Symmetry breaking bifurcation

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

This section focuses on steady and unsteady hydrodynamic modes that emerge as the rotational speed of the inner cylinder (expressed by Ta) and pressure-driven axial flow rate (scaled by Re) are varied, while the outer cylinder is kept fixed. These modes constitute primary, secondary and higher order bifurcations, which break the symmetry of the base helical Couette-Poiseuille (CP) flow and represent drastic changes in flow structure. Figure 4.4.2 presents a map of observed hydrodynamic modes in the (Ta, Re) space, and marks the domain where all of the hydrodynamic modes that interest us appear. We will return to this figure shortly. 

Once the door was opened to these new perspectives, the works multiplied rapidly. In 1968 an important paper by Prigogine and Rene Lefever was published On symmetry-breaking instabilities in dissipative systems (TNC.19). Clearly, not any nolinear mechanism can produce the phenomena described above. In the case of chemical reactions, it can be shown that an autocatalytic step must be present in the reaction scheme in order to produce the necessary instability. Prigogine and Lefever invented a very simple model of reactions which contains all the necessary ingerdients for a detailed study of the bifurcations. This model, later called the Brusselator, provided the basis of many subsequent studies. 

BIFURCATIONS AND SYMMETRY BREAKING IN FAR-FROM-EQUILIBRIUM SYSTEMS TOWARD A DYNAMICS OF COMPLEXITY... 

The irreversibility inherent in the equations of evolution of the state variables of a macroscopic system, and the maintenance of a critical distance from equilibrium, are two essential ingredients for this behavior. The former confers the property of asymptotic stability, thanks to which certain modes of behavior can be reached and maintained against perturbations. And the latter allows the system to reveal the potentialities hidden in the nonlinearity of its kinetics, by undergoing a series of symmetry breaking transitions across bifurcation points. 

For Fig. 4, the rate constant for the mutual inhibition (fo) was chosen as the bifurcation parameter however, all other parameters and the initial reactant concentrations serve as bifurcation parameters as well. Mirror-symmetry breaking in simulation is caused by the intrinsic instability of the autocat-alytic model. When placed into the proper parameter domain, this instability can be revealed by the inevitable machine round-off as well as by slightly... 

Fig. 4 Numerical simulation revealing mirror-symmetry breaking in which k.2 acts as a bifurcation parameter (same conditions as in Fig. 3 except [R]o + [S]o = 0). Each dot represents the final ee of an individual computer simulation. For k2 > 6 x 103 M-1 s-1 the system becomes optically active, where positive and negative values of the resulting ee are equally distributed...

Bifurcation, instability, multiple solutions, and symmetry-breaking states are all related to each other. Chemical cycles in living systems show asymmetry. The bifurcation of a solution indicates its instability, which is a general property of the solutions to nonlinear equations. 

This simple example also displays the importance of nonlinear dynamics. The unsealed elasticities are constant only for a linear system and hence result in the same behavior if a linear system is stable, it will remain stable when boundary conditions change, and if it is unstable, it will remain unstable. Under nonlinear equations, however, the values of elasticity depend on the system state, which varies with the boundary conditions. This may lead to a transition from a stable to unstable system, known as bifurcation or symmetry breaking. This means that nonlinearity causes a variety of new behaviors in a system. 

A very interesting series of studies of the influence of end effects in the rotating concentric cylinder problem has been published by Mullin and co-workers T. Mullin, Mutations of steady cellular flows in the Taylor experiment,J. Fluid Mech. 121, 207-18 (1982) T. B. Benjamin and T. Mullin, Notes on the multiplicity of flows in the Taylor experiment, J. Fluid Mech. 121, 219-30 (1982) K. A. Cliff and T. Mullin, A numerical and expwerimental study of anomalous modes in the Taylor experiment, J. Fluid Mech. 153, 243-58 (1985) G. Pfister, H. Schmidt, K. A. Cliffe and T. Mullin, Bifurcation phenomena in Taylor-Couette flow in a very short annulus, J. Fluid Mech. 191, 1-18 (1988) K. A. Cliffe, 1.1. Kobine, and T. Mullin, The role of anomalous modes in Taylor-Couette flow, Proc. R. Soc. London Ser. A 439, 341-57 (1992) T. Mullin, Y. Toya, and S. I. Tavener, Symmetry breaking and multiplicity of states in small aspect ratio Taylor-Couette flow, Phys. Fluids 14, 2778-87 (2002). 

Another well-studied example of symmetry-breaking bifurcation of the reaction path was provided by Ruedenberg and co-workers. They... 

An interesting situation also came to light in the limit of normal incidence. This case was impossible to analyze in the framework of the approximate model, as the modes become large quickly and violate the initial assumptions. It turned out that for a = 0 (which is a peculiar case, since the external symmetry breaking in the x direction vanishes), another stationary instability precedes the secondary Hopf bifurcation that spontaneously breaks the reflection symmetry with respect to x. It is shown by point A in Fig. 18. It is also seen from this figure, that the secondary pitchfork bifurcation is destroyed in tbe case of oblique incidence, which can be interpreted as an imperfect bifurcation with respect to the angle a [43]. 

Stationary and time-dependent solutions of the master equation of the Brusselator have been approximately established by Turner (1979). The main point of his procedure is that the stationary solution of the master equation can be seen as a time average over a period of the limit cycle of P y(t), the time-dependent solution. Some illustrative results of the symmetry-breaking bifurcations in the stochastic Brusselator model are shown in Fig. 5.6 (after Nicolis, 1984). 

AN ELEMENTARY EXAMPLE OF BIFURCATION AND SYMMETRY BREAKING Consider the equation... 

To date, no chemical reaction has produced chiral asymmetry in this simple manner. However, symmetry breaking does occur in the crystallization of NaC103 [9, 10] in far from equilibrium conditions. The simple model however leads to interesting conclusions regarding the sensitivity of bifurcation discussed below. 

Figure 19.6 A symmetry-breaking transition or bifurcation in the presence of a small bias that favors one of the bifurcating branches. It can be analyzed through the general equation (19.3.17) and the probability of the system making a transition to the favored branch is given by equation (19.3.18)...

Many other examples may be found in the literature. Bifurcations can be found in the behavior of social insects as well [7]. Imagine an ant nest connected to a food source by two paths, identical except for the directions of their two limbs (Fig. 20.5). At first, equal numbers of ants are traveling on the two paths. After some time, practically all the ants are found on the same path due to the catalytic effects of chemical substances called pheromones, produced by the ants. Note that which bridge will be used is unpredictable. This corresponds to a typical symmetry-breaking bifurcation. 

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