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In mathematics, specifically in the 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 the system's long-term dynamical behaviour. Bifurcations occur in both continuous systems (described by ODEs, DDEs or PDEs), and discrete systems (described by maps). For other meanings of mathematics or math, see mathematics (disambiguation). ...
In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...
Ode is a form of stately and elaborate lyrical verse. ...
In mathematics, delay differential equations are a type of differential equations. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
Bifurcation theory is the study of how and when such bifurcations can occur.
Types of bifurcation
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).
The point is called an equilibrium point for the differential equation if for all . ...
Local bifurcations
Phase portrait showing Saddle-node bifurcation.
 Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos. 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'). Image File history File links Saddlenode. ...
Image File history File links Saddlenode. ...
Image File history File links Download high resolution version (1024x328, 58 KB) Summary Period-halving bifurcations leading to order, followed by period doubling bifurcations leading to chaos. ...
Image File history File links Download high resolution version (1024x328, 58 KB) Summary Period-halving bifurcations leading to order, followed by period doubling bifurcations leading to chaos. ...
More technically, consider the continuous dynamical system described by the ODE
 A local bifurcation occurs at (x0,λ0) if the matrix  has an eigenvalue with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, if the eigenvalue is non-zero but purely imaginary, this is a Hopf bifurcation.
For discrete dynamical systems, consider the system
- xn + 1 = f(xn,λ)
Then a local bifurcation occurs at (x0,λ0) 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:
In bifurcation theory a fold or saddle-node bifurcation is a local bifurcation in which two fixed points of a dynamical system annihilate one another. ...
In bifurcation theory a saddle-node bifurcation is a local bifurcation in which two fixed points of a dynamical system annihilate one another. ...
In bifurcation theory, a field within mathematics, a transcritical bifurcation is a particular kind of zero-eigenvalue bifurcation, meaning that it is characterized by a collision between two fixed points. ...
In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a special case zero-eigenvalue bifurcation. ...
A Period doubling bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. ...
In bifurcation theory a Hopf or Andronov Hopf bifurcation 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 imaginary axis of the complex plane. ...
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:
Global bifurcations can also involve more complicated sets such as chaotic attractors. A homoclinic bifurcation is a global bifurcation which often occurs when a periodic orbit collides with a saddle point. ...
In mathematics, in the area of dynamical systems, 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 or as time approaches minus-infinity. ...
Plot of y = x3 with a saddle-point at (0,0). ...
In mathematics, particularly dynamical systems, a heteroclinic bifurcation is a global bifurcation involving a heteroclinic cycle. ...
In mathematics, in the area of dynamical systems, 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 or as time approaches minus-infinity. ...
In mathematics, in the area of dynamical systems, 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 or as time approaches minus-infinity. ...
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 Takens-Bogdanov bifurcation.
See also In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. ...
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. ...
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