**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Pitchfork bifurcation

Summary

In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from one fixed point to three fixed points. Pitchfork bifurcations, like Hopf bifurcations, have two types – supercritical and subcritical.
In continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry.
The normal form of the supercritical pitchfork bifurcation is
For , there is one stable equilibrium at . For there is an unstable equilibrium at , and two stable equilibria at .
The normal form for the subcritical case is
In this case, for the equilibrium at is stable, and there are two unstable equilibria at . For the equilibrium at is unstable.
An ODE
described by a one parameter function with satisfying:
(f is an odd function),
has a pitchfork bifurcation at . The form of the pitchfork is given
by the sign of the third derivative:
Note that subcritical and supercritical describe the stability of the outer lines of the pitchfork (dashed or solid, respectively) and are not dependent on which direction the pitchfork faces. For example, the negative of the first ODE above, , faces the same direction as the first picture but reverses the stability.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (43)

Related people (8)

Related units (1)

Related concepts (1)

Related MOOCs (2)

Related courses (5)

Related lectures (41)

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.

Neuronal Dynamics - Computational Neuroscience of Single Neurons

The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.

Neuronal Dynamics - Computational Neuroscience of Single Neurons

The activity of neurons in the brain and the code used by these neurons is described by mathematical neuron models at different levels of detail.

PHYS-460: Nonlinear dynamics, chaos and complex systems

The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during th

COM-502: Dynamical system theory for engineers

Linear and nonlinear dynamical systems are found in all fields of science and engineering. After a short review of linear system theory, the class will explain and develop the main tools for the quali

ME-466: Instability

This course focuses on the physical mechanisms at the origin of the transition of a flow from laminar to turbulent using the hydrodynamic instability theory.

Dynamical System Theory: Bifurcations

Covers the topic of bifurcations in dynamical system theory, focusing on the pitchfork bifurcation.

Equilibrium Points and Bifurcations

Introduces equilibrium points and bifurcations in differential equations, discussing their stability and relevance in various contexts.

Limit Cycles in Biological Systems

Explores the concept of limit cycles in biological systems and their significance in understanding system dynamics.

Edouard Boujo, Giuseppe Antonio Zampogna

We study the stability of laminar wakes past three-dimensional rectangular prisms. The width-to-height ratio is set to W/H = 1.2, while the length-to-height ratio 1/6 < L/H < 3 covers a wide range of geometries from thin plates to elongated Ahmed bodies. F ...

It is known that the pitchfork bifurcation of Kelvin-Helmholtz instability occurring at minimum gradient Richardson number Ri(m) similar or equal to 1/4 in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Pra ...

2021Christophe Ancey, Ivan Pascal, Bob de Graffenried, Raphaël Miazza

Although the importance of studying channel bifurcations is widely recognised, their hydraulic behaviour in shallow, rough mountain rivers has so far received little attention from researchers. Understanding the specific hydraulics of such units is essenti ...

2023