Takens-Bogdanov bifurcation

Figure 23. A calculated two-parameter bifurcation diagram for the formic acid model [Eq. (15)] showing the locations of the saddle-node (solid line), Hopf (dashed line), and saddle-loop bifurcations (dotted-dashed line). All three curves meet in a Takens-Bogdanov point close to the cusp. (Reprinted with permission from P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 107, 991-1003, 1997. Copyright 1997 American Institute of Physics.)...

Fig. 85. Phase portraits and a bifurcation diagram for the Takens-Bogdanov bifurcation.

A complete classification of the catastrophes in two state variables of codimension three has not yet been developed. Consequently, only the standard form of a catastrophe corresponding to the sensitive state of the Takens-Bogdanov bifurcation will be given. The generalized Takens--Bogdanov bifurcation has the corresponding standard form ... 

Note in the end that the Takens-Bogdanov bifurcation may occur in the Bykov model, see Section 5.5.3.4. The sensitive state of the Takens-Bogdanov bifurcation, Xt = X2 = 0, may be obtained by imposing in (6.121a), (6.121b) the following requirements on the constants k t, k2, k 3... 

Only those solutions have a physical meaning for which zx 0 whereas the sign of the z2 variable can be arbitrary. Note that the system (6.126) resembles a standard form of the Takens-Bogdanov bifurcation (Section 5.5.3.5), in which cl = — a, c2 = v, R1 = a, R2= 0. The difference lies in the fact that in the Takens-Bogdanov bifurcation the parameters Rt, R2 are fixed and only c2, c2 undergo changes. In the case of system (6.126) the... 

Fig. 11.1 Two-dimensional bifurcation diagram calculated by continuation from the Citri-Epstein mechanism for the chlorite-iodide system. Plot of 2D space of constraints is chosen to be the ratio of input concentrations, [CIO ]o/[I ]o> versus the logarithm of the reciprocal resideuce time, logfco-Notation SSI, SS2, region of steady states with high [1 ] and low [I ], respectively Osc, region of periodic oscillations Exc, region of excitahihty snl, sn2, curves of saddle-node bifurcations of steady states hp, curve of Hopf bifiucatious sup, ciuve of saddle-node bifurcations of periodic orbits swt, swallow tail (a small area of tristability) C, cusp point TB, Takens-Bogdanov point (terminus of hp on snl). (From [5].)...

Figure 5. Bifurcation diagram on the plane of the two control parameters p and a. The solid lines 1 and 2 mark the primary instability, where the homogeneous homeotropic orientation becomes unstable. At 1, the bifurcation is a stationary (pitchfork) bifurcation, and a Hopf one at 2. The two lines connect in the Takens-Bogdanov (TB) point. The solid lines 3 and 4 mark the first gluing bifurcation and the second gluing bifurcation respectively. The dashed lines 2b and 3b mark the lines of the primary Hopf bifurcation and the first gluing bifurcation when calculated without the inclusion of flow in the equations.

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. 

Theorem 10.3 If the well-mixed system is sufficiently close to a Takens-Bogdanov point, then there exists a set of dij such that a Turing bifurcation occurs for arbitrarily small e. 

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 ... 

It has been established [59] that any activated exothermic reaction in an open system can give rise to bistable, excitable and oscillatory behavior. The principle features can be obtained using just a single unimolecular step. A detailed bifurcation analysis of such a system has been carried out by Vance and Ross [60] the bifurcation set consists of an unfolding of a (codimension-3) Takens-Bogdanov-cusp point [61]. (The bifurcation fine... 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

Fig. 13.2.9. Bifurcation diagram of the Bogdanov-Takens point in the (/ii, /X2 )-parameter plane.

This bifurcation diagram for the equilibrium state with two zero characteristic exponents had been known for a long time. However, there remained a problem of proving the uniqueness of the limit cycle. In other words, one must prove additionally that there are no other bifurcational curves besides Li,...,L4 (namely, curves corresponding to semi-stable limit cycles). This problem was independently solved by Bogdanov [33] and Takens [146] with whom this bifurcation is often named after. 

C.7. 81. Following the same steps as in the study of the generic Bogdanov-Takens normal form, analyze the structure of the bifurcation set near the origin /xi = /i2 = 0 in the Khorozov-Takens normal form with reflection symmetry ... 

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