2007; Stiefs et al., 2008), but due to practical considerations, they remain limited to two or three adjustable bifurcation parameters.
The challenge to find parameter values for bifurcations in a high-dimensional parameter space is of particular relevance in the area of biological systems. This is due to the fact that biochemical systems contain a high number of model parameters, usually even more parameters than state variables. Due to the limited knowledge of the system, it is usually required to assume that most of these parameters are allowed to vary simultaneously over a wide range of values. In order to apply classical continuation methods in a multi-dimensional parameter space, it is necessary to pre-define a line in parameter space along which parameter values are varied during the continuation. A good choice of this line is essential for obtaining meaningful results. Yet, this choice is often made by intuitive understanding of the system in the better case or iterative trials in the worse. Often only a single parameter is varied at a time, but then again the choice of the varying parameter is not trivial and needs to be taken e.g. via sensitivity considerations.
In this chapter, we develop a new method to locate points in a possibly high-dimensional parameter space where a change in the stability properties of equilibrium points occurs.
Such points may then be used as an upper bound on the robustness measures derived in Chapter 4. To this end, we extend the results from Section 4.3.1 by further characterising the relationship between local bifurcations and the properties of the open loop transfer function G(χ, s) as defined in (4.20). With the obtained conditions, a numerical method for searching parameter values leading to a change in stability properties is developed. In theory, the method can be applied for parameter spaces of arbitrary dimension, as neither the conditions nor the algorithm we use depend on the dimension of the parameter space.
We consider only codimension one bifurcations, as they are the case usually encountered for generic parameter variations in non-linear systems.
The use of frequency domain methods for bifurcation analysis has already been intro-duced by Allwright (1977) (see also Mees and Chua, 1979), and relevant results have also been presented by Moiola and coworkers over the last decade (Moiola et al., 1991, 1997).
Several authors have also studied the problem of finding bifurcations in systems with many parameters using geometric tools. Based on a description of vectors normal to a bifurcation manifold (M¨onnigmann and Marquardt, 2002), a method to search for locally closest bifurcations from a given reference point has been developed by Dobson (2003).
These approaches can be seen as complementary to our results. A recent application of the geometric concept to biological systems has been discussed by Lu et al. (2006).
5.2 Loop breaking and steady state stability properties
5.2.1 Feedback loop breaking and critical frequencies
In this section, we directly build on the feedback loop breaking (Definition 4.6), linear approximation and transformation to the frequency domain as outlined in Section 4.3.
From these results, specifically Lemma 4.8, we see that the main object to be studied is the open loop transfer function G(χ, s). Let us recall the representation of G as a complex
rational function with real coefficients from (4.21):
G(χ, s) = Q(χ, s)
R(χ, s), (5.1)
where Q(χ, s), R(χ, s) are polynomials in s with real scalar functions of χ as coefficients.
A key result obtained in Section 4.3 to be reused in this chapter is Lemma 4.8, which allows to characterise imaginary eigenvalues jω of the Jacobian A(χ) by the condition G(χ, jω) = 1.
The following technical assumption is required for the next steps.
Assumption (A1): For any χ ∈ M, the Jacobian Ao(χ) of the open loop system (4.17) does not have an eigenvalue on the imaginary axis, and the transfer function G(χ, s) does not have zeros on the imaginary axis, i.e.
∀χ ∈ M ∀ω ∈ R : Q(χ, jω) 6= 0 and det(jωIn− Ao(χ)) 6= 0. (5.2) In addition, the degrees of Q(χ, s) and R(χ, s) in s are assumed to be constant with respect to χ ∈ M.
Starting from the premise that we are interested in stability changes produced by changing the characteristics of the feedback loop broken in Definition 4.6, this assumption is usually satisfied.
The notion of a critical frequency which is introduced in the next definition will be useful to compute possible eigenvalues of the closed loop Jacobian A(χ), as given by (4.19), on the imaginary axis.
Definition 5.1. ωc ∈ R is said to be a critical frequency for the transfer function G(χ, s), if
G(χ, jωc) ∈ R. (5.3)
Obviously, different values of χ will result in different critical frequencies. For a specific χ, all critical frequencies are given by the solutions of the equation
Im(Q(χ, jωc)R(χ, −jωc)) = 0, (5.4) which is a scalar polynomial equation in ωc, with coefficients which are real scalar functions of χ.
We define the set of all critical frequencies for a specific χ as
R(χ) = {ω ∈ R | Im(Q(χ, jω)R(χ, −jω)) = 0} . (5.5) In classical control theory, R is also called the realness locus of G(χ, jω) (Hinrichsen and Pritchard, 2005). Since only the imaginary part is considered, the polynomial in (5.4) is odd. The following properties of the set R(χ) can then be shown easily.
Proposition 5.2. For any χ ∈ M, the set R(χ) satisfies the conditions (i) 0 ∈ R(χ);
5.2 Loop breaking and steady state stability properties
(ii) ωc ∈ R(χ) implies that −ωc ∈ R(χ);
(iii) either R(χ) = R or R(χ) has finitely many elements.
