δ ∧ 162ϑv317 αv1 + 162ϑαv1 v38 + 162αv1 v38 δ + ϑ + 2ϑv39 δ + ϑ2 v318 δ + ϑv39 + 2ϑδ + 81αv1 v38 ϑδ + 81αv1 v317 ϑδ + δ 2 + ϑδ 2 + ϑ2 δ + ϑ2 + 2ϑ2 v39 + ϑ2 v318 + 6561α2 v12 v316 + 2ϑ2 v39 δ + δ + 81αv1 v38 + ϑv39 δ 2 − 9λ1 ϑv1 v38 δ = 0)) ∧ 0 < ϑ) ∧ 0 < γ0 ) ∧ 0 < µ) ∧ 0 < δ ∧ 0 < α)). Figure 2. The input formulae ϕ2 and ϕ9 generated by the Maple script in Figure 1. The typesetting v1 , . . . , v3 indicates Maple variables vv1, . . . , vv3 generated by the function DEHopfexistence in the script.
any of our existential sentences ∃ϕ2 , . . . , ∃ϕ10 . Systematically trying some REDLOG switches, we managed to determine special settings that deliver results for n = 4, 6, 8, 10. Table 1 summarizes these experiments, where all rows with unsuccessful nonstandard switch settings have been deleted. The keyword sigxcpu in the result column indicates that the computation has been automatically interrupted after exceeding our self-imposed limit of 10 minutes of CPU time. The switch rlsifaco (“simplifier factorization with ordering relations”) toggles the factorization of left hand side polynomials of ordering inequalities for the purpose of simplification; rlqeprecise (“quantifier elimination using precise test points”) avoids to some extent the substitution of non-standard symbols like ∞ or ε with virtual substitution; rlqevarseltry (“quantifier elimintion variable selection try systematically”)
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Table 1. Applying regular quantifier elimination to the existential closures ∃ϕ2 , . . . , ∃ϕ10 of the formulae in Section 3.1. We systematically try several switch settings in REDLOG. n rlsifaco 2 ◦ 3 ◦ 4 ◦ 4 ◦ 4 • 5 ◦ 6 ◦ 6 • 7 ◦ 8 ◦ 8 ◦ 8 • 9 ◦ 10 ◦ 10 •
rlqeprecise ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
rlqevarseltry ◦ ◦ ◦ • • ◦ ◦ • ◦ ◦ • • ◦ ◦ •
result sigxcpu sigxcpu sigxcpu false false sigxcpu sigxcpu false sigxcpu sigxcpu false false sigxcpu sigxcpu true
time (s) > 600 > 600 > 600 113.26 84.10 > 600 > 600 170.10 > 600 > 600 407.63 346.77 > 600 > 600 543.82
does not only apply the standard heuristics for choosing the next variable to be eliminated but additionally breaks ties by trying all heuristically best choices and proceeding with the smallest result. By default all theses switches are off – for good reason, since this provides in general the by far best performance. Altogether, we consider the results in Table 1 not at all satisfactory: a regular user of some software must not be expected to run lengthy experiments with switch settings in order to hopefully obtain some results. 3.1.2. Applying positive quantifier elimination. Our idea is now to improve the situation by making use of the observation that all variables and parameters except λ1 are known to be positive. This observation has in fact been used also in [1] by solving for λ1 in one of the linear equations involving this parameter, after which all remaining variables and parameters are known to be positive, and then performing a “hand analysis.” Our approach avoids any hand analysis, because by using the generally applicable case distinction approach on λ1 , cf. Section 2, we can solve each of the examples in less than half a minute of computation time, cf. the left hand side part of Table 2. Our results for n = 2, . . . , 9 confirm those in [1]; for our additional case n = 10, we discover the existence of a Hopf bifurcation fixed point. The computation times are the sums of times for all three respective eliminations. For this example, however, the cases λ1 < 0 and λ1 = 0 turn out to take no considerable time.
