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. Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps).
The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. 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').
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, molecular systems, and resonant tunneling diodes. Bifurcation theory has also been applied to the study of laser dynamics and a number of theoretical examples which are difficult to access experimentally such as the kicked top and coupled quantum wells. 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 work on quantum chaos. 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).
Fluid velocities were measured by laser Doppler velocimetry under conditions of pulsatile flow in a scale model of the human carotid bifurcation. Flow velocity and wall shear stress at five axial and four circumferential positions were compared with intimal plaque thickness at corresponding locations in carotid bifurcations obtained from cadavers. Velocities and wall shear stresses during diastole were similar to those found previously under steady flow conditions, but these quantities oscillated in both magnitude and direction during the systolic phase. At the inner wall of the internal carotid sinus, in the region of the flow divider, wall shear stress was highest (systole = 41 dynes/cm2, diastole = 10 dynes/cm2, mean = 17 dynes/cm2) and remained unidirectional during systole. Intimal thickening in this location was minimal. At the outer wall of the carotid sinus where intimal plaques were thickest, mean shear stress was low (-0.5 dynes/cm2) but the instantaneous shear stress oscillated between -7 and +4 dynes/cm2. Along the side walls of the sinus, intimal plaque thickness was greater than in the region of the flow divider and circumferential oscillations of shear stress were prominent. With all 20 axial and circumferential measurement locations considered, strong correlations were found between intimal thickness and the reciprocal of maximum shear stress (r = 0.90, p less than 0.0005) or the reciprocal of mean shear stress (r = 0.82, p less than 0.001). An index which takes into account oscillations of wall shear also correlated strongly with intimal thickness (r = 0.82, p less than 0.001). When only the inner wall and outer wall positions were taken into account, correlations of lesion thickness with the inverse of maximum wall shear and mean wall shear were 0.94 (p less than 0.001) and 0.95 (p less than 0.001), respectively, and with the oscillatory shear index, 0.93 (p less than 0.001). These studies confirm earlier findings under steady flow conditions that plaques tend to form in areas of low, rather than high, shear stress, but indicate in addition that marked oscillations in the direction of wall shear may enhance atherogenesis.
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.
This is not a comprehensive list of codimension one bifurcations. Additional types can be found in systems with quasiperiodic oscillations or chaotic dynamics. Moreover, there are subcases in the list above that deal withsuch issues as whether an Andronov-Hopf bifurcation is sub-critical or super-critical,and the implications of eigenvalue magnitudes for homoclinic bifurcation. 041b061a72