On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay

A transcendental equation $\lambda + \alpha - \beta\mathrm{e}^{-\lambda\tau} = 0$ 
with complex coefficients is investigated. 
This equation can be obtained from the characteristic equation of a linear differential equation 
with a single constant delay. 
It is known that the set of roots of this equation can be expressed by the Lambert W function. 
We analyze the condition on parameters for which all the roots have negative real parts 
by using the ``graph-like'' expression of the W function. 
We apply the obtained results to the stabilization of an unstable equilibrium solution 
by the delayed feedback control 
and the stability condition of the synchronous state in oscillator networks.

1. Introduction. In this paper, we investigate how the exponential stability of equilibrium solutions of a delay differential equation (DDE) in R n (n ≥ 2) is determined by this equation. Here f (x(t), x(t − τ )) is a smooth (nonlinear) function of (x(t), x(t − τ )) ∈ R n × R n , and τ > 0 is a "delay" parameter. By the exponential stability of an equilibrium solution x * (t) ≡ p ∈ R n , we mean that there exist δ, C and γ > 0 such that |x(t) − p| ≤ Ce −γt for all t ≥ 0 holds for any solutions x(t) satisfying an initial condition where | · | denotes the Euclidean norm. We have the following stability result called the principle of linearized stability.
Fact 1 (e.g. Diekmann et al. [4]). We consider the linearized equation of (1) along x * (t) ≡ p and its characteristic equation det(λI + A − e −λτ B) = 0 (I is the n × n identity matrix), where D 1 f (p, p) and D 2 f (p, p) denote the partial Jacobian matrices of f at (p, p). Then the following statements hold.
1.1. Transcendental equations and Lambert W function. Our concern in this paper is how to analyze the characteristic equation (2) to obtain the parameter dependence. See Stépán [14] for general results about the characteristic equations in DDEs. If matrices A and B have some (generalized) eigenvectors in common, then there is a chance that (2) is reduced to a transcendental equation where α and β are some eigenvalues of A and B, respectively. For the case α, β ∈ R, Hayes [7] obtained a necessary and sufficient condition on parameters for which all the roots have negative real parts. The condition on τ > 0 for each fixed α and β is easily obtained (cf. Cooke & Grossman [1]) from this. However, to the best of the author's knowledge, no condition on α, β and τ where α, β ∈ C has been obtained. We note that (3) can be viewed as a transcendental equation with real coefficients where p = α +ᾱ, q = |α| 2 , r(λ) = (β +β)λ + (αβ +ᾱβ) and s = |β| 2 by multiplying (3) and the equation with complex conjugate coefficients. To achieve our objective, we use the Lambert W function that is the multi-valued inverse of a complex function z → ze z (see Corless et al. [3]), that is W (ζ) = { z ∈ C : ze z = ζ } (ζ ∈ C).
The idea of this paper is to study this set by investigating the complex branches of the W function. The "graph-like" expression of these branches in some coordinate system of the complex plane C (Corollary 1 in Section 2) is the ingredient of this approach.
Definition 1.1. We call a complex function f to be graph-like on a subset D if there exists a real-valued function g such that we have a formula Here i is the imaginary unit.
The main result is the following: Let α ∈ C, β ∈ C \ {0}, and τ > 0. Then all the roots of (3) have negative real parts if and only if α, β and τ satisfy the following (a) or (b): The notation is given in Subsection 3.1.  [12] is a method to stabilize an orbitally unstable periodic solution of a given autonomous smooth ordinary differential equation (ODE). Assume that we have a smooth ODE x (t) = f (x(t)) in R n and there is an orbitally unstable periodic solution γ(t). Then we consider a DDE where K is a constant n × n real matrix and τ > 0 is an integer multiple of the minimal period of γ(t). We note that γ(t) is also a periodic solution of (5). The objective of the DFC is to find K and τ so that γ(t) is an orbitally stable periodic solution of (5). Pyragas showed the possibility of the stabilization by the DFC applying this to the Rössler system numerically, but this problem remains open mathematically. There is a positive result by Fiedler et al. [6] in which they showed that it is possible to stabilize unstable periodic orbits of the normal form of a subcritical Hopf bifurcation that emanate from the stable equilibrium by the DFC. Instead of a periodic solution, one can consider the stabilization of an unstable equilibrium solution by the DFC, in which we can choose an arbitrarily τ > 0 (e.g., see Hövel & Schöll [9]). By applying the results obtained from Theorem 1.2, we show that it is possible to stabilize an unstable equilibrium solution by the DFC on some assumption of its equilibrium (Theorem 4.1 in Subsection 4.1).

1.2.2.
Stability criterion of synchronous state in oscillator networks. Earl & Strogatz [5] studied networks of identical phase oscillators with delayed coupling in which the in-degrees of each vertices of the associated graph are uniform. For these oscillator networks, they obtained an analytic criterion of the linear stability of the synchronous state which does not depend on their connection topologies.
We consider a network of identical phase oscillators with delayed coupling. For this network, we assume that the associated graph Γ does not have self loops and the in-degrees of each vertices of Γ are equal. This model is obtained from a system of DDEsθ where θ i is the phase variable of the ith oscillator, ω is its natural frequency, K is the coupling strength, and f is a nonlinear 2π-periodic function. On the connection topology of Γ, k is the in-degree of vertices, and A = (a ij ) i,j is the adjacency matrix. The above assumptions mean that a ij = 0 or 1, a ii = 0, and N j=1 a ij = k holds for each i.
This statement means that the linear stability of the synchronous state does not depend on the connection topology of Γ, that is, the in-degree k and the adjacency matrix A of Γ.
We show that this condition in fact depends on the connection topology by applying Theorem 1.2 (Theorem 4.7 in Subsection 4.2). This paper is organized as follows. In Section 2, we show the "graph-like" expressions of the complex branches of the Lambert W function (Corollary 1). Section 3 is divided into two subsections. In Subsection 3.1, we prove Theorem 1.2 which gives a necessary and sufficient condition on α, β and τ for which all the roots of (3) have negative real parts. In Subsection 3.2, we investigate a condition on τ for each fixed α and β by using Theorem 1.2 and obtain Theorems 3.2 and 3.3. In Section 4, we apply these results to the stabilization of unstable equilibrium solutions by the delayed feedback control and the stability criterion of the synchronous states of oscillator networks. In Appendix A, some lemmas are proved.

