Bifurcation double-zero

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. 

The point S of figure 8 at which the Hopf bifurcation curve crosses the boundary of the multiplicity region is not a double zero degeneracy, for the upper steady state (i.e. that with the larger 0b) is undergoing the Hopf bifurcation at the same time as the lower steady-state undergoes a saddle-node bufurcation, i.e. the conditions trJ = 0 and detJ = 0 apply at different points. It does, however, serve to show the four combinations of the two most common... 

The stationary bifurcation and the Hopf bifurcation typically occur as one parameter is varied and are therefore known as codimension-one bifurcations. They represent the generic ways in which a steady state of a two-variable system can become unstable. It is sometimes possible to make the stationary and Hopf instability threshold coalesce by varying two parameters. Such an instability, where T = A = 0, is known as a Takens-Bogdanov bifurcation or a double-zero bifurcation, since Ai = A.2 = 0 at such a point [175], This bifurcation is a codimension-two bifurcation, since it requires the fine-tuning of two system parameters. 

The preceding theorem establishes that the well-mixed system being in the vicinity of a double-zero point is a necessary condition for a Turing bifurcation to occur in (10.1) with nearly equal diffusion coefficients. The next theorem establishes that this is also a sufficient condition. 

The proofs of Theorems 10.2, 10.3, and 10.4 are found in [348]. Equation (10.17) is of particular interest. Near the Takens-Bogdanov point, the frequency of the limit-cycle oscillations along the line of Hopf bifurcations, a = 0, is given by >h = see above. On the line of saddle-node bifurcations we have Aj = 0. An equation like (10.17) is expected from simple dimensional arguments. The only intrinsic length scales in reaction-diffusion systems come from the diffusion coefficients. The inverse time is determined by the rate coefficients of the reaction kinetics. Thus (10.17) provides an estimate of the intrinsic length of the Turing pattern near a double-zero point ... 

One can see that since the constant term is negative, it follows immediately from the Routh-Hurwitz criterion that the origin is an unstable equilibrium state. Furthermore, it may have no zero characteristic roots when a and b are positive. The codimension-2 point (a = b = 0) requires special considerations. We postpone its analysis to the last section, where we discuss the bifurcation of double zeros in systems with symmetry. 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

Fig. 6.6. Stability regions for the stationary states of coupled systems with detuning. LP and H denotes saddle-node and Hopf bifurcations, respectively. ZH is a codimension 2 Zero-Hopf (or Guckenheimer-Gavrilov) bifurcation point. PD - period-doubling bifurcation curve.

Figure 1 The canonical cusp catastrophe function, Az = -f + 6x, at different values of the parameter a. The left panel illustrates the symmetric behavior for h = 0 the right panel for 6 = 1 illustrates the asymmetric behavior which occurs whenever b is different from zero. In the horizontal plane (a, x) we have drawn the locus of maxima as dotted lines, of minima as dashed lines and of inflections as dashed-dotted lines. The triply degenerate (or catastrophe) point occurs at a = 0 and 6 = 0 and bifurcation between single and double well modes occurs there for symmetric systems. For asymmetric systems, the bifurcation occurs on the fold line (at a = —1.9 for 6 = 1) where the cusp function has a doubly degenerate horizontal inflection point at x = 0.8.

In the previous section we have reduced the problem of period-two orbits which were spawned from a fixed point with a multiplier —1 to a study of the mapping (11.4.17) analogous to the mapping (11.5.16). Therefore, the bifurcation diagram in this case is the same as in the period-doubling bifurcation with (fc — 1) zero Lyapunov values this consists of a union of the plane /io = 0 on which the equilibrium state at the origin loses its stability and that half of... 

It C ui also be shown that these bifurcations are non-degenerate, i.e. the first Lyapimov value is non-zero in both cases. Moreover, at the period-doubling bifurcation the Lyapunov value is positive. Therefore, a saddle orbit of period two is born at this bifurcation. 

Fig. C.2.3. A partial bifurcation diagram for the asymmetric Lorenz model. The point CP is a cusp, at BT the system has a double-degenerate equilibrium state with two zero characteristic exponents (see Sec. 13.2).

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