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Table 2. Applying positive quantifier elimination to the existential closures ∃ϕ2 , . . . , ∃ϕ10 of the formulae in Section 3.1. We use the standard switch settings in REDLOG; rlsifaco is switched on, which is a general recommendation for positive quantifier elimination. n 2 3 4 5 6 7 8 9 10
∃ϕn false false false false false false false true true
∃ϕn [λ1 ← −λ1 ] false false false false false false false false false
∃ϕn [λ1 ← 0] false false false false false false false false false
time (s) < 0.01 19.28 21.58 19.09 23.72 23.89 22.35 0.17 0.17
∃ϕ[λ1 ← λ1 − λ01 ] sigxcpu sigxcpu sigxcpu sigxcpu sigxcpu sigxcpu sigxcpu true true
time (s) > 600 > 600 > 600 > 600 > 600 > 600 > 600 69.10 69.19
The right hand side part of Table 2 summarizes the alternative approach introducing instead of a case distinction a new variable λ01 and then substituting λ1 ← λ1 − λ01 . It performs considerably worse on these examples. This might appear not surprising from the point of view that the complexity of our elimination procedure is exponential in the number of quantified variables. On the other hand, however, iterating case distinctions is exponential as well. 3.1.3. Obtaining sample points. For the cases in which a Hopf bifurcation fixed point exists the extended quantifier elimination [22] in REDLOG computes in addition a “sample point” fulfilling the existential quantifiers. For the following experiments, we turn on the switch rlqeaprecise (“quantifier elimination with answer using precise test points”), which is the analogue for extended quantifier elimination of rlqeprecise discussed in Section 3.1.1 above. It reduces the introduction of infinitesimals to some extent though not completely in general. In addition, we switch off rlsiexpla (“simplifier explode always”) so that atomic formulae are not split via polynomial factorization unless the Boolean structure is preserved. For the system with n = 9 we obtain within 0.11 s the following sample point: α = 1, γ0 =
−2 ·
√
1180986 − 2187 · ε1 + 2187 , 2187
δ = 1, λ1 = L(ε1 , ε2 ) , µ = M (ε1 , ε2 ) , ϑ = T (ε1 , ε2 ) ,
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√ −2 · 1180986 − 2187 · ε1 + 2187 v1 = , 43048908 v2 = 3 , √ 3 v3 = 3 . Here the ε1 , ε2 are positive infinitesimals, which have to be interpreted as follows: There are 0 < ε1 , ε2 ∈ R, which yield valid sample points, and moreover all positive choices less than ε1 and ε2 , respectively, yield valid sample points too. The symbols L, M , and T denote rather complicated algebraic expressions, which we present for the sake of completeness in Appendix A. In order to get an idea about approximate solutions we set ε1 = ε2 = 0, which yields: γ0 = 0.00618948546946 , ϑ = 0.0000571304095036 ,
λ1 = 9540696.11947 , v1 = 0.000000314442464411 .
For the case n = 10 the same procedure yields within 0.09 s the following approximate sample point, where again γ0 , λ1 , ϑ, and v1 are approximated by fixing two infinitesimals to 0: α = 1, γ0 = 0.0100554964908 , δ = 1, λ1 = 17617230.5528 , µ = 0, ϑ = 0.0000211443608455 , v1 = 0.000000170287832189 , v2 = 3 , v3 = 1.24573093962 . One can now systematically enumerate further solutions by adding to the input formulae conditions prohibiting for one or several variables or parameters the solutions already found and then rerunning extended quantifier elimination.
3.2. Extended models of genetic circuits 3.2.1. Reduced quasi-steady state approximation. In [2], which continues the work in [1] discussed so far, there are three extended families of models on genetic circuits introduced. Only one of the three families is, however, analyzed with respect to
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Table 3. Results of quantifier elimination on Hopf conditions for [2, Sys. (3)]. In analogy to the left hand side part of Table 2, we refer to the input formula generated for instance n by ϕn and treat the only non-positive variable λ1 occurring there by a case distinction on its sign. n 2 3 4 5 6 7 8 9 10
∃ϕn false false false false false false false true true
∃ϕn [λ1 ← −λ1 ] ∃ϕn [λ1 ← 0] false false false false false false false false false false false false false false false false false false
time (s) 0.01 15.17 21.74 22.64 8.70 12.11 9.56 0.41 0.41
Hopf bifurcation fixed points, viz. [2, Sys. (3)]: d G(t) = ϑ γ0 − G(t) − G(t)P (t)n , dt d M (t) = λ1 G(t) + γ0 µ − M (t) , dt d nα(γ0 − G(t) − G(t)P (t)n ) + δ(M (t) − P (t)) P (t) = . Pn−1 2 i dt i=0 (i + 1) Ki P
([2, Sys. (3)])
This system is in fact closely related to our [1, Sys. (1)] discussed in Section 3.1 above: In order to recover [1, Sys. (1)] from [2, Sys. (3)] it is sufficient to replace the right hand side of the last equation of [2, Sys. (3)] by its numerator. For the system [2, Sys. (3)] under consideration now, we generated the Hopf conditions for n = 2, 3, . . . , 9 using an auxiliary variable D for the denominator of the last equation, which is also known to be positive, and using the additional equality constraint n−1 X D= (i + 1)2 Ki P i . i=0
The results are given in Table 3. They coincide with the findings of the “hand reduction” performed in [2]: For the biologically relevant values of the parameters and variables a Hopf bifurcation fixed point exists for the same values of n as is the case in the system considered in [1], namely for n ≥ 9, whereas for n ≤ 8 no Hopf bifurcation fixed point exists. 3.2.2. Initial model. We have also investigated [2, Sys. (1)], which is the model directly derived from the chemical reaction system discussed in [2]. Notice that this model was not directly investigated with respect to Hopf bifurcations there.
