Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, 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.Bifurcations occur in both continuous systems (described by OD Es, DD Es or PD Es) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them with motif.

 Bifurcation types

         It is useful to divide bifurcations into two principal classes:

• Local bifurcations, which can be analyses entirely through changes in the local stability properties of equilibrium, 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 equilibrium of the system. They cannot be detected purely by a stability analysis of the equilibrium (fixed points).     
  

• 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 OD Es), this corresponds to a fixed point having a Eloquent 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 neighborhoods 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

  ${\displaystyle {\dot {x}}=f(x,\lambda )\quad f\colon \mathbb {R} ^{n}\times \mathbb {R} \rightarrow \mathbb {R} ^{n}.}$

  A local bifurcation occurs at ${\displaystyle (x_{0},\lambda _{0})}$ if the Jacobin matrix ${\displaystyle {\textrm {d}}f_{x_{0},\lambda _{0}}}$ has an eigenvalue 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 Ho pf bifurcation.

  For discrete dynamical systems, consider the system

  ${\displaystyle x_{n+1}=f(x_{n},\lambda )\,.}$

  Then a local bifurcation occurs at ${\displaystyle (x_{0},\lambda _{0})}$ if the matrix ${\displaystyle {\textrm {d}}f_{x_{0},\lambda _{0}}}$ 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
• Pitchfork bifurcation
• Period-doubling (flip) bifurcation
• Hopf bifurcation
• Neimark–Sacker (secondary Hopf) bifurcation

  Global bifurcations

  

  A phase portrait before, at, and after a homo clinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point. At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a homo clinic orbit. After the bifurcation there is no longer a periodic orbit. Left panel: For small parameter values, there is a saddle point at the origin and a limit cycle in the first quadrant. Middle panel: As the bifurcation parameter increases, the limit cycle grows until it exactly intersects the saddle point, yielding an orbit of infinite duration. Right panel: When the bifurcation parameter increases further, the limit cycle disappears completely.

  Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibrium. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighborhood, 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:

• Homo clinic bifurcation in which a limit cycle collides with a saddle point.Homo clinic bifurcations can occur super critically or sub critically. The variant above is the "small" or "type I" homo clinic bifurcation. In 2D there is also the "big" or "type II" homo clinic bifurcation in which the homo clinic orbit "traps" the other ends of the unstable and stable manifolds of the saddle. In three or more dimensions, higher co dimension bifurcations can occur, producing complicated, possibly chaotic dynamics.
• Hetero-clinic bifurcation in which a limit cycle collides with two or more saddle points; they involve a hetero-clinic cycle. Hetero-clinic bifurcations are of two types: resonance bifurcations and transverse bifurcations. Both types of bifurcation will result in the change of stability of the hetero-clinic cycle. At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the eigenvalues of the equilibrium in the cycle is satisfied. This is usually accompanied by the birth or death of a periodic orbit. A transverse bifurcation of a hetero-clinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibrium in the cycle passes through zero. This will also cause a change in stability of the hetero-clinic cycle.
• Infinite-period bifurcation in which a stable node and saddle point simultaneously occur 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 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 points.
• Blue sky catastrophe in which a limit cycle collides with a non hyperbolic cycle.
• Global bifurcations can also involve more complicated sets such as chaotic at tractors

 Co- dimension of a bifurcation:

The co-dimension of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This corresponds to the co-dimension of the parameter set for which the bifurcation occurs within the full space of parameters. Saddle-node bifurcations and Ho-pf bifurcations are the only generic local bifurcations which are really co-dimension-one (the others all having higher co-dimension). However, trans critical and pitchfork bifurcations are also often thought of as co-dimension-one, because the normal forms can be written with only one parameter.

An example of a well-studied co-dimension-two bifurcation is the Zhdanov–Ta-kens bifurcation.

 Applications in semi classical 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 Gutz willer points out in his classic work on quantum chaos.Many kinds of bifurcations have been studied with regard to links between classical and quantum dynamics including saddle node bifurcations, Ho-pf bifurcations, umbilici bifurcations, period doubling bifurcations, re-connection bifurcations, tangent bifurcations, and cusp bifurcations.