Bifurcation theory is the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
study of changes in the qualitative or topological structure of a given family. Examples of such families are the
integral curveIn mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
s of a family of
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
s or, the solutions of a family of
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. Most commonly applied to the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
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 behaviour.
Bifurcation theory is the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
study of changes in the qualitative or topological structure of a given family. Examples of such families are the
integral curveIn mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
s of a family of
vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...
s or, the solutions of a family of
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. Most commonly applied to the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
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 behaviour. Bifurcations occur in both continuous systems (described by
ODEIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....
s,
DDEIn mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
s or
PDEsIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables...
), and discrete systems (described by maps).
Bifurcation Types
It is useful to divide bifurcations into two principal classes:
- Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria
In mathematics, the point is an equilibrium point for the differential equationif for all .Similarly, the point is an equilibrium point for the difference equationif for ....
, periodic orbits or other invariant sets as parameters cross through critical thresholds; and
- Global bifurcations, which often occur when larger invariant sets of the system 'collide' with each other, or with equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed points).
Local bifurcations
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 (those described by maps rather than ODEs), 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').
More technically, consider the continuous dynamical system described by the ODE
A local bifurcation occurs at if the
JacobianIn vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space...
matrix
has an
eigenvalueIn mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix...
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 bifurcationIn bifurcation theory a Hopf or Andronov–Hopf bifurcation, named after Eberhard Hopf and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the...
.
For discrete dynamical systems, consider the system
Then a local bifurcation occurs at if the matrix
has an eigenvalue with modulus equal to one. If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to -1, it is a period-doubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation.
Examples of local bifurcations include:
- Saddle-node
In the mathematical area of bifurcation theory a saddle-node bifurcation or tangential bifurcation is a local bifurcation in which two fixed points of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical...
(fold) bifurcation
- Transcritical bifurcation
In bifurcation theory, a field within mathematics, a transcritical bifurcation is a particular kind of local bifurcation, meaning that it is characterized by an equilibrium having an eigenvalue whose real part passes through zero....
- Pitchfork bifurcation
In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations have two types - supercritical or subcritical....
- Period-doubling
In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system...
(flip) bifurcation
- Hopf bifurcation
In bifurcation theory a Hopf or Andronov–Hopf bifurcation, named after Eberhard Hopf and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the...
- Neimark (secondary Hopf) bifurcation
Global bifurcations
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').
Examples of global bifurcations include:
- Homoclinic bifurcation
In mathematics, a homoclinic bifurcation is a global bifurcation which often occurs when a periodic orbit collides with a saddle point.The image below shows a phase portrait before, at, and after a homoclinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point...
in which a limit cycle collides with a saddle pointIn mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
.
- Heteroclinic bifurcation
In mathematics, particularly dynamical systems, a heteroclinic bifurcation is a global bifurcation involving a heteroclinic cycle. Heteroclinic bifurcations come in two types, resonance bifurcations, and transverse bifurcations...
in which a limit cycle collides with two or more saddle points.
- Infinite-period bifurcation
In mathematics, an infinite-period bifurcation is a global bifurcation that can occur when two fixed points emerge on a limit cycle. As the limit of a parameter approaches a certain critical value, the speed of the oscillation slows down and the period approaches infinity. The infinite-period...
in which a stable node and saddle point simultaneously occur on a limit cycle.
- Blue sky catastrophe
The blue sky catastrophe is a type of bifurcation of a periodic orbit. In other words, it describes a sort of behaviour stable solutions of a set of differential equations can undergo as the equations are gradually changed...
in which a limit cycle collides with a nonhyperbolic cycle.
Global bifurcations can also involve more complicated sets such as chaotic attractors.
Codimension of a bifurcation
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 are the only generic local bifurcations which are really codimension-one (the others all having higher codimension). However, often transcritical and pitchfork bifurcations are also often thought of as codimension-one, because the normal forms can be written with only one parameter.
An example of a well-studied codimension-two bifurcation is the
Bogdanov-Takens bifurcationIn bifurcation theory, a field within mathematics, a Bogdanov-Takens bifurcation is a well-studied example of a bifurcation with co-dimension two, meaning that two parameters must be varied for the bifurcation to occur...
.
Applications in semiclassical and quantum physics
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 GutzwillerMartin C. Gutzwiller is a physicist. He is known for the invention of the variational wave-function describing electrons with strong localized interactions , for the first investigation of relations between classical and quantum mechanics in chaotic systems , as well as new solutions to...
points out in his classic work on
quantum chaosQuantum chaos is a branch of physics which studies how chaotic classical systems can be shown to be limits of quantum-mechanical systems. The phenomena covered by quantum chaos so far are mainly related to wave theory. An alternative name, proposed by Sir Michael Berry, is quantum chaology...
. 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.
See also
- Bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values of a system as a function of a bifurcation parameter in the system...
- Catastrophe theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....
- Feigenbaum constant
- Phase portrait
A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is representated by a different curve, or point....