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lown:= 2; upn := 12; for n from lown to upn do fcns1 := {G(t), H(t), M(t), seq((cat(’P’, i))(t), i = 1 .. n)}; eqnseq := diff(G(t),t) = theta*H(t) - alpha*G(t)*P||n(t), diff(H(t),t) = -theta*H(t) + alpha*G(t)*P||n(t), diff(M(t),t) = rhof*G(t) + rhob * H(t) - deltaM * M(t), diff(P1(t),t) = beta*M(t) - deltaP * P1(t) + 2*A1 + add((cat(’A’, i)), i = 2 .. n-1), seq(diff((cat(’P’, i))(t),t) = (cat(’A’, i)) - (cat(’A’, i-1)), i = 2 .. n-1), diff(P||n(t),t) = -(cat(’A’, n-1)) + theta*H(t) - alpha*G(t)*P||n(t); subAseq:=seq(A||i = km||i*P||(i+1)(t) - k||i*P||i(t)*P1(t), i= 1 .. n-1); eqnseqsubs := seq(lhs(eqnseq[j])=subs(subAseq,rhs(eqnseq[j])), j=1..nops([eqnseq])); funccondlist:=[G(t)>0, H(t)>0, H(t)+G(t)=gamma0, M(t)>0, seq((cat(’P’, n-i+1))(t) > 0, i = 1 .. n)]; paramseq := theta,alpha,gamma0,rhof,rhob,beta,deltaP,deltaM, seq(km||i,i=1..n),seq(k||i,i=1..n); paramcondseq:=theta>0,alpha>0,gamma0>0,rhof>0,rhob>0,beta>0, deltaP>0,deltaM>0, seq(km||i>0,i=1..n),seq(k||i>0,i=1..n); DEHopfexistence({eqnseqsubs},fcns1,[paramseq], funccondlist,[paramcondseq]); od;
Figure 3. A Maple script for generating the first-order input formulae for [2, Sys. (1)].
We used the Maple script given in Figure 3 to generate the system of ordinary differential equations corresponding to [2, Sys. (1)] for various n, and to generate first-order formulas describing the exact Hopf bifurcation condition. For n = 2 the script successfully generated a first-order formula which was shown to be equivalent to false by REDLOG within a few milliseconds. For larger values of n the sequence of Hurwitz determinants of the characteristic polynomial of the Jacobian matrix – which are computed for the Hopf conditions, cf. [9, Sect. 3.3] – quickly become so big that Maple could not even generate them. Using subconditions. If one suspects that there is no Hopf bifurcation fixed point (for the ranges of parameters and variables of interest), then one can use the following observations:
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• The first-order formula stating the exact condition for the existence of Hopf bifurcation fixed points is an existentially quantified conjunction of various subconditions. • In addition to the equations stating the equilibrium conditions and the constraints on the variables and parameters set by the users these subconditions consist of one equality and several inequality constraints involving Hurwitz determinants on the characteristic polynomial of the Jacobian matrix, cf. [9, Sect. 3.3]. • Thus one can conclude the following: If quantifier elimination returns false on an existentially quantified subconjunction of the formula then a fortiori the full existentially quantified conjunction must also be false. So we can leave out several of the equality or inequality constraints involving the Hurwitz determinants, and if quantifier elimination returns false, then one can nevertheless conclude that no Hopf bifurcation fixed point exists (fulfilling the conditions on the variables and parameters). If, however, quantifier elimination returns true, then no direct conclusion is possible but one has to go into another iteration adding some of the left out constraints involving Hurwitz determinants. Using this strategy we generated the “smallest” Hurwitz determinant on the characteristic polynomial. Maple could generate the characteristic polynomial of the Jacobian matrix of [2, Sys. (1)] for n ≤ 8; for n = 9 it failed after having allocated more than 1.5 GB of memory. A similar problem occurred with Mathematica. For all n ≤ 8 quantifier elimination reduced all generated formulae to false thus proving that no Hopf bifurcation fixed point exists for n ≤ 8 (and positive values of variables and parameters). So using our Maple library we could only proceed