Bifurcation mechanisms systems

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

In this section we analyze a classic problem from first-year physics, the bead on a rotating hoop. This problem provides an example of a bifurcation in a mechanical system. It also illustrates the subtleties involved in replacing Newton s law, which is a second-order equation, by a simpler first-order equation. 

As a simple example of imperfect bifurcation and catastrophe, consider the following mechanical system (Figure 3.6.7). 

Let us clarify the underlying bifurcation mechanisms, which cause the observed phenomenon. The inspection of the system dynamics and the continuation technique reveal that the blowup of the synchi onous attractor at j = 0.0284 occurs via a collision with a chaotic saddle. The chaotic saddle is located in the phase space as shown in Fig. 6.19(b). In order to detect the geometrical place of this unstable chaotic saddle, we used the continuation technique. We follow ed the unstable low-periodic orbits, which were born in the period-doubling cascade. 

Scheme 2. Kochi s catalytic cycle (lel t) for the Mn "(salcn)/PhlO catalytic system and Adam-Collman s bifurcated mechanism (right) [30,311...

In the study of the response of nonlinear systems to external periodic perturbations there exists a dual search, that for universal relations and that for responses specific to a particular reaction mechanism. System mathematicians are, of course, intrigued by commonalities and universal relations. As an example, the similarities of alkali atoms and irons are of course remarkable. However, the chemists and biologists must also face the task of differences in the behavior of the sequence in the periodic table. Lithium carbonate controls manic depressive illness effectively, whereas the other alkali carbonates do not (nor do other alkali salts other than lithium salts). We have the same duality of interest in complex reaction mechanisms. Bifurcations, limit cycles, critical slowing down, occur in many nonlinear systems and have common features and universal laws. To the extent that these hold we find out little about the specific reaction mechanism of a given system and we seek properties which are specific to such reaction mechanisms. 

A similar theoretical analysis can be reproduced for higher dimensional systems [48]. In a prospective paper [63], in collaboration with Pearson and Russo, we have reported preliminary results of a study of two-dimensional reaction-diffusion systems that model sustained front patterns observed in gel reactors. The linear [34, 35] and annular [21, 22, 38] gel reactors are strips of gel that are fed from the lateral boundaries. These reactors have a natural tendency to produce narrow front (linear or circular) structures away from the boundaries [39]. Stationary single-front and multi-front patterns have been observed experimentally [34-38]. According to the Hopf bifurcation mechanism reported in section 5, these front patterns are expected to destabilize into periodically oscillating structures. Since the Hopf mode is likely to be condensed in the active region at the front zones, the reaction-diffusion sys-... 

These examples have identified two types of catastrophe points, a distinction that arises as a corollary of a theorem on structural stability. This theorem, when used to describe structural changes in a molecular system, states that the structure associated with a particular geometry X in nuclear configuration space is structurally stable if p r X) has a finite number of cps such that (i) each cp is nondegenerate (ii) the stable and unstable manifolds of any pair of cps intersect transver-sally. The immediate consequence of this theorem is that a structural instability can be established solely through either of two mechanisms in the bifurcation mechanism the charge distribution exhibits a degenerate cp, while the conflict mechanism is characterized by the nontransversal intersection of the stable and unstable manifolds of cps in p(r X). 

Leine RI, Nijmeijer H (2004) Dynamics and bifurcations of non-smooth mechanical systems. Lecture notes in applied and computational mechanics, vol. 18. Springer, Heidelberg... 

Abstract. A model of the conformational transitions of the nucleic acid molecule during the water adsorption-desorption cycle is proposed. The nucleic acid-water system is considered as an open system. The model describes the transitions between three main conformations of wet nucleic acid samples A-, B- and unordered forms. The analysis of kinetic equations shows the non-trivial bifurcation behaviour of the system which leads to the multistability. This fact allows one to explain the hysteresis phenomena observed experimentally in the nucleic acid-water system. The problem of self-organization in the nucleic acid-water system is of great importance for revealing physical mechanisms of the functioning of nucleic acids and for many specific practical fields. 

