**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 with motif.

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).

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 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 non-zero but purely imaginary, this is a Hopf bifurcation.

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 (fold) bifurcation
- Transcritical bifurcation
- Pitchfork bifurcation
- Period-doubling (flip) bifurcation
- Hopf bifurcation
- Neimark–Sacker (secondary Hopf) bifurcation

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.^{ [3] }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 dynamics.**Heteroclinic bifurcation**in which a limit cycle collides with two or more saddle points; they involve a heteroclinic cycle.^{ [4] }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 of the equilibria in the cycle is satisfied. This is usually accompanied by the birth or death of a periodic orbit. 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.^{ [5] }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 nonhyperbolic cycle.

Global bifurcations can also involve more complicated sets such as chaotic attractors (e.g. crises).

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 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.

Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems,^{ [6] }^{ [7] }^{ [8] } molecular systems,^{ [9] } and resonant tunneling diodes.^{ [10] } Bifurcation theory has also been applied to the study of laser dynamics ^{ [11] } and a number of theoretical examples which are difficult to access experimentally such as the kicked top^{ [12] } and coupled quantum wells.^{ [13] } 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^{ [14] } work on quantum chaos.^{ [15] } 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.

- ↑ Blanchard, P.; Devaney, R. L.; Hall, G. R. (2006).
*Differential Equations*. London: Thompson. pp. 96–111. ISBN 978-0-495-01265-8. - ↑ Henri Poincaré. "
*L'Équilibre d'une masse fluide animée d'un mouvement de rotation*".*Acta Mathematica*, vol.7, pp. 259-380, Sept 1885. - ↑ Strogatz, Steven H. (1994).
*Nonlinear Dynamics and Chaos*. Addison-Wesley. p. 262. ISBN 0-201-54344-3. - ↑ Luo, Dingjun (1997).
*Bifurcation Theory and Methods of Dynamical Systems*. World Scientific. p. 26. ISBN 981-02-2094-4. - ↑ James P. Keener, "Infinite Period Bifurcation and Global Bifurcation Branches",
*SIAM Journal on Applied Mathematics*, Vol. 41, No. 1 (August, 1981), pp. 127–144. - ↑ 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/S0009-2614(97)00931-7. - ↑ 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. S2CID 120968155. - ↑ 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 non-autonomous systems. The case of quantum kicked top".
*Physics Letters A*.**368**(3–4): 206–214. arXiv: quant-ph/0702172 . Bibcode:2007PhLA..368..206S. doi:10.1016/j.physleta.2007.04.003. S2CID 15562617. - ↑ 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/S0034-4877(99)80148-7. - ↑ 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. S2CID 116944147. - ↑ Gutzwiller, Martin C. (1990).
*Chaos in Classical and Quantum Mechanics*. New York: Springer-Verlag. ISBN 978-0-387-97173-5.

In mathematics the **Lyapunov exponent** or **Lyapunov characteristic exponent** of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by

In theoretical physics, the term **renormalization group** (**RG**) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying force laws as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle.

**Quantum chaos** is a branch of physics which studies how 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 between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?

The **Lotka–Volterra equations**, also known as the **predator–prey equations**, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

The **competitive Lotka–Volterra equations** are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to include trophic interactions.

The **Rössler attractor** is the attractor for the **Rössler system**, a system of three non-linear ordinary differential equations originally studied by Otto Rössler. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor.

In probability theory and mathematical physics, a **random matrix** is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

**Møller–Plesset perturbation theory** (**MP**) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset.

In linear algebra, an **eigenvector** or **characteristic vector** of a linear transformation is a nonzero vector that changes by a scalar factor when that linear transformation is applied to it. The corresponding **eigenvalue**, often denoted by , is the factor by which the eigenvector is scaled.

In the mathematical area of bifurcation theory a **saddle-node bifurcation**, **tangential bifurcation** or **fold 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 systems. In discrete dynamical systems, the same bifurcation is often instead called a **fold bifurcation**. Another name is **blue sky bifurcation** in reference to the sudden creation of two fixed points.

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 stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.

In the mathematics of evolving systems, the concept of a **center manifold** was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling.

In mathematics, a **homoclinic orbit** is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.

In mathematics, in the phase portrait of a dynamical system, a **heteroclinic orbit** is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit.

The **Born rule** is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

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 Rifkat Bogdanov and Floris Takens, who independently and simultaneously described this bifurcation.

**Numerical continuation** is a method of computing approximate solutions of a system of parameterized nonlinear equations,

The **Andronov–Pontryagin criterion** is a necessary and sufficient condition for the stability of dynamical systems in the plane. It was derived by Aleksandr Andronov and Lev Pontryagin in 1937.

The **Kibble–Zurek mechanism** (**KZM**) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.

The light front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.

- Afrajmovich, V. S.; Arnold, V. I.; et al. (1994).
*Bifurcation Theory and Catastrophe Theory*. ISBN 978-3-540-65379-0. - 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
- Wiggins, Stephen (1988).
*Global bifurcations and Chaos: Analytical Methods*. New York: Springer. ISBN 978-0-387-96775-2.

- Nonlinear dynamics
- Bifurcations and Two Dimensional Flows by Elmer G. Wiens
- Introduction to Bifurcation theory by John David Crawford

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