for n ≤ 8. Remarkably, using the computer algebra system REDUCE already for computing the characteristic polynomial of the Jacobian for the case n = 9, we succeded, however, within some seconds of computation time with the function char poly and the switch setting on cramer. Similarly to our heuristical approach discussed so far, one can use other necessary conditions for the existence of a Hopf bifurcation fixed point. One of these conditions is that the Jacobian determinant must be non-zero at the equilibrium points. We have implemented a function which generates such a subcondition by generating, instead of conditions involving all the Hurwitz determinants on the Jacobian, a subformula stating that the Jacobian determinant is non-zero. When accordingly substituting the function DEHopfexistence in the script given in Figure 3 with a test for a non-zero Jacobian determinant at the equilibrium points (and obeying the parameter conditions), Maple could quickly generate corresponding formulae for n ≤ 12. It turned out that all Jacobian determinants are zero for [2, Sys. (1)]. Eliminating an algebraically constrained variable. Using the state equation H(t) + G(t) = γ0
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Table 4. Results of quantifier elimination for [2, Sys. (2)] on a subsystem of the Hopf condition omitting the equality constraint on the denominator. We refer to the input formula generated for instance n by ϕn . n ∃ϕn 2 false 3 false 4 false 5 false 6 false 7 false 8 false 9 true 10 true
time (s) 0.09 0.18 0.18 0.70 16.92 18.58 18.71 22.26 23.72
one can eliminate H(t) in the equations of [2, Sys. (1)]. When we analyze the obtained system with our algorithms we find for all values of n under consideration (n ≤ 12) that this system has a non-singular Jacobian (at the equilibrium points). Moreover the subcondition using the “smallest” Hurwitz determinant evaluates to true. So we have to add conditions involving the “second smallest” Hurwitz determinant. Notice that this already yields the full condition for these examples. Unfortunately, our runs of quantifier elimination on the formulas with that full condition did not terminate within one day of computation time even for n = 2 or n = 3. 3.2.3. Quasi-steady state approximation. Using a quasi-steady state approximation Boulier et al. [2] derive a much simpler system from their [2, Sys. (1)]: Similarly to the situation for [2, Sys. (1)] their [2, Sys. (2)] in the form presented in [2] uses state variables G(t) and H(t) obeying H(t) − G(t) = γ0 . When analyzing this system one therefore finds that it has singular Jacobians. When eliminating H(t) by γ0 − G(t) as above the formulae generated by the Maple script in Figure 4 gave the results and running times of the quantifier elimination given in Table 4. The Maple script in Figure 4 generates a subcondition for the full Hopf condition for [2, Sys. (2)], as the equality constraint for the denominator auxdenom occurring in [2, Sys. (2)] is omitted. However, for all cases, in which the quantifier elimination returns false we have a full proof that no Hopf bifurcation exists – so we have a proof, that no Hopf bifurcation fixed points exists (for positive values of variables and parameters) for n = 2, 3, . . . , 8. In the cases in which true is returned one has to check the full Hopf condition containing the equality constraint on the denominator. Unfortunately, we could not obtain successful runs of our quantifier elimination procedure for the full Hopf condition for n = 9. We hence only have a very strong indication that a Hopf bifurcation fixed point exists in this case but
