Bifurcation

A term in certain branches of mathematics, applied to situations in which some object $ \mathfrak D = \mathfrak D ( \lambda ) $ depends on a parameter $ \lambda $( which is not necessarily scalar) and which is such that in any neighbourhood of a certain value $ \lambda _ {0} $ of that parameter (a bifurcation value or a bifurcation point) the qualitative properties under consideration of the object $ \mathfrak D ( \lambda ) $ are not the same for all $ \lambda $. The corresponding rigorous definitions vary in different cases, but mainly follow (in a more or less modified form) the two variants below:

A) The qualitative properties of the object $ \mathfrak D $ being studied are the existence of other objects $ O $, associated with it in some manner. The bifurcation is characterized by the fact that when $ \lambda $ is varied, the objects $ O $ appear or disappear (in particular, they may coincide or else one object may generate several ones). See below (Paragraph 1)).

B) The first step is to determine under what circumstances two objects $ \mathfrak D ( \lambda ) $ are considered to be equivalent. (This definition must be such that all the qualitative properties of interest be identical for equivalent objects.) A change in the qualitative properties of $ \mathfrak D ( \lambda ) $ in a neighbourhood of a bifurcation point means, by definition, that $ \lambda $- values with non-equivalent $ \mathfrak D ( \lambda ) $ are found near this point. See below (Paragraph 2)).

1) In the theory of operators the initial object $ \mathfrak D ( \lambda ) $ is a non-linear operator $ \Phi (x, \lambda ) $ in a real Banach space, depending on a real parameter $ \lambda $, defined in a neighbourhood of the point $ x = 0 $ and such that $ \Phi (0, \lambda ) \equiv 0 $. For each fixed $ \lambda $ one associates new objects $ O $ to this $ \Phi $; they are the solutions $ x $ of the non-linear operator equation $ \Phi (x, \lambda ) = x $. A bifurcation point is a point at which a new, non-trivial, solution of this equation is generated. It is in fact a point $ \lambda _ {0} $ such that for any $ \epsilon > 0 $ there exists a $ \lambda $, $ | \lambda - \lambda _ {0} | < \epsilon $, for which the equation $ x = \Phi (x, \lambda ) $ has a solution $ x ( \lambda ) $ satisfying the conditions $ 0 < \| x ( \lambda ) \| < \epsilon $. If $ \Phi (x, \lambda ) \equiv \lambda Ax $, where $ A $ is a linear completely-continuous operator, the concept of a bifurcation point coincides with the concept of a characteristic value of $ A $.

If $ \Phi (x, \lambda ) $ is a non-linear completely-continuous operator which is continuously Fréchet differentiable and is such that $ \Phi _ {x} (0, \lambda ) \equiv \lambda A $, then only the characteristic values of $ A $ can be bifurcation points of $ \Phi $. It was found by a topological method [1], [2] that each odd-fold (in particular, simple) characteristic value of $ A $ is a bifurcation point of $ \Phi $. An analogous sufficient condition for the case of even-fold characteristic values is formulated using the concept of the rotation of a vector field.

If $ x = 0 $ is a non-isolated solution of the equation $ x = \Phi (x, \lambda ) $, then $ \lambda _ {0} $ is a bifurcation point of $ \Phi $. It was shown by a variational method [1], [2] that if $ \Phi (x) $ is a non-linear completely-continuous operator in a Hilbert space which is the gradient of a weakly continuous functional, while $ A = \Phi ^ \prime (0) $ is a completely-continuous self-adjoint operator, then all characteristic values of $ A $ are bifurcation points of $ \Phi $. The concept of a bifurcation point is also modified in the case of large solutions $ x ( \lambda ) \rightarrow \infty $ for $ \lambda \rightarrow \lambda _ {0} $. The importance of these concepts and results consists in the fact that, subject to relatively weak limitations, the branching of the solution $ x = 0 $ can be established; in particular, it is possible to prove that the solution of the non-linear problem is not unique. The methods of the theory of branching of solutions of non-linear equations often yield more accurate information [5].

2) The theory of smooth dynamical systems studies one-parameter (and sometimes also two-parameter [6]) families of flows (and cascades; only the former will be considered here), and the conditions under which the bifurcation is "typical" , i.e. preserves its character under a small change of the family in question [9]; both variants A) and B) above are useful. In the second variant two flows are considered equivalent if there exists a homeomorphism of the phase space which converts the trajectories of the one into trajectories of the other while preserving the direction of motion. There exists a completely satisfactory theory of bifurcations of one-parameter families of flows with a two-dimensional phase space [7], [9], and a local variant referring to a neighbourhood of an equilibrium position or of a periodic solution in the $ n $- dimensional case [6].

In variant A) the objects $ O $ under study, which are associated to the given dynamical system, are the positions of an equilibrium or periodic solution, and occasionally certain invariant manifolds (mainly tori) or hyperbolic invariant sets. The "growth" of these objects — taking place "locally" near an equilibrium position, or near a periodic solution, or "semi-locally" in a neighbourhood of a "closed contour" formed by several trajectories which tend to an equilibrium position or to a periodic solution — is considered. A case of bifurcation which is connected with a similar contour in a certain sense, but which takes place (with a change of the parameter $ \lambda $) prior to its arrival, is also possible . The growth of periodic solutions is often conveniently considered by rewriting the differential equation and the periodicity condition in the form of an integral equation to which suitable methods are applied [5].

3) Various bifurcations of different objects (both the initial objects and those associated to them) are encountered in the theory of singularities of mappings. As a result, this term (or rather a term derived from it) is used in several different ways [10], [6], [11], but it is more common to assign independent names to the corresponding concepts. They include, for instance, versal families (or deformations) [6], [11], [12], which describe, in a certain sense, all possible bifurcations which may take place under a small deformation of the object under consideration. These include, in particular, seven elementary catastrophes [12], which are represented by "typical" $ k $- parametric ( $ k \leq 4 $) families of functions which include a function with a degenerate critical point and which are defined in a neighbourhood of that point; they accordingly describe the corresponding bifurcation. In non-Soviet literature on the theory of singularities, the term "catastrophes" is often employed for "bifurcations" .

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A standard reference for bifurcation theory is [a1]. The presence of symmetry in a bifurcation problem can often be a powerful tool [a2]. For more precise details in the case of bifurcations of solutions to equations cf. Branching of solutions.

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