On a generalized Collatz-Wielandt formula and finding saddle-node bifurcations

03/27/2020
by   Yavdat Il'yasov, et al.
0

We introduce the nonlinear generalized Collatz-Wielandt formula λ^*= sup_x∈ Qmin_i:h_i(x) ≠ 0g_i(x)/ h_i(x),   Q ⊂R^n, and prove that its solution (x^*,λ^*) yields the maximal saddle-node bifurcation for systems of equations of the form: g(x)-λ h(x)=0,   x ∈ Q. Using this we introduce a simply verifiable criterion for the detection of saddle-node bifurcations of a given system of equations. We apply this criterion to prove the existence of the maximal saddle-node bifurcations for finite-difference approximations of nonlinear partial differential equations and for the system of power flow equations.

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1. Introduction

This paper is concerned with finding bifurcations for the system of equations of the form:

()

Here is an open domain in , is a real parameter,

are continuously differentiable vector functions and we suppose that

, in . By a solution of system (), we mean a pair which satisfies ().

A solution of () is called saddle-node bifurcation (or fold bifurcation, turning point) of () in if there exists a -map for some such that

  • satisfies () for and ,

  • and or , .

Bifurcation phenomena arise in many fields of science and technology, including quasi-classical and quantum physics, general relativity, analysis of voltage stability of power systems, neural networks, biology, ecology, and many others (see, e.g.,

[1, 16, 18, 22]).

This paper aims to address the following problems:

  • Finding general sufficient conditions under which a given system has a saddle-node bifurcation.

  • Finding an explicit variational formula for determining saddle-node bifurcations.

Note that a spectral point of a square matrix with can be considered as a saddle-node bifurcation. Indeed, the map , satisfies all the above conditions of the saddle-node bifurcation. We emphasize that the spectral theory of linear operators offers a number of well-developed methods for solving the above problems. Moreover, the spectral theory provides constructive methods of finding spectral points, including the method of the characteristic polynomial, the Courant–Fischer–Weyl and Collatz–Wielandt variational principles.

For nonlinear problems, however, the situation is more complicated. The existing, well-known approaches, such as the Crandall–Rabinowitz [7], Krasnoselskii [19] and Rabinowitz [21] theorems or the Vainberg–Trenogin branching method [24], etc., allow to prove the existence of bifurcations for nonlinear problems but omit the main question of how to find a point satisfying the necessary saddle-node bifurcation properties (see [16, 17, 20]).

The purpose of this article is to present a method which allows to obtain a complete answer to the question of the existence of saddle-node bifurcations for given systems of equations. Moreover, we are aimed to get a formula for determining bifurcations that would be useful in the numerical calculation of saddle-node bifurcations as well. In addition to this, we suppose to provide a contribution to the development of the Perron–Frobenius theory to the nonlinear problems.

Our approach is based on the extended functional method introduced in [11], however, the present paper is kept self-contained and full proofs are provided.

Let us state our main results. We shall look for a saddle-node bifurcation of () by means of the following nonlinear generalized Collatz-Wieland formula [11]

(1.1)

Here , , is called the functional of bifurcations. Denote by the Jacobian matrix of .

We assume the following condition:

(R):

is irreducible off-diagonal sign-constant matrix for all and .

Hereafter, a real square matrix is called off-diagonal sign-constant if one of the following is true: or, for all , . The matrix is irreducible if it cannot be conjugated into block upper triangular form by a permutation matrix (see [25]).

Our main result is as follows

Theorem 1.1.

Assume that (R) holds true and , in . Suppose .

:

Then () has no solutions in for any .

:

If there exists a maximizer of such that , , then is a maximal saddle-node bifurcation point of () in . Moreover,

and both right and left eigenvectors of

are strongly positive.

We call and the right and left eigenvectors of . A saddle-node bifurcation of () is said to be maximal in if for any other saddle-node bifurcation of () in .

