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

1. Introduction

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

(

f

)

f (x, λ) := g (x) - λ h (x) = 0, x \in Q .

Here $Q$ is an open domain in $R^{n}$ , $λ$ is a real parameter, $g, h : R^{n} \to R^{n}$

are continuously differentiable vector functions and we suppose that

h (x) \neq 0

h (x) \geq 0

Q

. By a solution of system (

f

), we mean a pair

(x, λ)

which satisfies (

f

A solution $(x^{*}, λ^{*})$ of ( $f$ ) is called saddle-node bifurcation (or fold bifurcation, turning point) of ( $f$ ) in $Q$ if there exists a $C^{1}$ -map $(- a, a) ∋ s ⟼ (x (s), λ (s)) \in Q \times R$ for some $a > 0$ such that

$(x (s), λ (s))$ satisfies ( $f$ ) for $s \in (- a, a)$ and $(x (0), λ (0)) = (x^{*}, λ^{*})$ ,
$\frac{d}{d s} λ (0) = 0$ and $λ (s) \in (- \infty, λ^{*}]$ or $λ (s) \in [λ^{*}, + \infty)$ , $\forall s \in (- a, a)$ .

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 $(ϕ^{*}, λ^{*})$ of a square matrix $A$ with $λ^{*} \in R$ can be considered as a saddle-node bifurcation. Indeed, the map $x (s) = s ϕ^{*}, λ (s) = λ^{*}$ , $s \in R$ 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 $(x^{*}, λ^{*})$ 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 ( $f$ ) by means of the following nonlinear generalized Collatz-Wieland formula [11]

(1.1)

λ^{*} = max x \in Q λ (x) \equiv max x \in Q min i : h_{i} (x) \neq 0 \frac{g_{i} (x)}{h_{i} (x)} .

Here $λ (x) := {min}_{i : h_{i} (x) \neq 0} \frac{g_{i} (x)}{h_{i} (x)}$ , $x \in Q$ , is called the functional of bifurcations. Denote by $J_{x} f (x, λ) = (\frac{\partial f}{\partial x_{i}})_{1 \leq i \leq n}$ the Jacobian matrix of $f (x, λ)$ .

We assume the following condition:

(R):: $J_{x} f (x, λ)$ is irreducible off-diagonal sign-constant matrix for all $x \in Q$ and $λ \in R$ .

Hereafter, a real square matrix $A$ is called off-diagonal sign-constant if one of the following is true: $a_{i j} \geq 0$ or, $a_{i j} \leq 0$ for all $i, j \in {1, \dots, n}$ , $i \neq j$ . The matrix $A$ is irreducible if it cannot be conjugated into block upper triangular form by a permutation matrix $P$ (see [25]).

Our main result is as follows

Theorem 1.1.

Assume that (R) holds true and $h (x) \neq 0$ , $h (x) \geq 0$ in $Q$ . Suppose $λ^{*} < + \infty$ .

$(1^{o})$ :: Then ( $f$ ) has no solutions in $Q$ for any $λ > λ^{*}$ .
$(2^{o})$ :: If there exists a maximizer $x^{*} \in Q$ of $λ (x)$ such that $h_{i} (x^{*}) \neq 0$ , $\forall i = 1, \dots, n$ , then $(x^{*}, λ^{*})$ is a maximal saddle-node bifurcation point of ( $f$ ) in $Q$ . Moreover, $d i m K e r (J_{x} f (x^{*}, λ^{*})) = 1$
and both right and left eigenvectors of
$J_{x} f (x^{*}, λ^{*})$ are strongly positive.

We call $ζ \in K e r J_{x} f (x, λ)$ and $ξ \in K e r (J_{x} f (x, λ))^{T}$ the right and left eigenvectors of $J_{x} f (x, λ)$ . A saddle-node bifurcation $(x^{*}, λ^{*})$ of ( $f$ ) is said to be maximal in $Q$ if $¯ λ \leq λ^{*}$ for any other saddle-node bifurcation $(¯ x, ¯ λ)$ of ( $f$ ) in $Q$ .

