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Bifurcation theory is a branch of mathematics that deals with the qualitative changes that occur in the solutions of a system of equations as a function of the parameters of the system. Bifurcation diagrams are used to visualize how the system changes as the parameter is varied.

The theory is named after the fact that, in many cases, the solutions of a system of equations "bifurcate" or "fork" as the parameters of the system are varied.

Bifurcation theory is concerned with the nature of the solutions of a system of equations and how these solutions change as a function of the parameters of the system.

The subject has its origins in the work of Henri Poincaré on celestial mechanics and the three-body problem.
Henri Poincare's work in bifurcation theory includes the development of the idea of a bifurcation diagram, which is a graphical way of representing the behaviour of a system as a function of a parameter. He also developed the concept of a saddle-node bifurcation, which is a type of bifurcation that can occur in systems with more than one dimension.

Henri Poincaré's work on celestial mechanics and the three-body problem was primarily focused on the stability of the Solar System. He developed the idea that the Solar System is stable because of the presence of resonances, which are regions where the orbital periods of the planets are in simple ratios with each other. He also showed that the three-body problem is not always solvable, and that in some cases the motion of the three bodies can be chaotic.

Bifurcation theory has applications in many fields of science, including physics, engineering, biology, and economics.In physics, bifurcation theory is used to study the stability of equilibrium points in dynamical systems.

In engineering, it is used to study the behavior of systems near critical points.

In biology, bifurcation theory is used to study the effect of mutations on the structure of populations.

In economics, it is used to study the behavior of markets near critical points.

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