Lambert W function.
To study an equation ze z = ζ of z, Wright [16] considered the (multi-valued) inverse of a complex function z → z + Log(z), where Log is the principal branch of the complex logarithm. Wright claimed that this inverse function is single-valued except for the branch cut. Corless & Jeffrey [2] named this inverse the Wright omega function.
In this section, we redefine the Wright omega function and investigate the relationship between the W function and the omega function to investigate the graphlike expression of the W function. The proofs of lemmas are given in Appendix A.
We first define two multi-valued functions.
Definition 2.1. For z ∈ C \ {0}, we define Arg W (z) and Log W (z) as and Log W (z) = ln |z| + i Arg W (z), respectively. Here Arg(z) represents the principal value of the argument of z ∈ C \ {0}, i.e., Arg(z) ∈ (−π, π] and z = |z| · e i Arg(z) . We call the multi-valued inverse The changes in definition of Arg W (z) are brought by the branch structure of the W function.
Function G(ρ, ·) is indeed single-valued and has monotonicity property.
Theorem 2.5. The omega function is single-valued, and ω(ζ) is expressed as follows: Proof. The proof is obvious from Lemmas 2.3 and 2.4.
Note that Theorem 2.5 gives a "graph-like" expression of the omega. In view of Lemma 2.2 and Theorem 2.5, the branches of the W function can be defined. Definition 2.6 (cf. Corless et al. [3]). For each m ∈ Z, we call the complex function W m (·) defined as The following is a corollary of Theorem 2.5.
By monotonicity of G(ρ, ·), the above expression gives an another proof of rela- where m ≥ 0 and m ≤ 0 respectively, which are stated in Shinozaki & Mori [13]. Note that holds in particular.

3.
A necessary and sufficient condition. In this section, we find a necessary and sufficient condition on parameters for which all the roots of transcendental equation (3) have negative real parts.
The following lemma is key for this purpose. hold.
Let f (x) = xe x . We investigate the condition in (I).
(If-part). We should reverse the argument.
This completes the proof.
3.1. Condition on α, β and τ . To study (3), we only have to consider the case β = 0. Denote by Arccos the inverse of cosine which is restricted on [0, π]. Let .
Proof of Theorem 1.2. From the representation of the set of roots of (3) by the W function, all the roots of (3) have negative real parts if and only if α, β and τ satisfy because W 0 is the right-most branch from (8). The conclusion is obtained from Lemma 3.1.

3.2.
Condition on τ . For each α and β, let all the roots of (3) have negative real parts }.
The following questions arise.
What is the condition for which I(α, β) is nonempty? (Q2) How is I(α, β) expressed as a function of α and β?
The next corollary is a direct consequence of Theorem 1.2.
We introduce some notation: hold.

4.
Applications. In this section, we apply the results obtained in Section 3 to the delayed feedback control and the stability of synchronous states in oscillator networks.
4.1. Delayed feedback control. Assume that the given ODE x (t) = f (x(t)) has an equilibrium solution x * (t) ≡ 0 ∈ R n , which is also an equilibrium solution of (5). The linearization of (5) along x * (t) gives the characteristic equation In the following, we see how (10) can be reduced to a transcendental equation (3).

Reduction to transcendental equation.
We take an eigenvalue µ of M and suppose that µ satisfies one of the following conditions: (E1) µ is not real, (E2) µ is a real eigenvalue whose algebraic multiplicity is even.
Let E µ be the sum E(µ)+ E(μ) of generalized eigenspaces E(µ) and E(μ) associated to µ andμ, respectively. Then the generalized eigenspace decomposition gives where V 0 is the sum of the generalized eigenspaces associated to the other eigenvalues. Let 2m = dim(E µ ). Note that we can take a real basis of E µ in both cases (E1) and (E2). We choose an n × n real matrix K so that σ(K| Eµ ) = {κ,κ} for some κ ∈ C and σ(K| V0 ) = 0, and the characteristic equation (10) becomes where I| V0 is the identity transformation on V 0 . We denote by K µ the set of such K having the above properties. Here σ(T ) represents the set of eigenvalues of a linear transformation T . Lastly, (10) is reduced to which is an equation (3) with if µ andμ are only eigenvalues satisfying (µ) ≥ 0.
Theorem 4.1. Suppose that the set of the eigenvalues of the equilibrium 0 whose real part is nonnegative is {µ,μ} for some µ ∈ C.
Proof. We use an identity µκ +μκ = 2( µ · κ). From this, we obtain the first inequality. The second inequality is obtained from the trigonometric identity.
Therefore, I(κ) consists of the following intervals: This completes the proof.
We next consider the case (µ) > 0, in which Theorem 3.3 can only be applied. Two conditions defining set D are related to r 1 (θ) and r 2 (θ), respectively.

Proof. A simple calculation and the trigonometric identities show
which leads us to the conclusion.
Proof. The expression of D is obtained from Propositions 3, 4 and (15).
Lemma 4.4 shows that D is the region enclosed by C 0 and C 1 except the line (κ) = (µ).
In this case, the left-hand side of (16) does not depend on r > 0.