Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.[1] Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps).
bifurcation
The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior.[2] Henri Poincaré also later named various types of stationary points and classified them with motif[clarify].
A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change. In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (described by maps), this corresponds to a fixed point having a Floquet multiplier with modulus equal to one. In both cases, the equilibrium is non-hyperbolic at the bifurcation point.The topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence 'local').
A local bifurcation occurs at ( x 0 , λ 0 ) \displaystyle (x_0,\lambda _0) if the Jacobian matrix d f x 0 , λ 0 \displaystyle \textrm df_x_0,\lambda _0 has an eigenvalue with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, but if the eigenvalue is non-zero but purely imaginary, this is a Hopf bifurcation.
Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibria. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as is the case with local bifurcations. In fact, the changes in topology extend out to an arbitrarily large distance (hence 'global').
The codimension of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This corresponds to the codimension of the parameter set for which the bifurcation occurs within the full space of parameters. Saddle-node bifurcations and Hopf bifurcations are the only generic local bifurcations which are really codimension-one (the others all having higher codimension). However, transcritical and pitchfork bifurcations are also often thought of as codimension-one, because the normal forms can be written with only one parameter.
Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems,[6][7][8] molecular systems,[9] and resonant tunneling diodes.[10] Bifurcation theory has also been applied to the study of laser dynamics[11] and a number of theoretical examples which are difficult to access experimentally such as the kicked top[12] and coupled quantum wells.[13] The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as Martin Gutzwiller points out in his classic[14] work on quantum chaos.[15] Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Hopf bifurcations, umbilic bifurcations, period doubling bifurcations, reconnection bifurcations, tangent bifurcations, and cusp bifurcations.
Objectives: The TRYTON (Prospective, Single Blind, Randomized Controlled Study to Evaluate the Safety & Effectiveness of the Tryton Side Branch Stent Used With DES in Treatment of de Novo Bifurcation Lesions in the Main Branch & Side Branch in Native Coronaries) bifurcation trial sought to compare treatment of de novo true bifurcation lesions using a dedicated bifurcation stent or SB balloon angioplasty.
Methods: We randomly assigned patients with true bifurcation lesions to a main vessel stent plus provisional stenting or the bifurcation stent. The primary endpoint (powered for noninferiority) was target vessel failure (TVF) (cardiac death, target vessel myocardial infarction, and target vessel revascularization). The secondary angiographic endpoint (powered for superiority) was in-segment percent diameter stenosis of the SB at 9 months.
Conclusions: Provisional stenting should remain the preferred strategy for treatment of non-left main true coronary bifurcation lesions. (Prospective, Single Blind, Randomized Controlled Study to Evaluate the Safety & Effectiveness of the Tryton Side Branch Stent Used With DES in Treatment of de Novo Bifurcation Lesions in the Main Branch & Side Branch in Native Coronaries [TRYTON]; NCT01258972).
Background: The optimal stenting strategy in coronary artery bifurcation lesions is unknown. In the present study, a strategy of stenting both the main vessel and the side branch (MV+SB) was compared with a strategy of stenting the main vessel only, with optional stenting of the side branch (MV), with sirolimus-eluting stents.
Methods and results: A total of 413 patients with a bifurcation lesion were randomized. The primary end point was a major adverse cardiac event: cardiac death, myocardial infarction, target-vessel revascularization, or stent thrombosis after 6 months. At 6 months, there were no significant differences in rates of major adverse cardiac events between the groups (MV+SB 3.4%, MV 2.9%; P=NS). In the MV+SB group, there were significantly longer procedure and fluoroscopy times, higher contrast volumes, and higher rates of procedure-related increases in biomarkers of myocardial injury. A total of 307 patients had a quantitative coronary assessment at the index procedure and after 8 months. The combined angiographic end point of diameter stenosis >50% of main vessel and occlusion of the side branch after 8 months was found in 5.3% in the MV group and 5.1% in the MV+SB group (P=NS).
Conclusions: Independent of stenting strategy, excellent clinical and angiographic results were obtained with percutaneous treatment of de novo coronary artery bifurcation lesions with sirolimus-eluting stents. The simple stenting strategy used in the MV group was associated with reduced procedure and fluoroscopy times and lower rates of procedure-related biomarker elevation. Therefore, this strategy can be recommended as the routine bifurcation stenting technique.
In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation (Rasband 1990).
More generally, a bifurcation is a separation of a structure into two branches or parts. For example, in the plot above, the function , where denotes the real part, exhibits a bifurcation along the negative real axis and .
where \(f\) is smooth. A bifurcation occurs at parameter \(\lambda = \lambda_0\) if there are parameter values \(\lambda_1\) arbitrarily close to \(\lambda_0\) with dynamics topologically inequivalent from those at \(\lambda_0\ .\) For example, the number or stability of equilibria or periodic orbits of \(f\) may change with perturbations of \(\lambda\)from \(\lambda_0\ .\) One goal of bifurcation theory is to produce parameter space maps or bifurcation diagramsthat divide the \(\lambda\) parameter space into regions of topologically equivalent systems. Bifurcations occur at points that do not lie in the interior of one of these regions.
Bifurcation theory provides a strategy for investigating the bifurcations that occur within a family. It does so by identifying ubiquitous patterns of bifurcations. Each bifurcation type or singularityis given a name; for example, Andronov-Hopf bifurcation. No distinction has been made in the literature between "bifurcation" and "bifurcation type," both being called "bifurcations."
Inequalities called non-degeneracy conditions are part of the specification of a bifurcation type.The bifurcation types and their normal forms serve as templates that facilitate construction of parameter space maps.Bifurcation theory analyzes the bifurcations within the normal forms and investigates the similarity of the dynamics within systems having a given bifurcation type. The "gold standard" forsimilarity of systems used by the theory is topological equivalence. In some cases, bifurcationtheory proves structural stability of a family. One of the principal objectives of bifurcation theory is to prove the structural stability of normal forms. Note, however, that there are bifurcation types for which structurally stable normal forms do notexist. An important aspect of the definition of structural stability in the context of bifurcation theoryis the specification of which perturbations of a family are allowed. For example, bifurcation types of systems possessing specifiedsymmetries have been studied extensively (Equivariant Bifurcation Theory).
One can view bifurcations as a failure of structural stability within a family. A starting point forclassifying bifurcation types is the Kupka-Smale theorem that lists three generic propertiesof vector fields:
Different ways that these Kupka-Smale conditions fail lead to different bifurcation types. Bifurcation theory constructs a layered graph of bifurcation types in which successive layers consist of types whosedefining equations specify more failure modes. These layers can be organized by the codimension of the bifurcation types, defined as the minimal number of parameters of families in which that bifurcation typeoccurs. Equivalently, the codimension is the number of equality conditions that characterize a bifurcation. 2ff7e9595c
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