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, $∥ \cdot ∥$ and $⟨ \cdot, \cdot ⟩$ stand for the Euclidean norm and the scalar product in $R^{n}$ , respectively; $\nabla_{x} = (\frac{\partial}{\partial x_{1}} \dots \frac{\partial}{\partial x_{n}})^{T}$ denotes the gradient. Furthermore, we write $h (x) \neq 0$ in $Q$ if $\forall x \in Q$ , $\exists i \in {1, . . ., n}$ s.t. $h_{i} (x) \neq 0$ . Denote $\lx@overaccentset∘Q=⋂ni=1{x∈Q: hi(x)≠0}$ . In what follows, we say that a solution $(¯ x, ¯ λ)$ of ( $f$ ) is maximal in $Q$ if ( $f$ ) has no solution $(x, λ)$ such that $λ > ¯ λ$ and $x \in Q$ .

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

Lemma 2.1.

Assume that $Q$ is an open domain in $R^{n}$ . Suppose that $(x^{*}, λ^{*})$ is a maximal solution of ( $f$ ) in $Q$ such that

d i m K e r (J_{x} f (x^{*}, λ^{*})) = 1 and \frac{\partial}{} \partial λ f (x^{*}, λ^{*}) \notin R (J_{x} f (x^{*}, λ^{*}))

Then $(x^{*}, λ^{*})$ is a maximal saddle-node bifurcation point of ( $f$ ) in $Q$ .

Let us introduce

N (x) = {i \in {1, \dots, n} : r_{i} (x) = λ (x), h_{i} (x) \neq 0}, x \in Q,

where $r_{i} (x) := \frac{g_{i} (x)}{h_{i} (x)}$ for $x \in Q$ s.t. $h_{i} (x) \neq 0$ , $i = 1, \dots, n$ . Denote by $| N (x) |$ the number of elements in $N (x)$ .

Remark 2.2.

For $x \in Q$ , the condition $| N (x) | = n$ (i.e., $N (x) = {1, \dots, n}$ ) is satified if and only if $(x, λ (x))$ is a solution of ( $f$ ) and $h_{i} (x) \neq 0$ , $\forall i \in {1, \dots, n}$ .

Introduce the following convex hull

\partial λ (x) := {z = \sum i \in N (x) ζ_{i} \nabla_{x} r_{i} (x) : \sum i \in N (x) ζ_{i} = 1, ζ_{i} \geq 0, i \in N (x)} .

Definition 2.3.

A point $^x \in Q$ is called stationary point of $λ (x)$ if $0 \in \partial λ (^x)$ .

For $h_{i} (x) \neq 0$ we have

(2.1)

\nabla_{x} r_{i} (x) = \frac{1}{h_{i} (x)} (\nabla_{x} g_{i} (x) - r_{i} (x) \nabla_{x} h_{i} (x)) = \frac{1}{h_{i} (x)} \nabla_{x} f_{i} (x, λ) |_{λ = r_{i} (x)},

and thus, for stationary point $^x$ of $λ (x)$ we have

\sum i \in N (^x) ζ_{i} \nabla_{x} r_{i} (^x) |_{λ = λ (^x)} = (J_{x} f (^x, λ (^x)))^{T}^ξ = 0,

where ${^ξ}_{i} = ζ_{i} / h_{i} (^x)$ for $i \in N (^x)$ , ${^ξ}_{i} = 0$ for $i∈{1,…,n}∖N(^x)$ and $\sum_{i \in N (^x)} ζ_{i} = 1, ζ_{i} \geq 0, i \in N (^x)$ .

A point $^x \in Q$ is called local maximizer of $λ (x)$ in $Q$ if there exists a neighbourhood $U (^x) \subset Q$ of $^x$ such that $λ (^x) \leq λ (x)$ for any $x \in U (^x)$ .

Lemma 2.4.

