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.^{[1]} Bifurcations occur in both continuous systems (described by ODEs, DDEs or PDEs) 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.^{[2]} Henri Poincaré also later named various types of stationary points and classified them.
YouTube Encyclopedic

1/5Views:8 83033 26756 86041416 571

✪ Mathematical Biology. 21: Hopf Bifurcations

✪ Bifurcations of a differential equation

✪ ODE  Bifurcation diagrams

✪ Concepts of Bifurcation: Introduction

✪ Saddlenode bifurcation
Transcription
Contents
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 (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 nonhyperbolic 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 Jacobian matrix 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 nonzero but purely imaginary, this is a Hopf bifurcation.
For discrete dynamical systems, consider the system
Then a local bifurcation at 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 saddlenode (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to −1, it is a perioddoubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation.
Examples of local bifurcations include:
 Saddlenode (fold) bifurcation
 Transcritical bifurcation
 Pitchfork bifurcation
 Perioddoubling (flip) bifurcation
 Hopf bifurcation
 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 collides with a saddle point.
 Heteroclinic bifurcation in which a limit cycle collides with two or more saddle points.
 Infiniteperiod bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle.
 Blue sky catastrophe 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).
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. Saddlenode bifurcations and Hopf bifurcations are the only generic local bifurcations which are really codimensionone (the others all having higher codimension). However, transcritical and pitchfork bifurcations are also often thought of as codimensionone, because the normal forms can be written with only one parameter.
An example of a wellstudied codimensiontwo bifurcation is the Bogdanov–Takens bifurcation.
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,^{[3]}^{[4]}^{[5]} molecular systems,^{[6]} and resonant tunneling diodes.^{[7]} Bifurcation theory has also been applied to the study of laser dynamics^{[8]} and a number of theoretical examples which are difficult to access experimentally such as the kicked top^{[9]} and coupled quantum wells.^{[10]} 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 points out in his classic^{[11]} work on quantum chaos.^{[12]} 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
Notes
 ^ Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006). Differential Equations. London: Thompson. pp. 96–111. ISBN 9780495012658.
 ^ Henri Poincaré. "L'Équilibre d'une masse fluide animée d'un mouvement de rotation". Acta Mathematica, vol.7, pp. 259380, Sept 1885.
 ^ Gao, J.; Delos, J. B. (1997). "Quantum manifestations of bifurcations of closed orbits in the photoabsorption spectra of atoms in electric fields". Phys. Rev. A. 56 (1): 356–364. Bibcode:1997PhRvA..56..356G. doi:10.1103/PhysRevA.56.356.
 ^ Peters, A. D.; Jaffé, C.; Delos, J. B. (1994). "Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model". Phys. Rev. Lett. 73 (21): 2825–2828. Bibcode:1994PhRvL..73.2825P. doi:10.1103/PhysRevLett.73.2825. PMID 10057205.
 ^ Courtney, Michael; Jiao, Hong; Spellmeyer, Neal; Kleppner, Daniel; Gao, J.; Delos, J. B.; et al. (1995). "Closed Orbit Bifurcations in Continuum Stark Spectra". Phys. Rev. Lett. 74 (9): 1538–1541. Bibcode:1995PhRvL..74.1538C. doi:10.1103/PhysRevLett.74.1538. PMID 10059054.
 ^ Founargiotakis, M.; Farantos, S. C.; Skokos, Ch.; Contopoulos, G. (1997). "Bifurcation diagrams of periodic orbits for unbound molecular systems: FH2". Chemical Physics Letters. 277 (5–6): 456–464. Bibcode:1997CPL...277..456F. doi:10.1016/S00092614(97)009317.
 ^ Monteiro, T. S. & Saraga, D. S. (2001). "Quantum Wells in Tilted Fields:Semiclassical Amplitudes and Phase Coherence Times". Foundations of Physics. 31 (2): 355–370. doi:10.1023/A:1017546721313.
 ^ Wieczorek, S.; Krauskopf, B.; Simpson, T. B. & Lenstra, D. (2005). "The dynamical complexity of optically injected semiconductor lasers". Physics Reports. 416 (1–2): 1–128. Bibcode:2005PhR...416....1W. doi:10.1016/j.physrep.2005.06.003.
 ^ Stamatiou, G. & Ghikas, D. P. K. (2007). "Quantum entanglement dependence on bifurcations and scars in nonautonomous systems. The case of quantum kicked top". Physics Letters A. 368 (3–4): 206–214. arXiv:quantph/0702172. Bibcode:2007PhLA..368..206S. doi:10.1016/j.physleta.2007.04.003.
 ^ Galan, J.; Freire, E. (1999). "Chaos in a Mean Field Model of Coupled Quantum Wells; Bifurcations of Periodic Orbits in a Symmetric Hamiltonian System". Reports on Mathematical Physics. 44 (1–2): 87–94. Bibcode:1999RpMP...44...87G. doi:10.1016/S00344877(99)801487.
 ^ Kleppner, D.; Delos, J. B. (2001). "Beyond quantum mechanics: Insights from the work of Martin Gutzwiller". Foundations of Physics. 31 (4): 593–612. doi:10.1023/A:1017512925106.
 ^ Gutzwiller, Martin C. (1990). Chaos in Classical and Quantum Mechanics. New York: SpringerVerlag. ISBN 9780387971735.
References
 Afrajmovich, V. S.; Arnold, V. I.; et al. (1994). Bifurcation Theory and Catastrophe Theory. ISBN 9783540653790.
 Wiggins, Stephen (1988). Global bifurcations and Chaos: Analytical Methods. New York: Springer. ISBN 9780387967752.
External links
 Nonlinear dynamics
 Bifurcations and Two Dimensional Flows by Elmer G. Wiens
 Introduction to Bifurcation theory by John David Crawford