It is important to note that variational formula (1.1) allows finding the saddle-node bifurcations numerically. Such investigations with related numerical experiments were initiated in [14, 15]. Notice that problem (1.1) belongs to a class of nonsmooth optimization problems. The theory of nonsmooth optimization has been intensively developed over the past few decades, and, at present, there are various powerful numerical methods for solving such problems (see, e.g., [2, 5, 6] and references therein). Thus, along with the well-known methods for numerically finding bifurcations ([16, 20, 22]), we can use the full range of nonsmooth optimization methods.

Remark 1.2.

To the best of our knowledge, the results on the existence of saddle-node bifurcations for finite-difference approximations of nonlinear boundary value problems proved in Section 4 have not been known before.

Remark 1.3.

As noted above, the nonlinear generalized Collatz-Wieland formula (1.1) is a finite-dimensional version of an infinite-dimensional minimax principle [11]. This minimax principle has been used to solve various theoretical problems from the nonlinear partial differential equations [3, 10] including problems which are not directly related to the finding of bifurcations (see e.g. [12, 13]).

The paper is organized as follows. In Section 2, we give some preliminaries. Theorems 1.1 is proved in Sections 3. In Section 4, we present some examples of applications of Theorem 1.1.

2. Preliminaries

Hereafter, and stand for the Euclidean norm and the scalar product in , respectively; denotes the gradient. Furthermore, we write in if , s.t. . Denote . In what follows, we say that a solution of () is maximal in if () has no solution such that and .

The proof of the next lemma can be obtained from many sources (see, e.g., [7, 16, 17, 24]).

Lemma 2.1.

Assume that is an open domain in . Suppose that is a maximal solution of () in such that

Then is a maximal saddle-node bifurcation point of () in .

Let us introduce

where for s.t. , . Denote by the number of elements in .

Remark 2.2.

For , the condition (i.e., ) is satified if and only if is a solution of () and , .

Introduce the following convex hull

Definition 2.3.

A point is called stationary point of if .

For we have

(2.1)

and thus, for stationary point of we have

where for , for and .

A point is called local maximizer of in if there exists a neighbourhood of such that for any .

Lemma 2.4.

Let be a local maximizer of the bifurcation functional in and . Then there exist real numbers , such that , for and , for , and

Proof.

Let be a local maximizer of . Then

for some neighbourhood . Then is a maximizer of the following constraint maximization problem

(2.2)

Hence, the Lagrange multiplier rule (see Theorem 48.B in [23]) implies that there exist such that and

(2.3)
(2.4)
(2.5)
(2.6)

From (2.3) and (2.5) it follows that . Thus, we may assume that and consequently, . Set , . Then , and by (2.6), for any . Finally, taking into account (2.3), (2.4) we conclude the proof of Lemma 2.4. ∎

3. Proof of Theorems 1.1

Suppose . Let us show that () has no solutions in for any . Conversely, suppose that there exists a solution of () in with . Then (1.1) implies . Hence, there exists such that or, equivalently which is a contradiction.

Let be a maximizer in (1.1) such that . Then Lemma 2.4 implies that there exist real numbers , such that for , and

where we set for and for . Due to assumption (R), this is possible only if and for all . Thus and the equality implies that is a solution of (). Now taking into account that () has no solutions in for we conclude that is a maximal solution of () in .

It is known that any irreducible off-diagonal sign-constant matrix has a unique strictly positive eigenvector, with a real simple eigenvalue [25]. Hence, by (R), and the right and left eigenvectors , of are strongly positive. From this and since , , we obtain

whence since , it follows that

Thus, all the assumptions of Lemma 2.1 are satisfied. Consequently, is a maximal saddle-node bifurcation point of () in .

4. Examples of application

4.1. Example 1.

Consider the so-called system of power flow equations with two buses [1]

(4.1)

Here and are unknown variables, are given and is the so-called load parameter [1]. The saddle-node bifurcation of system (4.1) corresponds to the so-called maximum load capacity of power system, and its finding plays a crucial role in the control of the voltage stability of power systems [1].