Let $^x$ be a local maximizer of the bifurcation functional $λ (x)$ in $Q$ and $^x∈\lx@overaccentset∘Q$ . Then there exist real numbers $ζ_{i}$ , $i = 1, \dots, n$ such that $ζ_{i} \geq 0$ , for $i \in N (^x)$ and $ζ_{i} = 0$ , for $i∈{1,…,n}∖N(^x)$ , $\sum_{i \in N (^x)} ζ_{i} = 1$ and

\sum i \in N (^x) ζ_{i} \nabla_{x} r_{i} (^x) = 0.

Proof.

Let $^x∈\lx@overaccentset∘Q$ be a local maximizer of $λ (x)$ . Then

^λ = λ (^x) = max x \in U (^x) λ (x) = max x \in U (^x) min i r_{i} (x) .

for some neighbourhood $U(^x)⊂\lx@overaccentset∘Q$ . Then $(^λ,^x)$ is a maximizer of the following constraint maximization problem

(2.2)

⎧ ⎨ ⎩ \begin{matrix} maximize λ, subject to: r_{i} (x) \geq λ, i = 1, \dots, n, x \in U (^x), λ \in R . \end{matrix}

Hence, the Lagrange multiplier rule (see Theorem 48.B in [23]) implies that there exist $μ_{0}, μ_{1}, \dots, μ_{n} \in R$ such that $\sum_{i} | μ_{i} | \neq 0$ and

(2.3)		$μ_{0} \frac{d}{d λ} λ + n \sum i = 1 μ_{i} \frac{d}{d λ} (r_{i} (^x) - λ) = 0 \Leftrightarrow μ_{0} - n \sum i = 1 μ_{i} = 0,$
(2.4)		$n \sum i = 1 μ_{i} \nabla_{x} r_{i} (^x) = 0,$
(2.5)		$μ_{i} \leq 0, i = 0, \dots, n,$
(2.6)		$μ_{i} (r_{i} (^x) - λ) = 0, i = 1, \dots, n .$

From (2.3) and (2.5) it follows that $μ_{0} \neq 0$ . Thus, we may assume that $μ_{0} = - 1$ and consequently, $\sum_{i = 1}^{n} μ_{i} = - 1$ . Set $ζ_{i} = - μ_{i}$ , $i = 1, \dots, n$ . Then $ζ_{i} \geq 0$ , $i = 1, \dots, n$ and by (2.6), $ζ_{i} = 0$ for any $i∈{1,…,n}∖N(^x)$ . Finally, taking into account (2.3), (2.4) we conclude the proof of Lemma 2.4. ∎

3. Proof of Theorems 1.1

$(1^{o})$ Suppose $λ^{*} < + \infty$ . Let us show that ( $f$ ) has no solutions in $Q$ for any $λ > λ^{*}$ . Conversely, suppose that there exists a solution $(x_{λ}, λ)$ of ( $f$ ) in $Q$ with $λ > λ^{*}$ . Then (1.1) implies ${min}_{i : h_{i} (x) \neq 0} r_{i} (x_{λ}) \leq λ^{*} < λ$ . Hence, there exists $i_{0} \in {1, \dots, n}$ such that $r_{i_{0}} (x_{λ}) < λ$ or, equivalently $g_{i_{0}} (x_{λ}) - λ h_{i_{0}} (x_{λ}) < 0$ which is a contradiction.

$(2^{o})$ Let $x^{*} \in Q$ be a maximizer in (1.1) such that $x∗∈\lx@overaccentset∘Q$ . Then Lemma 2.4 implies that there exist real numbers $ζ_{i}$ , $i \in N (x^{*})$ such that $ζ_{i} \geq 0$ for $i \in N (x^{*})$ , $\sum_{i} ζ_{i} = 1$ and

\sum i \in N (x^{*}) ζ_{i} \nabla_{x} r_{i} (x^{*}) = (J_{x} f (x^{*}, λ (x^{*})))^{T} ξ^{*} = 0,

where we set $ξ_{i}^{*} = ζ_{i} / h_{i} (x^{*})$ for $i \in N (x^{*})$ and $ξ_{i}^{*} = 0$ for $i \in {1, \dots, n} ∖ N (x^{*})$ . Due to assumption (R), this is possible only if $| N (x^{*}) | = n$ and $ξ_{i}^{*} > 0$ for all $i = 1, \dots, n$ . Thus $ξ^{*} \in K e r (J_{x} f (x^{*}, λ^{*}))^{T}$ and the equality $| N (x^{*}) | = n$ implies that $(x^{*}, λ^{*})$ is a solution of ( $f$ ). Now taking into account that ( $f$ ) has no solutions in $Q$ for $λ > λ^{*}$ we conclude that $(x^{*}, λ^{*})$ is a maximal solution of ( $f$ ) in $Q$ .

