Local bifurcations

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

The maximum number of solutions of equation (1) is r+1 next to such a singular point. Moreover, all the local bifurcation diagrams of the function F can be determined by the analysis of the simpler polynomial function G. 

The qualitative features of the local bifurcation diagrams (0 vs. Da) of Eq. (13) in the neighborhoood of any singular point defined by (14) are same as those of its universal unfolding... 

Eq. (18) can have for any N, either zero, two, four,. . . or 2N bifurcation points. All the possible local bifurcation diagrams can be constructed by a method described in [1]. Moreover, it can be proven [JJ that any global bifurcation diagram of Eq. (13) must be similar to one of the local bifurcation diagrams of Eq. (18). 

The steady-state equations describing lumped parameter systems in which several reactions occur simultaneously contain a very large number of parameters. Thus, it is impractical to conduct an exhaustive parametric study to determine their features. The new technique presented here predicts qualitative features of these systems such as the maximum number of solutions, parameter values for which these solutions exist and all the local bifurcation diagrams. Construction of the three varieties enables the division of the global parameter space into regions with different bifurcation diagrams. 

It turns out that the language of normal and local modes that emerged from the bifurcation analysis of the Darling-Dennison Hamiltonian is not sufficient to describe the general Fermi resonance case, because the bifurcations are qualitatively different from the normal-to-local bifurcation in figure Al.2.10. For example, in 2 1 Fermi systems, one type of bifurcation is that in which resonant collective modes are bom [54]. The resonant collective modes are illustrated in figure A12.11 their difference from the local modes of the Darling-Dennison system is evident. Other types of bifurcations are also possible in Fermi resonance systems a detailed treatment of the 2 1 resonance can be found in [44]. 

Vol. 1893 H. HanOmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems, Results and Examples (2007)... 

Several papers have shown the direct relation between saddle-node bifurcations and voltage collapse problems, e.g., (Canizares and Alvarado, 1993 Canizares, 1995). Saddle-node bifurcations, also known as turning points, are generic codimension one local bifurcations of nonlinear dynamical systems of the form ... 

The predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

The key methods in our presentation of local bifurcations are based on the center manifold theorem and on the invariant foliation technique (see Sec. 5.1. of Part I). The assumption that there are no characteristic exponents to the right of the imaginary axis (or no multipliers outside the unit circle) allows us to conduct a smooth reduction of the system to a very convenient standard form. We use this reduction throughout this book both in the study of local bifurcations on the stability boundaries themselves and in the study of global bifurcations on the route over the stability boundaries (Chap. 12).These... 

In the general case where there are both stable and unstable characteristic exponents, or stable and unstable multipliers in the spectrum, the local bifurcation problem does not cause any special difficulties, thanks to the reduction onto the center manifold. Consequently, the pictures from Chaps. 9-H will need only some slight modifications where unstable directions replace stable ones, or be added to existing directions in the space. However, the reader must... 

An explicit mathematical formulation to the finite-parameter approach to the local bifurcations was given by Arnold [19], based on the notion of versal families. Roughly speaking, versality is a kind of structural stability of the family in the space of families of dynamical systems. Different versions of such stability are discussed in detail in [97]. 

LOCAL BIFURCATIONS ON THE ROUTE OVER STABILITY BOUNDARIES... 

The answers to these questions are settled by the theory of bifurcations. In this chapter, we consider only local bifurcations, i.e. those which occur near critical equilibrium states, and near fixed points of a Poincare map. We restrict our study to the simplest but key bifurcations which have an immediate connection to the critical cases are discussed in the two last chapters. 

---