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eq1n:=diff(G(t),t)= (theta*(gamma0-G(t))-alpha*K[n-1]*P(t)^n*G(t)); eq3n:=diff(M(t),t)= rhob*(gamma0-G(t))+rhof*G(t)-deltaM*M(t); eq4n:=diff(P(t),t)= (n*theta*(gamma0-G(t))-n*alpha*K[n-1]*P(t)^n*G(t) - deltaP * P(t) + beta*M(t)) / auxdenom; fcns1:={G(t),P(t),M(t)}; lown:=2; upn:=10; for nn from lown to upn do eq11a:=eval(subs(n=nn,eq1n)); eq31a:=eval(subs(n=nn,eq3n)); eq41a:=eval(subs(n=nn,eq4n)); seqK := seq(K||i,i = 1 .. nn-1); seqsubKiki := seq(K[i]=K||i, i = 1 .. nn-1); eq11 := eval(subs(seqsubKiki,eq11a)); eq31 := eval(subs(seqsubKiki,eq31a)); eq41 := eval(subs(seqsubKiki,eq41a)); params1p:=[auxdenom,gamma0,theta,alpha,rhob,rhof,deltaM, deltaP,beta,seqK]; funccondlist:=[G(t)>0, P(t)>0, M(t)>0]; seqkcond :=seq(K||i>0,i=1..nn-1); paramcondseq:=auxdenom>0,gamma0>0,theta>0,alpha>0,rhob>0, rhof>0,deltaM>0,deltaP>0,beta>0,seqkcond; paramcondlist:=[paramcondseq] ; DEHopfexistence({eq11,eq31,eq41},fcns1,params1p, funccondlist,paramcondlist); od;
Figure 4. A Maple script for generating the first-order input formulae for [2, Sys. (2)] with H(t) being eliminated by γ0 − G(t). not really a full proof. To conclude the discussion of this system, we would like to remind the reader that in [2] there is no direct analysis of [2, Sys. (2)] given at all. 3.2.4. Reducing the quasi-steady state approximation. When reducing the equations of [2, Sys. (2)] using the variable substitutions and renamings given in [2] we obtain the following system: d G(t) = ϑ γ0 − G(t) − G(t)P (t)n , dt d M (t) = λ1 G(t) + γ0 µ − M (t) , dt α
δ
z}|{ z}|{ d n ϑ (γ0 − G(t) − G(t)P (t)n ) + β (M (t) − P (t)) P (t) = . Pn−1 2 i dt i=0 (i + 1) Ki P
(1)
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eq11 := diff(x1(t),t)= k21*x1(t)^2*x2(t)+k46-k64*x1(t)-k34*x1(t)+k43*x3(t); eq12 := diff(x2(t),t)=-k21*x1(t)^2*x2(t)+k56-k65*x2(t); eq13 := diff(x3(t),t)=k34*x1(t)-k43*x3(t); fcns1 := {x1(t),x2(t),x3(t)}; params1 := [k21,k46,k64,k34,k43,k56,k65]; paramcondlist1 := [k21>0,k46>0,k64>0,k34>0,k43>0,k56>0,k65>0]; funccondlist1 := [x1(t)>0, x2(t)>0, x3(t)>0]; DEHopfexistence({eq11,eq12,eq13}, fcns1,params1,funccondlist1,paramcondlist1);
Figure 5. A Maple script for generating the first-order input formula ϕ for Section 3.3.
This system slightly differs from [2, Sys. (3)]; we give the differences in overbraces. Although it is a priori not clear that both variants have the same behavior with respect to Hopf bifurcations, we find by applying our algorithms that also the system (1) has no Hopf bifurcation fixed points for n ≤ 8 but does have such a fixed point for n = 9, 10, . . . 3.3. Mass action systems As an example of a mass action system we consider the system investigated in [11, Example 2.1]. It is described by the following system of differential equations: d x1 (t) = k21 x1 (t)2 x2 (t) + k46 − k64 x1 (t) − k34 x1 (t) + k43 dt d x2 (t) = −k21 x1 (t)2 x2 (t) + k56 − k65 x2 (t) dt d x3 (t) = k34 x1 (t) − k43 x3 (t) . (2) dt Again, as is the case for most chemical and biological systems, only positive values for the variables are of interest. The automatic generation of a first-order formula stating the question on the existence of Hopf bifurcation fixed points and taking into account the positivity conditions is straightforward. Figure 5 shows our Maple script for this. The script produces the first-order input formula in Figure 6, which gives the exact (i.e. both necessary and sufficient) condition for the existence of a Hopf bifurcation fixed point. The original literature [11], in contrast, does less: it first derives a necessary condition, and then in a another analysis a sufficient one. To illustrate the difference, notice that true is always necessary and false is always sufficient but neither of them is generally necessary and sufficient. Furthermore, our formulation uses the original coordinates in contrast to a coordinate transform as in [11]. Of course, such a coordinate transform could also be realized with our approach in a preprocessing step.