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), $d i m K e r J_{x} f (x^{*}, λ^{*}) = d i m K e r (J_{x} f (x^{*}, λ^{*}))^{T} = 1$ and the right and left eigenvectors $ζ^{*}$ , $ξ^{*}$ of $J_{x} f (x^{*}, λ^{*})$ are strongly positive. From this and since $h (x^{*}) \neq 0$ , $h (x^{*}) \geq 0$ , we obtain

⟨ \frac{d}{d λ} f (x^{*}, λ^{*}), ξ^{*} ⟩ = ⟨ h (x^{*}), ξ^{*} ⟩ \neq 0,

whence since $s p a n (ξ^{*}) = K e r (J_{x} f (x^{*}, λ^{*}))^{T}$ , it follows that

\frac{\partial}{\partial λ} f (x^{*}, λ^{*}) \notin R (J_{x} f (x^{*}, λ^{*})) .

Thus, all the assumptions of Lemma 2.1 are satisfied. Consequently, $(x^{*}, λ^{*})$ is a maximal saddle-node bifurcation point of ( $f$ ) in $Q$ .

4. Examples of application

4.1. Example 1.

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

(4.1)

{\begin{matrix} - v sin (θ) = λ p, v cos (θ) - v^{2} = λ q, \end{matrix}

Here $θ \in (- π / 2, π / 2)$ and $v > 0$ are unknown variables, $p, q \geq 0$ are given and $λ \in R$ 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 $p, q > 0$ . Denote $Q = (- π / 2, π / 2) \times R^{+}$ and introduce

r_{1} (θ, v) = - \frac{v}{p} sin (θ), r_{2} (θ, v) = \frac{1}{q} (v cos (θ) - v^{2}) .

Consider

(4.2)

λ^{*} = sup (θ, v) \in Q min {- \frac{v}{p} sin (θ), \frac{1}{q} (v cos (θ) - v^{2})}

Since $min {r_{1} (0, 1 / 2), r_{2} (0, 1 / 2)} = 1 / 4 q > 0$ , we infer that $λ^{*} \geq 1 / 4 q > 0$ . Using this it is not hard to show that there exists a maximizer $(θ^{*}, v^{*}) \in Q$ of (4.2). Moreover, $(θ^{*}, v^{*})$ is an internal point in $Q$ , i.e., $θ^{*} \neq - π / 2, π / 2$ and $v^{*} \neq 0$ . Observe, the Jacobian matrix

J_{θ, v} f (θ, v, λ) = ⎛ ⎝ \begin{matrix} - \frac{v}{p} cos (θ) & - \frac{1}{p} sin (θ) - \frac{v}{q} sin (θ) & \frac{1}{q} (cos (θ) - 2 v) \end{matrix} ⎞ ⎠

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

Lemma 4.1.

Assume $p, q > 0$ . Then $λ^{*} < + \infty$ and there exists a maximizer $(θ^{*}, v^{*})$ of (4.2). Furthermore, $(θ^{*}, v^{*}, λ^{*})$ is a maximal saddle-node bifurcation point of (4.1) in $(- π / 2, π / 2) \times R^{+}$ . Moreover, $d i m K e r (J_{θ, v} f (θ^{*}, v^{*}, λ^{*})) = 1$ and both right and left eigenvectors of $J_{θ, v} f (θ^{*}, v^{*}, λ^{*})$ 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)

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} - Δ u = λ u^{q} + p (u) % in Ω, u \geq 0 in Ω, u = 0 on \partial Ω, \end{matrix}

where $Ω$ is a bounded smooth domain in $R^{N}$ , $N \geq 1$ . Assume that $0 < q < 1$ , $p \in C^{1} (R)$ and

(H):: $lim t \to + \infty \frac{p (t)}{t^{q}} = + \infty$ , $lim t \to 0 \frac{p (t)}{t} = 0$ .

