View source for Branching of solutions

''bifurcation of solutions, of non-linear equations''

A phenomenon that causes a given solution of a non-linear equation to disappear completely or to become a number of solutions if minor changes in the parameters of the equation are introduced. More precisely, let the non-linear equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175102.png" /> (which need not be numerical) have the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175103.png" /> for a given value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175104.png" /> of the parameter. Then, if the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175105.png" /> are close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175106.png" />, equation (*) may have more than one solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175107.png" /> close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175108.png" />. One then speaks of branching (bifurcation) of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b0175109.png" />, and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751010.png" /> is said to be a branching (bifurcation) point of equation (*).

Example: The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751013.png" /> are complex variables, has the branching point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751014.png" /> because there exists a two-valued solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751015.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751016.png" /> is small, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751017.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751018.png" />) branches out into two small non-trivial solutions.

The modern theory of branching of solutions is based on the ideas of A.M. Lyapunov [[#References|[1]]] and E. Schmidt [[#References|[2]]] and has mostly been developed for non-linear equations in Banach spaces.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751020.png" /> be complex Banach spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751021.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751022.png" /> be a complex variable, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751023.png" /> be a non-linear operator which, together with its [[Fréchet derivative|Fréchet derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751024.png" />, is continuous in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751025.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751026.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751027.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751028.png" /> into a neighbourhood of zero of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751030.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751031.png" /> is a [[Fredholm-operator(2)|Fredholm operator]].

It is required to find, in the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751032.png" />, where the radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751033.png" /> is sufficiently small, all the solutions of equation (*) that are continuous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751035.png" /> is also sufficiently small. In other words, this is a problem of local extension of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751036.png" /> in the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751037.png" />. If the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751038.png" /> exists, the problem has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751040.png" />. If, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751041.png" /> does not exist, then the null space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751043.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751044.png" />. In such a case the problem may be reduced to an analogous finite-dimensional problem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751045.png" /> denote the projection (cf. [[Projector|Projector]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751047.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751048.png" /> denote the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751049.png" /> onto the domain of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751051.png" /> is the identity operator. Equation (*) may be written as the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751052.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751053.png" />. The implicit operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751054.png" /> is found from the first equation of the system. Its substitution into the second equation of the system yields the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751055.png" /></td> </tr></table>

from which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751056.png" /> may be found; it is said to be the branching (bifurcation) equation. The complete solution of the problem — to find in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751057.png" /> of sufficiently small radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751058.png" /> all solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751059.png" /> of the branching equation that are continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751061.png" /> is sufficiently small — yields the complete solution of the initial problem, since all its solutions can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751062.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751063.png" /> is a certain solution of the branching equation.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751064.png" /> be an [[Analytic operator|analytic operator]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751065.png" />. The choice of bases in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751066.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751068.png" /> makes it possible to write the branching equation in the form of the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751069.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751070.png" />, is an analytic function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751071.png" />, while all partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751072.png" /> vanish at this point. This system can be studied using the theory of exclusion, the method of the [[Newton diagram|Newton diagram]] and other methods [[#References|[3]]], [[#References|[4]]], [[#References|[5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751073.png" />, the complete analysis is effected by the method of the Newton diagram. As regards the study of the branching equation, i.e. of the original problem as well, the following three cases are the only ones possible: a) the problem has no solution; b) the problem has a finite number of solutions, all of which can be represented by converging series of integer or rational powers of the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751074.png" />; or c) the problem has a finite number of families of solutions, each one of which depends on a finite number of small free parameters and, possibly, on the finite number of solutions given in b).

For case b) to occur, it is sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751075.png" /> to be an isolated solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751076.png" />. In case b) it is expedient to look for solutions by the method of indefinite coefficients in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751077.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751078.png" /> are the coefficients to be found, while the possible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751079.png" /> may be preliminarily found using the branching equation. The substitution of such a series in (*) yields a recurrent system for finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751080.png" />. Problems of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751081.png" /> are obtained, and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751082.png" /> is determined up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751083.png" /> arbitrary constants, which follow from the stipulated solvability of the successive equations. All resulting series converge in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751084.png" />. An estimate from below of the radius of the neighbourhood may be obtained by the construction of majorants [[#References|[6]]].

For case c) to occur, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751085.png" /> must be a non-isolated solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751086.png" />. Here the application of the method of indefinite coefficients may result in divergent series (formal solutions). If the problem is invariant with respect to a continuous group of linear operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751087.png" />, the use of group considerations will make it possible in several cases to reduce the number of the equations and of the unknowns involved, and thus simplify the problem or even reduce it to case b) [[#References|[7]]], [[#References|[8]]].

Equation (*) may also have solutions that are defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751088.png" /> only. Such solutions are possible only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751089.png" /> is a non-isolated solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751090.png" />; they are found using the branching equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751091.png" />. The determination of all its multi-parameter families of solutions yields all the solutions of equation (*) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751092.png" />.

If the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751094.png" /> are real, the branching equation is studied in the complex domain, after which real solutions are taken. Some of them may prove to be defined in half-neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751095.png" />.

This procedure is also partly applicable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751096.png" /> is a sufficiently smooth operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751097.png" /> is a [[Noetherian operator|Noetherian operator]] and the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751098.png" /> is an element of another Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b01751099.png" /> (branching points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b017510100.png" /> may be replaced by lines and surfaces). It is also used in the study of certain related problems: the problem of finding large solutions (equation (*) can have solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b017510101.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b017510102.png" />), the problem of branching of eigen values and eigen elements of linear operators, etc. [[#References|[3]]]. The special case when

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017510/b017510103.png" /></td> </tr></table>

was also studied by topological and variational methods and methods involving the use of cones in a Banach space. The concept of a [[Bifurcation|bifurcation]] point is very important in this circle of problems. Problems regarding branching of solutions which do not belong to this scheme are also encountered. They include, for example, degenerate differential equations [[#References|[9]]], [[#References|[10]]] and problems involving long and solitary waves [[#References|[11]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   "On equilibrium figures, deviating slightly from ellipses, rotating in a homogeneous mass of fluid" , ''Collected Works'' , '''4''' , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Schmidt,   "Zur Theorie der linearen und nichtlinearen Integralgleichungen III"  ''Math. Ann.'' , '''65'''  (1908)  pp. 370–399</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,   V.A. Trenogin,   "Theory of branching of solutions of non-linear equations" , Noordhoff  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Vainberg,   V.A. Trenogin,   "The methods of Lyapunov and Schmidt in the theory of non-linear equations and their further development"  ''Russian Math. Surveys'' , '''17''' :  2  (1962)  pp. 1–60  ''Uspekhi Mat. Nauk'' , '''17''' :  2  (1962)  pp. 13–75</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K.T. Akhmedov,   "The analytic method of Nekrasov–Nazarov in non-linear analysis"  ''Uspekhi Mat. Nauk'' , '''12''' :  4  (1957)  pp. 135–153  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.I. Yudovich,   "Free convection and bifurcation"  ''J. Appl. Math. Mech.'' , '''31'''  (1967)  pp. 103–114  ''Priklad. Mat. Mekh.'' , '''31'''  (1967)  pp. 101–111</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.V. Loginov,   V.A. Trenogin,   "On the application of continuous groups in the theory of branching"  ''Soviet Math. Dokl.'' , '''12''' :  2  (1971)  pp. 404–408  ''Dokl. Akad. Nauk SSSR'' , '''197''' :  1  (1971)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  K.T. Akhmedov,   "The Cauchy problem for a class of non-linear equations in function spaces"  ''Dokl. Akad. Nauk SSSR'' , '''115''' :  1  (1957)  pp. 9–12  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  N.A. Sidorov,   "Branching of solutions of the Cauchy problem for a certain class of nonlinear integro-differential equations"  ''Differential equations N.Y.'' , '''3''' :  9  (1967)  pp. 830–834  ''Differentsial'nye Uravneniya'' , '''3''' :  9  (1967)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A.M. Ter-Krikorov,   V.A. Trenogin,   "Long wave solutions for quasi-linear elliptic equations in an unbounded strip"  ''Differential Equations N.Y.'' , '''3''' :  3  (1967)  pp. 496–508  ''Differentsial'nye Uravneniya'' , '''3''' :  3  (1967)</TD></TR></table>



====Comments====
For a classification of types of bifurcation (branching) points see [[Singularities of differentiable mappings|Singularities of differentiable mappings]]. The technique of reducing solving equation (*) to a finite-dimensional problem, the bifurcation equation, is often called the Lyapunov–Schmidt method.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.-N. Chow,   J.K. Hale,   "Methods of bifurcation theory" , Springer  (1982)</TD></TR></table>