Note that R(χ) = R whenever Im(Q(χ, jω)R(χ, −jω)) is the zero polynomial, which in turn is the case whenever only the even powers of s in the polynomials Q(χ, s) and R(χ, s) have non-zero coefficients. This case is highly non-generic and usually does not occur in the applications we are interested in. Consequently, this case is not considered specifically.
The concept of critical frequencies can be understood intuitively when considering the Nyquist curve of the transfer function G(χ, s). A critical frequency is any value ωcat which the Nyquist curve crosses the real axis. This is obviously a necessary condition for having G(χ, jωc) = 1, which corresponds to the existence of an eigenvalue on the imaginary axis as shown in Lemma 4.8. Our concept is thus closely related to the idea of the gain margin for robustness analysis of linear control systems (Skogestad and Postlethwaite, 1996). An illustration of critical frequencies in the Nyquist plot is shown in Figure 5.1 on page 64.
Since a variation of the steady state–parameter pair χ influences the polynomial equa-tion (5.4), the set of critical frequencies R(χ) may change significantly with χ. In partic-ular, the number of elements in R(χ) generally does not have to be constant with respect to χ, which complicates the analysis. However, one can show that there is a minimal number of critical frequencies of the transfer function G(χ, s). As we will show below, this minimal number depends on the number of open loop poles and zeros and whether they are located in the right or the left half complex plane. To this end, define the number β = |n˜ p+− np−+ nz−− nz+|, (5.6) where np+ (np−) is the number of poles of G(χ, s), and nz+ (nz−) is the number of zeros of G(χ, s) in the right (left) half complex plane. Under Assumption (A1), ˜β is constant with respect to χ ∈ M. The number of elements in the set of critical frequencies can now be characterised by ˜β.
Proposition 5.3. Let ˜β be defined by (5.6) and assume that (A1) is satisfied. Then, for any χ ∈ M, R(χ) has at least ˜β distinct elements, if ˜β is odd, and at least ˜β − 1 distinct elements, if ˜β is even.
The proof is presented in the appendix, Section A.2. Proposition 5.3 is used to formulate the property of minimality for the set of critical frequencies.
Definition 5.4. Under Assumption (A1), the set of critical frequencies R(χ) is called minimal, if it contains exactly β elements, where
β =
( β,˜ if ˜β is odd
β − 1,˜ if ˜β is even. (5.7)
The concept of minimality is applied in the second assumption made to derive the main results.
Assumption (A2): The set of critical frequencies R(χ) is minimal for any χ ∈ M.
Given a transfer function G(χ, s) and a corresponding set of critical frequencies R(χ), Proposition 5.3 can be used to easily check the minimality of R(χ). Graphically, a suf-ficient condition for minimality of R(χ) is that the Nyquist curve G(χ, jω) encircles the origin monotonically as ω varies from −∞ to ∞. This is e.g. satisfied for stable trans-fer functions without zero dynamics, or with an anti-stable zero dynamics. In practice, minimality of R(χ) is satisfied by many stable transfer functions.
If Assumption (A2) holds, we can label the roots of the polynomial equation (5.4) in a consistent manner, writing
R(χ) =ω1c(χ), ωc2(χ), . . . , ωβc(χ) , (5.8) where the ωci are continuous functions of the steady state–parameter pair χ and can be identified with different solution branches of the polynomial equation (5.4).
5.2.2 Topological equivalence of equilibria
Changes in stability properties of equilibrium points are most easily studied using the concept of topological equivalence. Here, we consider topological equivalence of hyperbolic equilibrium points only, for which by definition the Jacobian does not have eigenvalues on the imaginary axis (Kuznetsov, 1995).
Definition 5.5. Let χ1, χ2 ∈ M be two hyperbolic steady state–parameter pairs of the system (4.10). χ1 and χ2 are said to be topologically equivalent, if the Jacobians ∂F∂x(χ1) and ∂F∂x(χ2) have the same number of eigenvalues in the left and right half complex plane, respectively.
It is a well known result from dynamical systems theory that the topological equivalence of all equilibria in two systems is a necessary condition for topological equivalence of the flows (Kuznetsov, 1995). Let us consider two variants of the system (4.10), one with parameter values p1 and the other with parameter values p2. In the simple case when there is only one steady state in each variant of the system, corresponding to the pairs χ1 and χ2, topological equivalence of χ1 and χ2 is a necessary condition for topological equivalence of the flows. In applications, we often aim at finding parameter values p2 such that the system (4.10) changes its dynamical behaviour when parameters are varied from initial values p1 to p2. For this problem, it is sufficient to find pairs χ1 and χ2 which are not topologically equivalent. Due to the continuous dependence of eigenvalues on parameters, this can only happen when the Jacobian ∂F∂x(χc) has eigenvalues on the imaginary axis for some critical point χc ∈ M. At this point, we can make use of the methodology developed in the previous subsection.
To this end, consider the set of critical frequencies R(χ) for a specific value of the steady state–parameter pair χ. Define the number β∗(χ) to be the number of elements ω in R(χ) such that G(χ, jω) > 1, i.e.
β∗(χ) = card {ω ∈ R(χ) | G(χ, jω) > 1} , (5.9) where card A denotes the number of elements in the set A.