An example of the function $p$ is as follows $p (s) = s^{γ}$ , $s \in R^{+}$ where $1 < γ < + \infty$ . Thus, in this case, the nonlinearity $λ u^{q} + u^{γ}$ is true convex-concave.

We use finite differences to approximate this problem and for the sake of simplicity, we restrict ourselves to the case $N = 1$ .

Assume that $Ω = (0, L)$ , $L > 0$ . Set $x_{i} = i \cdot τ$ , $u_{i} = u (x_{i})$ , $1 \leq i \leq n$ , where $τ = L / (n + 1)$ . For the second derivatives at $n$ mesh points we used a standard second-order finite difference approximation. This yields for (4.3) the following approximating system of $n$ nonlinear equations

(4.4)

{\begin{matrix} - \frac{u_{i + 1} - 2 u_{i} + u_{i - 1}}{τ^{2}} - p (u_{i}) - λ u_{i}^{q} = 0, 1 \leq i \leq n, \end{matrix}

where $u = (u_{1}, \dots, u_{n}) \in S := {u \in R^{n} : u_{i} > 0, i = 1, \dots, n}$ , $u_{0} = u_{n + 1} = 0$ .

Then we have

	$f_{1} (u, λ) = - \frac{u_{2} - 2 u_{1}}{τ^{2}} - p (u_{1}) - λ u_{1}^{q},$
	$f_{n} (u, λ) = - \frac{- 2 u_{n} + u_{n - 1}}{τ^{2}} - p (u_{n}) - λ u_{n}^{q},$
	$f_{i} (u, λ) = - \frac{u_{i + 1} - 2 u_{i} + u_{i - 1}}{τ^{2}} - p (u_{i}) - λ u_{i}^{q}, i = 2, . . ., n - 1,$

Introduce

	$r_{1} (u) = \frac{- u_{2} + 2 u_{1} - τ^{2} p (u_{1})}{τ^{2} u_{1}^{q}}, r_{n} (u) = \frac{2 u_{n} - u_{n - 1} - τ^{2} p (u_{n})}{τ^{2} u_{n}^{q}},$
	$r_{i} (u) = \frac{- u_{i + 1} + 2 u_{i} - u_{i - 1} - τ^{2} p (u_{i})}{τ^{2} u_{i}^{q}}, i = 2, . . ., n - 1.$

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

(4.5)

λ^{*} = sup u \in S λ (u) = sup u \in S min i : u_{i} \neq 0 r_{i} (u),

Lemma 4.3.

Assume that (H) holds true. Then $λ^{*} < + \infty$ and there exists a maximizer $u^{*} \in S$ of (4.5). Furthermore, $(u^{*}, λ^{*})$ is a maximal saddle-node bifurcation point of (4.4) in $S$ . Moreover, $d i m K e r (J_{u} f (u^{*}, λ^{*})) = 1$ and both right and left eigenvectors of $J_{u} f (u^{*}, λ^{*})$ 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 $λ^{*} > 0$ . Take $u (δ) = (u_{1} (δ), \dots, u_{n} (δ))$ such that $u_{1} (δ) = δ > 0$ , $u_{j} (δ) = 0$ , $j = 2, \dots, n$ . Then $u (δ) \in ¯ ¯¯ ¯ S ∖ 0$ and

λ^{*} \geq λ (u (δ)) = min i : u_{i} \neq 0 r_{i} (u (δ)) = r_{1} (δ) = δ^{1 - q} (\frac{2}{τ^{2}} - \frac{p (δ)}{δ}) > 0

for sufficiently small $δ$ , since assumption (H) implies that $\frac{p (δ)}{δ} \to 0$ as $δ \to 0$ .