Let us show that (4.1) has a saddle-node bifurcation. Assume that . Denote and introduce

Consider

(4.2)

Since , we infer that . Using this it is not hard to show that there exists a maximizer of (4.2). Moreover, is an internal point in , i.e., and . Observe, the Jacobian matrix

satisfies condition (R). Hence applying Theorem 1.1 we deduce

Lemma 4.1.

Assume . Then and there exists a maximizer of (4.2). Furthermore, is a maximal saddle-node bifurcation point of (4.1) in . Moreover, and both right and left eigenvectors of are strongly positive.

Remark 4.2.

For this simple case a similar result can be obtained directly by solving system (4.1) (see, e.g., [1], Sec.1.3.1). However, we are not sure that this approach is simpler than what is proposed above (cf. [1]). Moreover, we conjecture that our approach can be generalized to systems of power flow equations of large dimensions.

4.2. Example 2.

Consider the boundary value problem with the so-called nonlinearity of convex-concave type

(4.3)

where is a bounded smooth domain in , . Assume that , and

(H):

,   .

An example of the function is as follows , where . Thus, in this case, the nonlinearity is true convex-concave.

We use finite differences to approximate this problem and for the sake of simplicity, we restrict ourselves to the case .

Assume that , . Set , , , where . For the second derivatives at mesh points we used a standard second-order finite difference approximation. This yields for (4.3) the following approximating system of nonlinear equations

(4.4)

where , .

Then we have

Introduce

The nonlinear generalize Collatz-Wieland formula for (4.4) is as follows:

(4.5)
Lemma 4.3.

Assume that (H) holds true. Then and there exists a maximizer of (4.5). Furthermore, is a maximal saddle-node bifurcation point of (4.4) in . Moreover, and both right and left eigenvectors of are strongly positive.

Proof.

To prove the lemma it is sufficient to verify for (4.4) that all the assumptions of Theorem 1.1 are satisfied.

Let us show that . Take such that , , . Then and

for sufficiently small , since assumption (H) implies that as .

For , , , we have

(4.6)

Let be a maximizer sequence of (4.5), i.e., as . Suppose that . Then there exists a subsequence, again denoted by , such that as . Indeed, since , there exists and a subsequence such that as . Hence, (H) implies that as and consequently, by (4.6) we have as . However, by assumption we have . Thus is bounded and therefore there exists a limit point of . Passing to a subsequence if it’s necessary, we may assume that . We claim that . Indeed, assume . First, let us suppose that there exists such that , and . Then from (4.6) it follows that as which contradictions to . In the case for some , we have

and similarly, if for . Thus, we get a contradiction and . In the same manner we can see that .

Observe that the Jacobian matrix of has the following tridiagonal form:

Hence, we see that is a irreducible off-diagonal sign-constant matrix for any and . Thus condition (R) is satisfied for (4.4). Hence we see that all the assumptions of Theorem 1.1 are satisfied and this completes the proof of the lemma. ∎

4.3. Example 3.

Consider the Liouville-Bratu-Gelfand problem [4, 8, 9]

(4.7)

For the sake of simplicity, we restrict ourselves to the case so that we take , . Set , , , where . Using a standard second-order finite difference approximation we derive for (4.7) the following approximating system of nonlinear equations

(4.8)

where and .

The nonlinear generalize Collatz-Wieland formula now reads as follows:

(4.9)

where

Lemma 4.4.

and there exists a maximizer of (4.9). Furthermore, is a maximal saddle-node bifurcation point of (4.8) in . Moreover, there holds and both right and left eigenvectors of are strongly positive.

Proof.

Take so that , . Then and .

Let be a maximizer sequence of (4.9), i.e., as . Suppose that . Then we see at once that as which contradicts to the strong inequality . Thus, is bounded and there exists a limit point of and by continuity, . Observe that if for some , then , which implies a contradiction. Thus, for every .

Similar to the proof of Lemma 4.3, one can check that the Jacobian matrix has the tridiagonal form and satisfies condition (R). Hence and since , the proof of the lemma follows from Theorem 1.1.

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