Geometrically, if R(χ) is minimal, β∗(χ) gives the winding number of the graph of G(χ, jω) around the point 1 in the complex plane (see Lemma A.3 on page 116). The
5.2 Loop breaking and steady state stability properties
Argument Principle (Whittaker and Watson, 1965) can then be used to characterise topo-logically equivalent steady state–parameter pairs of the system (4.10) via the number β∗(χ). This characterisation is given in the following result.
Theorem 5.6. Assume that (A1) is satisfied. Let χ1, χ2 ∈ M be two hyperbolic steady state–parameter pairs of (4.10) such that R(χ1) and R(χ2) are minimal. Then χ1 and χ2 are topologically equivalent, if and only if
β∗(χ1) = β∗(χ2).
The proof is given in the appendix, Section A.3. It makes use of several intermediate statements which are given as lemmas in Section A.3. There are two main steps in the proof. First, we observe that minimality of the set of critical frequencies implies that the graph of G(χ, jω) cuts the positive and negative parts of the real axis in an alternating manner. The second step is to show that β∗(χ) is equal to the winding number of the graph of G(χ, jω) around the point 1.
5.2.3 Existence of marginally stable equilibria
Let us now turn to the problem of how to find parameter values for which a change in stability properties of equilibria can occur. This is equivalent to searching for critical points χc at which the Jacobian ∂F∂x(χc) has an eigenvalue on the imaginary axis. This typically means that χc is part of a submanifold of M which separates regions of topo-logical equivalence. In view of Theorem 5.6, there are typically steady state–parameter pairs χ1 and χ2 close to χc such that β∗(χ1) 6= β∗(χ2). Equivalently, for a specific critical frequency branch ωci, the transfer function value G(χ, jωic(χ)) on that branch has to cross the value 1 when χ is varied continuously along a path from χ1 to χ2. These observations are formalised in the following theorem.
Theorem 5.7. Assume that Assumptions (A1) and (A2) are satisfied, and that M is connected. There exists a critical point χc ∈ M such that ∂F∂x(χc) has an eigenvalue on the imaginary axis, if and only if there exist χ1, χ2 ∈ M such that, for some i ∈ {1, 2, . . . , β}, G(χ1, jωci(χ1)) ≤ 1 ≤ G(χ2, jωci(χ2)), (5.10) where ωci(χ) ∈ R(χ). In that case, ±jωci(χc) is an eigenvalue of ∂F∂x(χc).
Proof. With Lemma 4.8, Assumption (A1) assures that a point χc is critical if and only if G(χc, jωci(χc)) = 1.
Necessity. Under the condition G(χc, jωc) = 1, take χ1 = χ2 = χc and (5.10) follows trivially.
Sufficiency. Let χ1 and χ2 be such that (5.10) holds. Connectivity of M implies that there is a path from χ1 to χ2 in M. Continuity of the critical frequency ωci(χ) and the transfer function coefficients result in continuity of G(χ, jωci(χ)) with respect to χ. This implies existence of χc such that G(χc, jωci(χc)) = 1 along any path from χ1 to χ2.
A graphical illustration of Theorem 5.7 is given in Figure 5.1. The figure also illustrates the relation to the classical Nyquist stability criterion.
critical frequencies
Re Im
1 G(χ1, jωc1(χ1))
G(χ2, jωc1(χ2))
Figure 5.1: Illustration of Theorem 5.7 in the Nyquist plot. Full line: G(χ1, jω), dashed line:
G(χ2, jω), both drawn only for ω ≥ 0. The theorem asserts existence of a critical point χc on any path between χ1 and χ2 in M.
The proof shows that the critical point χc is usually far from unique. It may be unique if M is of dimension one, i.e. there is only one free parameter to vary. In general, one will expect that there is a submanifold of critical points in M separating regions which represent different topological equivalence classes, where χ1is an element of one such class, and χ2 is an element of the other class. On this submanifold, the bifurcations which can be encountered generically are codimension one bifurcations. Therefore, the bifurcation condition provided by Theorem 5.7 is mainly useful in the search for codimension one bifurcations, although condition (5.10) also holds for bifurcations of higher codimension.
Using Theorem 5.7, it is easily possible to distinguish dynamical bifurcations from static bifurcations. In fact, if i is chosen such that the critical frequency ωci(χ) = 0 is considered, A(χc) has a zero eigenvalue, which generically corresponds to a saddle-node bifurcation. If a critical frequency ωci(χ) 6= 0 is considered, A(χc) has conjugated imaginary eigenvalues, and generically a Hopf bifurcation will occur.
To conclude this section, we give an illustrative example for the application of the loop breaking approach to a small biochemical network. In this example, all the computations can be done analytically.
Example 5.8. Consider a biochemical reaction network described by the ODE
˙x1 = 1
1 + xδ3 − x1
˙x2 = x1−1 2x2
˙x3 = x2− x3,
(5.11)
with one uncertain parameter δ > 0 (Griffith, 1968a). It is easily verified that x0 = (12, 1, 1)T is a steady state for any value of δ. We apply a loop breaking such that the