For $u = t v$ , $t = ∥ u ∥$ , $∥ v ∥ = 1$ , we have

(4.6)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} r_{1} (t v) = t^{1 - q} \frac{- v_{2} + 2 v_{1}}{τ^{2} (v_{1})^{q}} - \frac{p (t v_{1})}{τ^{2} (t v_{1})^{q}}, r_{n} (t v) = t^{1 - q} \frac{2 v_{n} - v_{n - 1}}{τ^{2} (v_{n})^{q}} - \frac{p (t v_{n})}{τ^{2} (t v_{n})^{q}}, r_{i} (t v) = t^{1 - q} \frac{- v_{i + 1} + 2 v_{i} - v_{i - 1}}{τ^{2} (v_{i})^{q}} - \frac{p (t v_{i})}{τ^{2} (t v_{i})^{q}}, i = 2, . . ., n - 1, \end{matrix}

Let $(u^{k})$ be a maximizer sequence of (4.5), i.e., $λ (u^{k}) \to λ^{*}$ as $k \to \infty$ . Suppose that $| | u^{k} | | := t^{k} \to \infty$ . Then there exists a subsequence, again denoted by $(u^{k})$ , such that $λ (u^{k}) \to - \infty$ as $k \to \infty$ . Indeed, since $| | v^{k} | | = 1$ , there exists $i \in {1, \dots, n}$ and a subsequence $v_{i}^{k_{j}}$ such that $v_{i}^{k_{j}} \to δ > 0$ as $k_{j} \to + \infty$ . Hence, (H) implies that $p (t^{k_{j}} v_{i}^{k_{j}}) / τ^{2} (t^{k_{j}} v_{i}^{k_{j}})^{q} \to - \infty$ as $k_{j} \to + \infty$ and consequently, by (4.6) we have $λ (u^{k_{j}}) := {min}_{i} r_{i} (u^{k_{j}}) \to - \infty$ as $k_{j} \to \infty$ . However, by assumption we have $λ (u^{k_{j}}) \to λ^{*} > 0$ . Thus $| | u^{k} | |$ is bounded and therefore there exists a limit point $u^{*} \in ¯ ¯¯ ¯ S$ of $(u^{k})$ . Passing to a subsequence if it’s necessary, we may assume that $u^{*} = {lim}_{k \to \infty} u^{k}$ . We claim that $u^{*} \neq 0$ . Indeed, assume $t^{k} := | | u^{k} | | \to 0$ . First, let us suppose that there exists $σ > 0$ such that $| v_{i}^{k} | > σ > 0$ , $\forall i = 1, \dots, n$ and $k = 1, \dots$ . Then from (4.6) it follows that $λ (u^{k}) \to 0$ as $k \to \infty$ which contradictions to $λ^{*} > 0$ . In the case $v_{i}^{k} \to 0$ for some $i \in {2, \dots, n - 1}$ , we have

r_{i} (t^{k} v^{k}) \leq (t^{k})^{1 - q} \frac{2 v_{i}^{k}}{τ^{2} (v_{i}^{k})^{q}} \to 0,

and similarly, ${lim}_{k \to \infty} r_{i} (t^{k} v^{k}) \leq 0$ if $v_{i}^{k} \to 0$ for $i = 1, n$ . Thus, we get a contradiction and $u^{*} \neq 0$ . In the same manner we can see that $u^{*} \in S$ .

Observe that the Jacobian matrix of $f (u, λ)$ has the following tridiagonal form:

Hence, we see that $J_{u} f (u, λ)$ is a irreducible off-diagonal sign-constant matrix for any $u \in S$ and $λ \in R$ . 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)

{\begin{matrix} - Δ u = λ e^{u} in Ω, u = 0 on \partial Ω . \end{matrix}

For the sake of simplicity, we restrict ourselves to the case $N = 1$ so that we take $Ω = (0, L)$ , $L > 0$ . Set $x_{i} = i \cdot τ$ , $u_{i} = u (x_{i})$ , $1 \leq i \leq n$ , where $τ = L / (n + 1)$ . Using a standard second-order finite difference approximation we derive for (4.7) the following approximating system of $n$ nonlinear equations

(4.8)

{\begin{matrix} - \frac{u_{i + 1} - 2 u_{i} + u_{i - 1}}{τ^{2}} - λ e^{u_{i}} = 0, 1 \leq i \leq n, \end{matrix}

where $u = (u_{1}, \dots, u_{n}) \in S$ and $u_{0} = u_{n + 1} = 0$ .