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∃v3 ∃v2 ∃v1 (((0 < v1 ∧ 0 < v2 ) ∧ 0 < v3 ) ∧ ((((((((((k21 v12 v2 + k46 − k64 v1 − k34 v1 + k43 v3 = 0 ∧ −k21 v12 v2 + k56 − k65 v2 = 0) ∧ k34 v1 − k43 v3 = 0) ∧ (0 < k64 k65 k43 + k64 k21 v12 k43 − 2k21 v1 v2 k65 k43 ∧ 2 2 2 k21 v12 k43 + k21 v14 k43 + k21 v14 k34 + 2k34 k65 k43 + 2k64 k34 k65 + 2k64 k65 k43 + 2 2 2 2 2 2 k64 k21 v12 + k21 v14 k64 + k34 k21 v12 + k34 k64 k43 + k64 k43 + k65 k43 + k64 k65 + 2 2 2 2 k65 k43 + k34 k65 + k64 k65 + k64 k43 + 2k64 k21 v12 k43 + 2k34 k21 v12 k43 +
2k21 v12 k64 k65 + 2k21 v12 k65 k43 + 2k21 v12 k34 k65 + 2k64 k34 k21 v12 − 2 2 2 2 2 4k21 v13 v2 k43 − 2k21 v13 v2 k64 − 2k21 v13 v2 k34 + 4k21 v12 v22 k43 + 4k21 v12 v22 k65 − 2 2 2 2 2k21 v13 v2 k65 − 2k21 v1 v2 k43 − 2k21 v1 v2 k65 + k34 k65 − 4k21 v1 v2 k65 k43 −
4k21 v1 v2 k64 k65 − 4k21 v1 v2 k64 k43 − 4k21 v1 v2 k34 k65 − 2k34 k21 v1 v2 k43 = 0)) ∧ 0 < k21 ) ∧ 0 < k46 ) ∧ 0 < k64 ) ∧ 0 < k34 ) ∧ 0 < k43 ) ∧ 0 < k56 ) ∧ 0 < k65 ))
Figure 6. The formula ϕ generated by the Maple script in Figure 5. The typesetting v1 , . . . , v3 indicates Maple variables vv1, . . . , vv3 generated by the function DEHopfexistence in the script.
Tables 5 and 6 show the timings and results for regular and for positive quantifier elimination, respectively, on the existential closure of ϕ. Most surprisingly, on this example regular quantifier elimination performs considerably better than its positive variant. We consider this a rare exception. Anyway it will not be completely ignored: We expect from a careful analysis of the various quantifier elimination runs some insights and ideas for improvements of the procedures. There is another irritating fact, which cannot be ignored: In the previous section we had found that switching on rlsifaco (“simplifier factorization with ordering relations”) and switching off rlqeprecise (“quantifier elimination using precise test points”) and rlqevarseltry (“quantifier elimination variable selection try systematically”) should be the default choice for positive quantifier elimination. The 5th row in Table 6 shows that this very choice fails on this particular example. Notice, however, that things look much better when experimentally switching on rlqevarseltry. This indicates that the unusual behavior is mainly caused by the fact that good orders for the successive elimination of the variables cannot be easily heuristically determined. Anyway, good news is that we can decide the existence of a Hopf bifurcation fixed point within only a few seconds. As the answer is affirmative, also sample answer points can be computed. For this we use regular extended quantifier elimination with the most efficient switch settings given in the 4th row of Table 5. We
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Table 5. Applying regular quantifier elimination to the existential closures ∃ϕ of the formula in Section 3.3. We systematically try several switch settings in REDLOG. rlsifaco ◦ ◦ ◦ ◦ • • • •
rlqeprecise ◦ ◦ • • ◦ ◦ • •
rlqevarseltry ◦ • ◦ • ◦ • ◦ •
result sigxcpu true true true sigxcpu true true true
time (s) > 600 261.72 3.03 1.54 > 600 264.73 3.35 1.79
Table 6. Applying positive quantifier elimination to the existential closures ∃ϕ of the formula in Section 3.3. We systematically try several switch settings in REDLOG. rlsifaco ◦ ◦ ◦ ◦ • • • •
rlqeprecise ◦ ◦ • • ◦ ◦ • •
rlqevarseltry ◦ • ◦ • ◦ • ◦ •
result true true sigxcpu sigxcpu sigxcpu true sigxcpu true
time (s) 20.79 32.20 > 600 > 600 > 600 32.54 > 600 8.03
obtain after 3.05 s: k21 = 32 ,
k34 = 2/3 ,
k64 = 14/3 ,
k65 = 1/2 ,
k43 = 1 , v1 = 1/8 ,
k46 = 1/12 , v2 = 1 ,
k56 = 1 , v3 = 1/12 .
Notice that for this example, the switch rlqeaprecise is powerful enough to entirely avoid the introduction of infinitesimals. Again, further sample points can be enumerated by explicitly excluding in the input formula the ones already found.
4. Conclusions and future work Using generic methods to reduce the Hopf bifurcation problem to a quantifier elimination problem and then using quantifier elimination methods available in REDLOG we can reconstruct most of the results given in the literature within less than a minute of computation time.