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

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. 

MSN. 151.1. Prigogine, Why irreversibility The formulation of classical and quantum mechanics for non-integrable systems, Int. J. Bifurcation and Chaos 5, 3-16 (1995). 

One of the main themes of this volume is the influence of the environment on chemical reactivity. Such a question is of special interest for chemical systems in far from equilibrium conditions. It is today well known that, far from equilibrium, chemical systems involving catalytic mechanisms may lead to dissipative structures.1-2 It has also been shown—and this is one of the main themes of this discussion—that dissipative structures are very sensitive to global features characterizing the environment of chemical systems, such as their size and form, the boundary conditions imposed on their surface, and so on. All these features influence in a decisive way the type of instabilities, called bifurcations, that lead to dissipative structures. 

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978 A. J. Lieberman and A. J. Lichienberg, Regular and Stochastic Motion, Springer, New York, 1983 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. 

A comparative study was done by Kevrekidis and published as I. G. Kevrekidis, L. D. Schmidt, and R. Aris. Some common features of periodically forced reacting systems. Chem. Eng. Sci. 41,1263-1276 (1986). See also two papers by the same authors Resonance in periodically forced processes Chem. Eng. Sci. 41, 905-911 (1986) The stirred tank forced. Chem. Eng. Sci. 41,1549-1560 (1986). A full study of the Schmidt-Takoudis vacant site mechanism is to be found in M. A. McKamin, L. D. Schmidt, and R. Aris. Autonomous bifurcations of a simple bimolecular surface-reaction model. Proc. R. Soc. Lond. A 415,363-387 (1988) Forced oscillations of a self-oscillating bimolecular surface reaction model. Proc. R. Soc. Lond. A 415,363-388 (1988). 

The detailed study of these bifurcations (which we shall describe in subsequent publications) is important for several reasons. It will be helpful in understanding some of the mechanisms that synchronize chemical oscillators, by breaking the pattern of higher-dimensional tori that exist when they are weakly coupled. It also provides a very convenient setting for studying on practically any oscillating system phenomena that are, by comparison, less frequently observed in autonomous systems. 

A periodically forced system may be considered as an open-loop control system. The intermediate and high amplitude forced responses can be used in model discrimination procedures (Bennett, 1981 Cutlip etal., 1983). Alternate choices of the forcing variable and observations of the relations and lags between various oscillating components of the response will yield information regarding intermediate steps in a reaction mechanism. Even some unstable phase plane components of the unforced system will become apparent through their role in observable effects (such as the codimension two bifurcations described above where they collide and annihilate stable, observable responses). 

The results surveyed in the preceding two sections provide a first clue to the origin of chirality chiral patterns can emerge spontaneously in an initially uniform and isotropic medium, through a mechanism of bifurcations far from thermodynamic equilibrium (see Figs. 4 and 5). On the other hand, because of the invariance properties of the reaction-diffusion equations (1) in such a medium, chiral solutions will always appear by pairs of opposite handedness. As explained in Sections III.B and III.C this implies that in a macroscopic system symmetry will be restored in the statistical sense. We are left therefore with an open question, namely, the selection of forms of preferred chirality, encompassing a macroscopic space region and maintained over a macroscopic time interval. 

Dynamics effects, which were described in previous sections, on reaction pathways, concerted-stepwise mechanistic switching, and path bifurcation have in most cases been examined for isolate systems without medium effects. Since energy distribution among vibrational and rotational modes and moment of inertia of reacting subfragment are likely to be modified by environment, it is intriguing to carry out simulations in solution. The difference or similarity in the effect of dynamics in the gas phase and in solution may be clarified in the near future by using QM/MM-MD method. Such study would provide information that is comparable with solution experiment and help us to understand reaction mechanisms in solution. 

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