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

(4.9)

λ^{*} = sup u \in S λ (u) = sup u \in S min i : u_{i} \neq 0 r_{i} (u),

where

	$r_{1} (u) = \frac{- u_{2} + 2 u_{1}}{τ^{2} e^{u_{1}}}, r_{n} (u) = \frac{2 u_{n} - u_{n - 1}}{τ^{2} e^{u_{n}}},$
	$r_{i} (u) = \frac{- u_{i + 1} + 2 u_{i} - u_{i - 1}}{τ^{2} e^{u_{i}}}, i = 2, . . ., n - 1.$

Lemma 4.4.

$λ^{*} < + \infty$ and there exists a maximizer $u^{*}$ of (4.9). Furthermore, $(u^{*}, λ^{*})$ is a maximal saddle-node bifurcation point of (4.8) in $S$ . Moreover, there holds $d i m K e r (J_{u} f (u^{*}, λ^{*})) = 1$ and both right and left eigenvectors of $J_{u} f (u^{*}, λ^{*})$ are strongly positive.

Proof.

Take $¯ v = (v_{1}, \dots, v_{n})$ so that $v_{i} = s i n (i \cdot τ)$ , $i = 1, . . ., n$ . Then $¯ v \in S$ and $λ^{*} \geq λ (¯ v) = {min}_{i = 1}^{n} r_{i} (¯ v) > 0$ .

Let $(u^{k})$ be a maximizer sequence of (4.9), i.e., $λ (u^{k}) \to λ^{*}$ as $k \to \infty$ . Suppose that $| | u^{k} | | := t^{k} \to \infty$ . Then we see at once that $λ (u^{k}) \to 0$ as $k \to \infty$ which contradicts to the strong inequality $λ^{*} > 0$ . Thus, $| | u^{k} | |$ is bounded and there exists a limit point $u^{*} \in ¯ ¯¯ ¯ S$ of $(u^{k})$ and by continuity, $λ^{*} = λ (u^{*})$ . Observe that if $u_{i}^{*} = 0$ for some $i = 1, \dots, n$ , then $r_{i} (u^{*}) \leq 0$ , which implies a contradiction. Thus, $u_{i}^{*} > 0$ for every $i = 1, \dots, n$ .

Similar to the proof of Lemma 4.3, one can check that the Jacobian matrix $J_{u} f (u, λ)$ has the tridiagonal form and satisfies condition (R). Hence and since $u^{*} \in S$ , the proof of the lemma follows from Theorem 1.1.

