Zero-eigenvalue bifurcation

We also have the hint of a new type of degeneracy associated with systems possessing multiple stationary states. It is possible for both the trace and the determinant of the Jacobian matrix to become zero simultaneously this gives the system two eigenvalues which are both equal to zero. These double-zero eigenvalue situations are important because they represent conditions at which a Hopf bifurcation point with an associated homoclinic orbit first appears. In the present case, tr(J) = det(J) = 0 only when k2 = Vg, but then the isola has shrunk to a point. 

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

The double-zero eigenvalue points, such as M, represent the coalescence of Hopf bifurcation and stationary-state turning points. As mentioned above, they thus represent the points at which the Hopf bifurcation loci begin and end. They also have other significance. Such points correspond to the beginning or end of loci of homoclinic orbits. For the present model, with the given choices of k1 and k2, there are two curves of homoclinic orbit points, one connecting M to N, the other K to L, as shown schematically in Fig. 12.7. 

FIGURE 8 Detailed bifurcation diagram for variations in the reactant partial pressures (a, and a2) when y, = 0.001 and y2 = 0.002. The middle diagram is the superposition of the four surrounding bifurcation curves. The points K, L, M and N correspond to double zero eigenvalues and the points G and H are metacritical Hopf points. 

In the case of steady state bifurcations, certain eigenvalues of the linear-approximation matrix reduce to zero. If we consider relaxations towards a steady state, then near the bifurcation point their rates are slower. This holds for the linear approximation in the near neighbourhood of the steady state. Similar considerations are also valid for limit cycles. But is it correct to consider the relaxation of non-linear systems in terms of the linear approximations To be more precise, it is necessary to ask a question as to whether this consideration is sufficient to get to the point. Unfortunately, it is not since local problems (and it is these problems that can be solved in terms of the linear approximations) are more simple than global problems and, in real systems, the trajectories of interest are not always localized in the close neighbourhood of their attractors. 

In other words, at a Hopf bifurcation the eigenvalues Xni 2 of the Jacobian matrix are complex and the real part of the eigenvalues passes through zero. Denoting the Jacobian matrix of the homogenous system by... 

In all of the examples above, the bifurcation occurs when A = 0, or equivalently, when one of the eigenvalues equals zero. More generally, the saddle-node, transcritical, and pitchfork bifurcations are all examples of zero-eigenvalue bifurcations. (There are other examples, but these are the most common.) Such bifurcations always involve the collision of two or more fixed points. 

Prove that at any zero-eigenvalue bifurcation in two dimensions, the nullclines always intersect tangentially. (Hint Consider the geometrical meaning of the rows in the Jacobian matrix.)... 

Besides single bifurcations, there is also the po.ssibility of triple bifurcations (two eigenvalues of the Hessian become zero) and even higher bifurcations. In principle, it is not difficult to detect bifurcation points and algorithms have been proposed to determine the location of a bifurcation point. However, it is somewhat more difficult to correctly follow a bifurcating path. 

This approach is a rather general one. Its advantage is that when the rescaling procedure has been carried out, many resonant monomials disappear. The most trivial example is a saddle-node bifurcation with a single zero eigenvalue. In this case the center manifold is one-dimensional. The Taylor expansion of the system near the equilibrium state may be written in the following form... 

C.4. 53. Below we present (following [185]) a list of asymptotic normal forms which describe the trajectory behavior of a triply-degenerate equilibrium state near a stability boundary in systems with discrete symmetry. We say there is a triple instability when a dynamical system has an equilibrium state such that the associated linearized problem has a triplet of zero eigenvalues. In such a case, the analysis is reduced to a three-dimensional system on the center manifold. Assuming that (x, y, z) are the coordinates in the three-dimensional center manifold and a bifurcating equilibrium state resides at the origin, we suppose also that our system is equivariant with respect to the transformation (x,y,z) <- (-X, -y, z). 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Note that this involves the derivative of the real parts -of the eigenvalues Alt 2 or, equivalently, of the trace of J evaluated at the Hopf bifurcation point. We know that Reflj 2) is passing through zero at this point. 

Of particular interest is the special case of a complex pair of principal eigenvalues whose real parts are passing through zero. This is the situation which we have seen corresponding to a Hopf bifurcation in the well-stirred systems examined previously. Hopf bifurcation points locate the conditions for the emergence of limit cycles. Using the CSTR behaviour as a guide it is relatively easy to find conditions for Hopf bifurcations, and then locally values of the diffusion coefficient for which a unique stationary state is unstable. Indeed the stationary-state profile shown in Fig. 9.5 is such a... 

We have specified that the conditions of interest here are those lying within the Hopf bifurcation locus, so tr(U) will be greater than zero. This means that the eigenvalues appropriate to this uniform state must have positive real parts. The system is unstable to uniform perturbations. We must exclude this n = 0 mode from any perturbations in the remainder of this section. 

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