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We have set the focus on examples already treated in the literature by other symbolic methods. On some examples we have extended the analysis given in the literature to related systems, which have not been analyzed before. We presume that our generic methods are applicable to many other problems not only in theory but also in practice. As no tedious hand computations are involved the use of these generic methods will hopefully be a useful tool for investigating a wide variety of other examples. In future work we will test our generic methods for parametrically investigating Hopf bifurcation fixed points on examples treated in the literature by numerical computations or pure hand calculations, e.g. on the systems described in [10, 12, 15, 18]. As there is the option to have some parameters fixed and only to investigate few others symbolically – still considerably extending the possibilities of pure numeric investigations – one can also apply these techniques to larger systems, which cannot be expected to be amenable for a full parametric analysis, e.g. the systems described in [14, 19]. We found that in cases, in which one suspects that no Hopf bifurcation fixed point exists for biologically relevant values of parameters and system variables, the range of computations can be extended considerably if one restricts the logical and algebraic investigations to finding subconditions being inconsistent with the hypothesis of the existence of Hopf bifurcation fixed points. On the long term we envision to include our method into a consistency checker, in which results in the literature on the existence of oscillations of a certain system can be automatically checked versus the existence of Hopf bifurcations. On the quantifier elimination side, the challenge is to improve the procedure such that the original elimination problems, in contrast to the existential closures, become tractable. There is actually good hope for success: In principle the elimination for the closure is a harder problem than that for the original formula, but we apparently benefit from the greater freedom in choosing the variable order for elimination. The fact that this is successful indicates that the problems are not inherently hard.
Acknowledgements We are grateful to A. Dolzmann, C. Groß, A. Reitelmann, and U. Waldmann for helpful discussions, and to the anonymous reviewers for detailed comments and suggestions.
References ¨ upl¨ [1] F. Boulier, M. Lefranc, F. Lemaire, P. Morant, and A. Urg¨ u. On proving the absence of oscillations in models of genetic circuits. In H. Anai, H. Horimoto, and T. Kutsia, editors, Algebraic Biology (AB 2007), volume 4545 of Lecture Notes in Computer Science, pages 66–80. Springer-Verlag, 2007.
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[2] F. Boulier, M. Lefranc, F. Lemaire, and P.-E. Morant. Applying a rigorous quasisteady state approximation method for proving the absence of oscillations in models of genetic circuits. In K. Horimoto, G. Regensburger, M. Rosenkranz, and H. Yoshida, editors, Algebraic Biology (AB 2008) – Third International Conference, volume 5147 of Lecture Notes in Computer Science, pages 56–64, Castle of Hagenberg, Austria, August 2008. Springer-Verlag. [3] C. W. Brown. QEPCAD B: A system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bulletin, 38(1):23–24, 2004. [4] C. W. Brown and C. Groß. Efficient preprocessing methods for quantifier elimination. In V. G. Ganzha, Ernst W. Mayr, and Evgenii V. Vorozhtsov, editors, Computer Algebra in Scientific Computing (CASC ’06), volume 4194 of Lecture Notes in Computer Science, pages 89–100, Chisinau, Moldova, September 2006. Springer-Verlag. [5] G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3):299–328, September 1991. [6] A. Dolzmann and T. Sturm. REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bulletin, 31(2):2–9, June 1997. [7] A. Dolzmann and T. Sturm. Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation, 24(2):209–231, August 1997. [8] A. Dolzmann, T. Sturm, and V. Weispfenning. A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning, 21(3):357–380, 1998. [9] M. El Kahoui and A. Weber. Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. Journal of Symbolic Computation, 30(2):161– 179, August 2000. [10] G. F. Fussmann, S. P. Ellner, K. W. Shertzer, and N. G. Hairston, Jr. Crossing the Hopf bifurcation in a live predator-prey system. Science, 290(5495):1358–1360, 2000. [11] K. Gatermann, M. Eiswirth, and A. Sensse. Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. Journal of Symbolic Computation, 40(6):1361–1382, 2005. [12] M. Mincheva and M. R. Roussel. Graph-theoretic methods for the analysis of chemical and biochemical networks. I. multistability and oscillations in ordinary differential equation models. Journal of Mathematical Biology, 55(1):61–86, 2007. [13] W. Niu and D. Wang. Algebraic approaches to stability analysis of biological systems. Mathematics in Computer Science, 1(3):507–539, 2008. [14] B. Novak, Z. Pataki, A. Ciliberto, and J. J. Tyson. Mathematical model of the cell division cycle of fission yeast. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(1):277–286, 2001. [15] A. Sensse, M. J. B. Hauser, and M. Eiswirth. Feedback loops for Shilnikov chaos: The peroxidase-oxidase reaction. The Journal of Chemical Physics, 125:014901–1– 12, 2006. [16] T. Sturm. Redlog online resources for applied quantifier elimination. Acta Academiae Aboensis, Ser. B, 67(2):177–191, 2007. [17] T. Sturm and A. Weber. Investigating generic methods to solve Hopf bifurcation problems in algebraic biology. In K. Horimoto, G. Regensburger, M. Rosenkranz, and H. Yoshida, editors, Algebraic Biology – Third International Conference (AB 2008),
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volume 5147 of Lecture Notes in Computer Science, Castle of Hagenberg, Austria, 2008. Springer-Verlag. [18] M. Sun, L. Tian, and J. Yin. Hopf bifurcation analysis of the energy resource chaotic system. International Journal of Nonlinear Science, 1(1):49–53, 2006. [19] J. J. Tyson, K. Chen, and B. Novak. Network Dynamics and Cell Physiology. Nat Rev Mol Cell Biol, 2(12):908–916, 2001. [20] A. Weber. Quantifier elimination on real closed fields and differential equations. In B. L¨ owe, editor, Algebra, Logic, Set Theory – Festschrift f¨ ur Ulrich Felgner zum 65. Geburtstag, volume 4 of Studies in Logic, pages 291–315. College Publications, 2007. [21] V. Weispfenning. Quantifier elimination for real algebra – the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing, 8(2):85– 101, February 1997. [22] V. Weispfenning. Simulation and optimization by quantifier elimination. Journal of Symbolic Computation, 24(2):189–208, August 1997. Special issue on applications of quantifier elimination.
