Pitchfork bifurcation

Pitchfork bifurcation is a type of bifurcation in which, as a parameter is varied, a stable equilibrium point becomes unstable and two new stable equilibrium points are created, symmetrically with respect to the original point. In a supercritical pitchfork bifurcation, the original equilibrium point is asymptotically stable and becomes unstable as the bifurcation parameter increases, while the two new equilibrium points are also asymptotically stable. This type of bifurcation can occur in various systems, including fluid dynamics, and can result in the emergence of multiple solutions.From: Feedback Control in Systems Biology [2019], Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature [2020]

Continuous Models Using Ordinary Differential Equations

View Chapter

Purchase Book

Published in Sandip Banerjee, Mathematical Modeling, 2021

Sandip Banerjee

A pitchfork bifurcation is a particular type of local bifurcation (possible in dynamical systems) that has symmetry. In such cases, equilibrium points appear and disappear in symmetrical pairs. The bifurcation diagram looks like a pitchfork, hence the name pitchfork bifurcation. There are two types of pitchfork bifurcations, namely supercritical and subcritical. A pitchfork bifurcation is called supercritical if a stable solution branch bifurcates into two new stable branches as the parameter μ is increased. It is called subcritical if two unstable and one stable equilibria collapse to produce one stable one.

Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature

View Article

Journal Information

Published in International Journal of Computational Fluid Dynamics, 2020

Martin W. Hess, Annalisa Quaini, Gianluigi Rozza

Each curved wall is defined by a second-order polynomial, interpolating three prescribed points. While the points at the domain boundary y=0 and y=3 are kept fixed, the inner points are moved towards x=0 in order to create an increasing curvature. The viscosity varies in the interval . We recall that the Reynolds number Re (6) depends on the kinematic viscosity. As Re is varied for each fixed geometry, a supercritical pitchfork bifurcation occurs: for Re higher than the critical bifurcation point, three solutions exit. Two of these solutions are stable, one with a jet towards the top wall and one with a jet towards the bottom wall, and one is unstable. The unstable solution is symmetric to the horizontal centreline at y=1.5, while the jet of the stable solutions is said to undergo the Coanda effect.

Pitchfork bifurcation

Explore chapters and articles related to this topic

Continuous Models Using Ordinary Differential Equations

Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature