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

$$\tag{* } F (x, \lambda ) = 0$$

with parameter $\lambda$( which need not be numerical) have the solution $x _ {0}$ for a given value $\lambda _ {0}$ of the parameter. Then, if the values of $\lambda$ are close to $\lambda _ {0}$, equation (*) may have more than one solution $x ( \lambda )$ close to $x _ {0}$. One then speaks of branching (bifurcation) of the solution $x _ {0}$, and the pair $(x _ {0} , \lambda _ {0} )$ is said to be a branching (bifurcation) point of equation (*).

Example: The equation $x ^ {2} - \lambda = 0$, where $x$ and $\lambda$ are complex variables, has the branching point $(x _ {0} , \lambda _ {0} ) = (0, 0)$ because there exists a two-valued solution $x = \sqrt \lambda$, i.e. if $\lambda \neq 0$ is small, the solution $x = 0$( for $\lambda = 0$) branches out into two small non-trivial solutions.

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

Let $E _ {1}$ and $E _ {2}$ be complex Banach spaces, $x \in E _ {1}$, let $\lambda$ be a complex variable, and let $F(x, \lambda )$ be a non-linear operator which, together with its Fréchet derivative $F _ {x} (x, \lambda )$, is continuous in a neighbourhood $\Omega$ of the point $(x _ {0} , \lambda _ {0} )$. Let $F(x, \lambda )$ map $\Omega$ into a neighbourhood of zero of the space $E _ {2}$ such that $F(x _ {0} , \lambda _ {0} ) = 0$, and suppose that $F _ {x} (x _ {0} , \lambda _ {0} ) \equiv B$ is a Fredholm operator.

It is required to find, in the sphere $\| x - x _ {0} \| < r$, where the radius $r$ is sufficiently small, all the solutions of equation (*) that are continuous if $| \lambda - \lambda _ {0} | < \rho$, where $\rho$ is also sufficiently small. In other words, this is a problem of local extension of the solution $x _ {0}$ in the parameter $\lambda$. If the inverse operator $B ^ {-1}$ exists, the problem has a unique solution $x ( \lambda )$, and $x( \lambda _ {0} ) = x _ {0}$. If, on the other hand, $B ^ {-1}$ does not exist, then the null space $N(B)$ of $B$ has dimension $n \geq 1$. In such a case the problem may be reduced to an analogous finite-dimensional problem. Let $P$ denote the projection (cf. Projector) of $E _ {1}$ on $N(B)$ and let $I - Q$ denote the projection of $E _ {2}$ onto the domain of values of $B$, where $I$ is the identity operator. Equation (*) may be written as the system

$$\left . \begin{array}{c} (I - Q) F (x _ {0} + u + v, \lambda ) = 0, \\ QF (x _ {0} + u + v, \lambda ) = 0, \\ \end{array} \ \right \}$$

where $u=(I - P)(x - x _ {0} ), v=P(x - x _ {0} )$. The implicit operator $u = f(v, \lambda )$ is found from the first equation of the system. Its substitution into the second equation of the system yields the equation

$$QF (x _ {0} + f (v, \lambda ) + v , \lambda ) = 0,$$

from which $v$ may be found; it is said to be the branching (bifurcation) equation. The complete solution of the problem — to find in a sphere $\| v \| < r$ of sufficiently small radius $r$ all solutions $v( \lambda )$ of the branching equation that are continuous for $| \lambda - \lambda _ {0} | < \rho$, where $\rho$ is sufficiently small — yields the complete solution of the initial problem, since all its solutions can be represented in the form

$$x ( \lambda ) = \ x _ {0} + v ( \lambda ) + f [v ( \lambda ), \lambda ],$$

where $v$ is a certain solution of the branching equation.

Let $F(x, \lambda )$ be an analytic operator in $\Omega$. The choice of bases in the $n$- dimensional subspaces $PE _ {1} = N(B)$ and $QE _ {2}$ makes it possible to write the branching equation in the form of the system

$${\mathcal L} _ {i} ( \xi _ {1} \dots \xi _ {n} , \lambda ) = 0,\ \ i = 1 \dots n,$$

where ${\mathcal L} _ {i} , i = 1 \dots n$, is an analytic function at the point $(0 \dots 0, \lambda _ {0} )$, while all partial derivatives $\partial {\mathcal L} _ {i} / \partial \xi _ {j}$ vanish at this point. This system can be studied using the theory of exclusion, the method of the Newton diagram and other methods [3], [4], [5]. If $n = 1$, 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 $\lambda - \lambda _ {0}$; 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 $x _ {0}$ to be an isolated solution of the equation $F(x, \lambda _ {0} ) = 0$. In case b) it is expedient to look for solutions by the method of indefinite coefficients in the form

$$x ( \lambda ) = \ x _ {0} + \sum _ { k=1 } ^ \infty x _ {k} ( \lambda - \lambda _ {0} ) ^ {k/p} ,$$

where $x _ {k}$ are the coefficients to be found, while the possible values of $p$ may be preliminarily found using the branching equation. The substitution of such a series in (*) yields a recurrent system for finding $x _ {1} , x _ {2} , . . .$. Problems of the type $Bx _ {k} = H(x _ {1} \dots x _ {k-1} )$ are obtained, and each $x _ {k}$ is determined up to $n$ arbitrary constants, which follow from the stipulated solvability of the successive equations. All resulting series converge in some neighbourhood of $\lambda _ {0}$. An estimate from below of the radius of the neighbourhood may be obtained by the construction of majorants [6].

For case c) to occur, $x _ {0}$ must be a non-isolated solution of the equation $F(x, \lambda _ {0} ) = 0$. 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 $E _ {1}$, 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) [7], [8].

Equation (*) may also have solutions that are defined for $\lambda = \lambda _ {0}$ only. Such solutions are possible only if $x _ {0}$ is a non-isolated solution of the equation $F(x, \lambda _ {0} ) = 0$; they are found using the branching equation for $\lambda = \lambda _ {0}$. The determination of all its multi-parameter families of solutions yields all the solutions of equation (*) with $\lambda = \lambda _ {0}$.

If the spaces $E _ {1}$ and $E _ {2}$ 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 $\lambda _ {0}$.

This procedure is also partly applicable if $F(x, \lambda )$ is a sufficiently smooth operator, $B$ is a Noetherian operator and the parameter $\lambda$ is an element of another Banach space $E$( branching points in $E$ 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 $x ( \lambda ) \rightarrow \infty$ as $\lambda \rightarrow \lambda _ {0}$), the problem of branching of eigen values and eigen elements of linear operators, etc. [3]. The special case when

$$E _ {1} = E _ {2} , \ F (x, \lambda ) \equiv \ x - \Phi (x, \lambda ),\ \Phi (0, \lambda ) \equiv 0,$$

was also studied by topological and variational methods and methods involving the use of cones in a Banach space. The concept of a 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 [9], [10] and problems involving long and solitary waves [11].

References

 [1] A.M. Lyapunov, "On equilibrium figures, deviating slightly from ellipses, rotating in a homogeneous mass of fluid" , Collected Works , 4 , Moscow (1959) (In Russian) [2] E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen III" Math. Ann. , 65 (1908) pp. 370–399 MR1511472 Zbl 39.0399.03 [3] M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian) MR0344960 Zbl 0274.47033 [4] 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 [5] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian) MR385655 Zbl 0231.41024 [6] K.T. Akhmedov, "The analytic method of Nekrasov–Nazarov in non-linear analysis" Uspekhi Mat. Nauk , 12 : 4 (1957) pp. 135–153 (In Russian) [7] V.I. Yudovich, "Free convection and bifurcation" J. Appl. Math. Mech. , 31 (1967) pp. 103–114 Priklad. Mat. Mekh. , 31 (1967) pp. 101–111 Zbl 0173.28803 [8] 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) Zbl 0227.47049 [9] 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) [10] 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) MR221255 Zbl 0233.45029 [11] 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)