∎

References

[1] V. Ajjarapu, Computational techniques for voltage stability assessment and control, Springer Science & Business Media, 2007.
[2] A. Bagirov, N. Karmitsa and M. M. Mäkelä, Introduction to Nonsmooth Optimization: Theory, Practice and Software, Springer, 2014.
[3] V. Bobkov, Y. Il’yasov, Maximal existence domains of positive solutions for two-parametric systems of elliptic equations, Complex Variables and Elliptic Equations, 61(5), 2016, pp. 587–607.
[4] G. Bratu, Sur les equation integrals non-lineaires, Bull. Math. Soc. France 42 (1914), pp. 113–142.
[5] V.F. Demyanov and V.N. Malozemov, Introduction to Minimax, John Wiley & Sons, New York, 1974.
[6] V.F. Demyanov, G. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos, Quasidifferentiability and nonsmooth modelling in mechanics, engineering and economics, Kluwer Academic, Dordrecht, 1996.
[7] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), pp. 161–180.
[8] I.M. Gelfand, Some problems in the theory of quasi-linear equations, Trans. Amer. Math. Soc. Ser. 2 (1963) 295–381.
[9] D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources. Archive for Rational Mechanics and Analysis, 49(4), (1973) pp.241-269.
[10] Y. Il’yasov, On positive solutions of indefinite elliptic equations, Comptes Rendus de l’Academie des Sciences-Series I, 333(6) (2001), pp. 533–538.
[11] Y. S. Il’yasov, Bifurcation calculus by the extended functional method, Functional Analysis and Its Applications, vol. 41, no. 1,(2007), pp. 18–30.
[12] Y. Il’yasov, A duality principle corresponding to the parabolic equations, Physica D, 237(5) (2008), pp. 692–698.
[13] Y.Sh. Il’yasov and T. Runst, Positive solutions of indefinite equations with p-Laplacian and supercritical nonlinearity, Complex Var. Elliptic Equ., 56(10-11) (2011), pp. 945–954.
[14] Y.S. Il’yasov, A.A. Ivanov, Finding bifurcations for solutions of nonlinear equations by quadratic programming methods, Computational Mathematics and Mathematical Physics, 53(3) (2013), pp. 350–364.
[15] Y. Il’yasov, A. Ivanov, Computation of maximal turning points to nonlinear equations by nonsmooth optimization, Optimization Methods and Software, vol. 31, no. 1, (2016), pp. 1–23.
[16] H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Application of bifurcation theory, P. Rabinowitz, ed., Academic Press, New York, 1977, pp. 359–384.
[17] H. Kielhöfer, Bifurcation theory: An introduction with applications to PDEs. Vol. 156. Springer Science & Business Media, 2006.
[18] I.S. Kohli, M.C. Haslam, Einstein’s field equations as a fold bifurcation. Journal of Geometry and Physics, 123, pp.434-437. 2018.
[19] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], Gosudarstv. Izdat. Tehn.-Teor. Lit. (GITTL), Moscow, 1956.
[20] Y. A. Kuznetsov, Elements of applied bifurcation theory, Vol. 112. Springer Science & Business Media, 2013.
[21] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487–513.
[22] R. Seydel, Practical Bifurcation and Stability Analysis, Interdisciplinary Applied Mathematics Vol. 5, Springer, 2010.
[23] E. Zeidler, Nonlinear functional analysis and its applications: III: variational methods and optimization. Springer Science & Business Media, 2013. https://doi.org/10.1007/978-1-4612-5020-3
[24] M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations, Nauka, Moscow, 1969;
[25] R. S. Varga, Iterative analysis. Prentice Hall, Englewood Cliffs, NJ, 1962.

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

Authors

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systems

FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network

An Upwind Generalized Finite Difference Method for Meshless Solution of Two-phase Porous Flow Equations

Refined RBF-FD analysis of non-Newtonian natural convection

Diversified Top-k Partial MaxSAT Solving

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

1. Introduction

Theorem 1.1.

Remark 1.2.

Remark 1.3.

2. Preliminaries

Lemma 2.1.

Remark 2.2.

Definition 2.3.

Lemma 2.4.

Proof.

3. Proof of Theorems 1.1

4. Examples of application

4.1. Example 1.

Lemma 4.1.

Remark 4.2.

4.2. Example 2.

Lemma 4.3.

Proof.

4.3. Example 3.

Lemma 4.4.

Proof.

References

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

Authors

Related Research

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systems

FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network

An Upwind Generalized Finite Difference Method for Meshless Solution of Two-phase Porous Flow Equations

Refined RBF-FD analysis of non-Newtonian natural convection

Diversified Top-k Partial MaxSAT Solving

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

This week in AI

1. Introduction

Theorem 1.1.

Remark 1.2.

Remark 1.3.

2. Preliminaries

Lemma 2.1.

Remark 2.2.

Definition 2.3.

Lemma 2.4.

Proof.

3. Proof of Theorems 1.1

4. Examples of application

4.1. Example 1.

Lemma 4.1.

Remark 4.2.

4.2. Example 2.

Lemma 4.3.

Proof.

4.3. Example 3.

Lemma 4.4.

Proof.

References