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Bifurcation theory is the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
study of changes in the qualitative or
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
structure of a given
family of curves In geometry, a family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more c ...
, such as the
integral curve In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpreta ...
s of a family of vector fields, and the solutions of a family of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. Most commonly applied to the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
study of
dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, 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 Ordinary or The Ordinary often refer to: Music * ''Ordinary'' (EP) (2015), by South Korean group Beast * ''Ordinary'' (Every Little Thing album) (2011) * "Ordinary" (Two Door Cinema Club song) (2016) * "Ordinary" (Wayne Brady song) (2008) * ...
,
delay Delay (from Latin: dilatio) may refer to: Arts, entertainment, and media * ''Delay 1968'', a 1981 album by German experimental rock band Can * ''The Delay'', a 2012 Uruguayan film People * B. H. DeLay (1891–1923), American aviator and acto ...
or
partial Partial may refer to: Mathematics * Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
differential equations) and discrete systems (described by maps). The name "bifurcation" was first introduced by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
in 1885 in the first paper in mathematics showing such a behavior.
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
also later named various types of
stationary points In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
and classified them .


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, 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 (described by maps), this corresponds to a fixed point having a
Floquet multiplier Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with \displaystyle A(t) a piecewise continuous periodic function ...
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 ordinary differential equation (ODE) :\dot x=f(x,\lambda)\quad f\colon\mathbb^n\times\mathbb\rightarrow\mathbb^n. A local bifurcation occurs at (x_0,\lambda_0) if the Jacobian matrix \textrmf_ has an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
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 In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses ...
. For discrete dynamical systems, consider the system :x_=f(x_n,\lambda)\,. Then a local bifurcation occurs at (x_0,\lambda_0) if the matrix \textrmf_ 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 (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. A transcritical ...
*
Pitchfork bifurcation 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 ...
* Period-doubling (flip) bifurcation *
Hopf bifurcation In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses ...
* Neimark–Sacker (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 which a
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
collides with a
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
. Homoclinic bifurcations can occur supercritically or subcritically. The variant above is the "small" or "type I" homoclinic bifurcation. In 2D there is also the "big" or "type II" homoclinic bifurcation in which the homoclinic orbit "traps" the other ends of the unstable and stable manifolds of the saddle. In three or more dimensions, higher codimension bifurcations can occur, producing complicated, possibly
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
dynamics. *Heteroclinic bifurcation in which a limit cycle collides with two or more saddle points; they involve a
heteroclinic cycle In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching t ...
. Heteroclinic bifurcations are of two types: resonance bifurcations and transverse bifurcations. Both types of bifurcation will result in the change of stability of the heteroclinic cycle. At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of the equilibria in the cycle is satisfied. This is usually accompanied by the birth or death of a
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
. A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero. This will also cause a change in stability of the heteroclinic cycle. * Infinite-period bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle. As the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of a parameter approaches a certain critical value, the speed of the oscillation slows down and the period approaches infinity. The infinite-period bifurcation occurs at this critical value. Beyond the critical value, the two fixed points emerge continuously from each other on the limit cycle to disrupt the oscillation and form two
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
s. *
Blue sky catastrophe The blue sky catastrophe is a form of orbital indeterminacy, and an element of bifurcation theory. Orbital dynamics Blue sky catastrophe is a type of bifurcation of a periodic orbit. In other words, it describes a sort of behaviour stable solutio ...
in which a limit cycle collides with a nonhyperbolic cycle. Global bifurcations can also involve more complicated sets such as chaotic attractors (e.g.
crises A crisis ( : crises; : critical) is either any event or period that will (or might) lead to an unstable and dangerous situation affecting an individual, group, or all of society. Crises are negative changes in the human or environmental affair ...
).


Codimension of a bifurcation

The
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the ...
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. An example of a well-studied codimension-two bifurcation is the
Bogdanov–Takens bifurcation In 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. It is named after Rifka ...
.


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 diode A resonant-tunneling diode (RTD) is a diode with a resonant-tunneling structure in which electrons can tunnel through some resonant states at certain energy levels. The current–voltage characteristic often exhibits negative differential resistan ...
s. Bifurcation theory has also been applied to the study of
laser dynamics A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
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 Martin may refer to: Places * Martin City (disambiguation) * Martin County (disambiguation) * Martin Township (disambiguation) Antarctica * Martin Peninsula, Marie Byrd Land * Port Martin, Adelie Land * Point Martin, South Orkney Islands Austral ...
points out in his classic work on
quantum chaos Quantum chaos is a branch of physics which studies how chaos theory, chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship betwee ...
. 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 values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
*
Bifurcation memory Bifurcation memory is a generalized name for some specific features of the behaviour of the dynamical system near the bifurcation. General information The phenomenon is known also under the names of "''stability loss delay for dynamical bifur ...
*
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. Bifurcation theory studies and classifies phenomena cha ...
*
Feigenbaum constants In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. Hist ...
*
Geomagnetic reversal A geomagnetic reversal is a change in a planet's magnetic field such that the positions of magnetic north and magnetic south are interchanged (not to be confused with geographic north and geographic south). The Earth's field has alternated ...
*
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 represented by a different curve, or point. Phase portraits are an invaluable tool in studying dyn ...
*
Tennis racket theorem The tennis racket theorem or intermediate axis theorem is a result in classical mechanics describing the movement of a rigid body with three distinct principal moments of inertia. It is also dubbed the Dzhanibekov effect, after Soviet cosmonau ...


Notes


References

* *Guardia, M.; Martinez-Seara, M.; Teixeira, M. A. (2011)
Generic bifurcations of low codimension of planar Filippov Systems
"Journal of differential equations", Febrer 2011, vol. 250, núm. 4, pp. 1967–2023. DOI:10.1016/j.jde.2010.11.016 *


External links


Nonlinear dynamics

Bifurcations and Two Dimensional Flows
by Elmer G. Wiens
Introduction to Bifurcation theory
by John David Crawford {{DEFAULTSORT:Bifurcation Theory Nonlinear systems