Appendix A. Exact solutions with infinitesimals for Section 3.1.3 q √ √ 4 · 1180986 + 2187 · ε1 · 393662 · ε2 q √ √ √ + 22238501759909568 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 · ε1 · ε2 q √ √ √ − 22238501759909568 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 · ε2 √ − 2402002240184651776 · 1180986 · ε22
M (ε1 , ε2 ) =
√ 61010978765184 · ε1 ·
− 2626589449641916717056 · ε1 · ε22 2 + 2626589449641916717056 · ε2 q √ √ √ 4649045868 · ε1 · 4 · 1180986 + 2187 · ε1 · 393662 · ε1 q √ √ √ − 4649045868 · ε1 · 4 · 1180986 + 2187 · ε1 · 393662 q √ √ √ + 847288609443 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 · ε21 q √ √ √ − 1694577218886 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 · ε1 q √ √ √ + 1684121117211 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 √ − 564859072962 · 1180986 · ε21
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√ − 366065872591104 · 1180986 · ε1 · ε2 √ √ + 1106473861368 · 1180986 · ε1 + 366065872591104 · 1180986 · ε2 √ − 727577567910 · 1180986 − 205891132094649 · ε31 − 200146515839186112 · ε21 · ε2 + 604964583702954 · ε21 + 400293031678372224 · ε1 · ε2 − 1202306669284833 · ε1 − 397823091334329024 · ε2 + 790681240245960 q √ √ √ T (ε1 , ε2 ) = −6561 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 √ + 1458 · 1180986 + 1594323 · ε1 + 1549839424 · ε2 − 1495912 1549839424 q √ √ √ L(ε1 , ε2 ) = −200156684335646976 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 · ε2 q √ √ √ − 22877954716428 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 √ + 5083989936984 · 1180986 + 5559342996092004 · ε1 + 23640506047897342779392 · ε22 + 5404230477062468352 · ε2 − 5216187623191776 q √ √ √ 1062882 · ε1 · 4 · 1180986 + 2187 · ε1 · 393662 q √ √ √ + 387420489 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 · ε1 q √ √ √ − 387420489 · ε1 · 4 · 1180986 + 2187 · ε1 · 3 √ √ − 172186884 · 1180986 · ε1 − 83691328896 · 1180986 · ε2 √ + 166872690 · 1180986 − 94143178827 · ε21 − 91516468147776 · ε1 · ε2 + 182475286515 · ε1 + 91516468147776 · ε2 − 181313497440
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Investigating Algorithms for Hopf Bifurcation Problems
Thomas Sturm Departamento de Matem´ aticas Estad´ıstica y Computaci´ on Facultad de Ciencias Universidad de Cantabria E-39071 Santander Spain e-mail:
[email protected] [email protected] Andreas Weber and Essam O. Abdel-Rahman Institut f¨ ur Informatik II Universit¨ at Bonn R¨ omerstr. 164 D-53117 Bonn Germany e-mail:
[email protected] [email protected] M’hammed El Kahoui Department of Mathematics Faculty of Sciences Semlalia Cadi Ayyad University P.O. Box 2390 Marrakech Morocco e-mail:
[email protected] Received: September 29, 2008. Revised: December 14, 2008. Accepted: December 17, 2008.
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