# On stability regions of the modified midpoint method for a linear delay differential equation

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## Abstract

The paper deals with stability regions of a certain discretization of a linear differential equation with constant delay. The main aim of the paper is to analyze the regions of asymptotic stability of the modified midpoint method applied to a linear differential equation with constant delay. Obtained results are compared with other known results, particularly for Euler discretization. The relation between asymptotic stability conditions in the discrete case and continuous case is discussed, too.

### Keywords

Difference Equation Asymptotic Stability Stability Region Delay Differential Equation Left Endpoint## 1 Introduction

and it is asymptotically stable if and only if $-\pi /2<b\tau <0$.

$\ell \ge 1$, $\ell \in \mathbb{Z}$ was discussed by Kuruklis [4].

**Theorem 1**

*Let*$\alpha \ne 0$,

*β*

*be arbitrary reals*.

*Equation*(5)

*is asymptotically stable if and only if*$|\alpha |<(\ell +1)/\ell $,

*and*

*where* $\varphi \in (0,\pi /(\ell +1))$ *is a solution of the auxiliary equation* $sin(\ell x)/sin((\ell +1)x)=1/|\alpha |$.

We note that for $\alpha =0$ the necessary and sufficient condition for asymptotic stability of (5) becomes $|\beta |<1$. We remark that the conditions in this famous result have an implicit form with respect to *ℓ*. Another equivalent set of conditions in an explicit form with respect to *ℓ* is introduced in [5].

where $\alpha ,\beta \in \mathbb{R}$ and $\ell \ge 1$, $\ell \in \mathbb{Z}$.

**Theorem 2**

*Let*

*α*,

*β*

*be arbitrary reals such that*$\alpha \beta \ne 0$.

- (i)
*Let**ℓ**be even and*$\beta {(-\alpha )}^{\ell /2+1}<0$.*Then*(6)*is asymptotically stable if and only if*$|\alpha |+|\beta |<1.$(7) - (ii)
*Let**ℓ**be even and*$\beta {(-\alpha )}^{\ell /2+1}>0$.*Then*(6)*is asymptotically stable if and only if either*$|\alpha |+|\beta |\le 1,$(8)

*or*

*holds*.

- (iii)
*Let**ℓ**be odd and*$\alpha <0$.*Then*(6)*is asymptotically stable if and only if*(7)*holds*. - (iv)
*Let**ℓ**be odd and*$\alpha >0$.*Then*(6)*is asymptotically stable if and only if either*(8),*or*${\beta}^{2}<1-\alpha <|\beta |,\phantom{\rule{2em}{0ex}}\ell <2arcsin\frac{1-{\alpha}^{2}-{\beta}^{2}}{2|\alpha \beta |}/arccos\frac{{\alpha}^{2}-{\beta}^{2}+1}{2|\alpha |}$(10)

*holds*.

Recently, Ren [7] also gave an equivalent system of necessary and sufficient conditions for asymptotic stability of (6), but his formulation needs to solve a nonlinear auxiliary equation, similarly to the result of Kuruklis mentioned above. A description of the stability boundary for (6) in terms of some straight lines and certain parametric curves can be found in Kipnis and Nigmatullin [8].

with positive integers *m*, *ℓ* was investigated.

The above mentioned results can be utilized to describe stability regions (*i.e.*, sets of pairs $(a,b)\in \mathbb{R}\times \mathbb{R}$, for which the given discretization is asymptotically stable considering given stepsize) for various numerical schemes, which solve an initial value problem for (1). For more details about numerical background, methods and their stability theory, see, *e.g.*, Bellen and Zennaro [10] and in’ t Hout [11].

where the stepsize *h* satisfies $ah\ne 1$. The value $Y(n)$ then represents a numerical approximation of solution *y* of delay differential equation (1) at the nodal point ${t}_{n}$.

The paper is organized as follows. Section 2 presents the set of necessary and sufficient conditions for asymptotic stability of (11). In Section 3 we discuss some important properties of obtained results and compare them with the results known for another discretization as well as with the asymptotic stability conditions for the corresponding differential equation. Section 4 concludes the paper by final remarks.

## 2 Main result

*k*. For an effective and clear formulation of the main result, we introduce the symbols

which are utilized in these two parts, respectively.

**Theorem 3**(I)

*Let*$k\ge 2$

*be even*.

*Then*(11)

*is asymptotically stable if and only if one of the following conditions holds*:

- (II)
*Let*$k\ge 3$*be odd and*$m=(k-1)/2$.*Then*(11)*is asymptotically stable if and only if one of the following conditions holds*:$a\le b<-a,\phantom{\rule{2em}{0ex}}|bh|<1,$(14)$|b|+a<0,\phantom{\rule{2em}{0ex}}{(-1)}^{m}bh=1,$(15)$b+|a|<0,\phantom{\rule{2em}{0ex}}bh>-1,\phantom{\rule{2em}{0ex}}\tau <{\tau}_{2}^{\ast}(h),$(16)${(-1)}^{m}b+a<0,\phantom{\rule{2em}{0ex}}{(-1)}^{m}bh>1,\phantom{\rule{2em}{0ex}}\tau <{\tau}_{2}^{\ast}(h),$(17)${(-1)}^{m}b+a>0,\phantom{\rule{2em}{0ex}}{(-1)}^{m+1}bh>1,\phantom{\rule{2em}{0ex}}\tau <{\tau}_{2}^{\ast}(h).$(18)

*Proof*The proof is based on the application of Theorem 2 to (11) and the ensuing analysis of the obtained conditions. In particular, if we consider (11) in the form of (6), the coefficients

*α*and

*β*of (6) are given by

and the indices *ℓ* and *k* are in the relation $\ell =k-1$.

*Case* (I): Investigating the case of *k* even, we utilize parts (iii) and (iv) of Theorem 2. Firstly, we focus on condition (iii): considering the coefficients (19), the assumption $\alpha <0$ implies $|ah|<1$. Thus, 7 is equivalent to $|b|+a<0$. Therefore, condition (iii) coincides with $|bh|<-ah<1$.

_{1}can be read as $2<2{b}^{2}{h}^{2}<1-ah$. Furthermore, the restriction (10)

_{2}becomes

Since $|ah|>1$ and $k=\tau /h$, it can be written as $\tau <{\tau}_{1}^{\ast}(h)$. Therefore, condition (iv) is satisfied if and only if either $|bh|\le 1$, $ah<-1$ or (13).

*i.e.*, $ah=-1$ or $b=0$). In our case we do not consider the eventuality $b=0$ with respect to the fact that we deal with the discretization of (1). Accordingly, for $ah=-1$, equation (11) turns to

and the necessary and sufficient condition for its asymptotic stability is given by Theorem 1 as $|bh|<1$. Summarizing the above discussion, we conclude that if *k* is even, (11) is asymptotically stable if either (12) or (13) holds.

*Case*(II): For

*k*odd, we consider conditions (i) and (ii) of Theorem 2. Condition (i) can be rewritten as

*m*odd and (21),

for *m* even.

*m*odd, (22) is equivalent to

*m*even, condition (22) is satisfied if and only if (23) or

holds. The above discussion of the Case (i), the part of (ii) considering (8) and including the case $\alpha =0$ (*i.e.*, $ah=-1$, $|bh|<0$, see *Case* (I)) gives (14)-(15).

*i.e.*,

*m*odd and (24), (25),

for *m* even. These conditions are jointly expressed by (16)-(18). In fact, (16) coincides with (24), (25). Condition (17) is equivalent to (28), (29) and (32), (33) for *m* odd and *m* even, respectively. Finally, (18) is the same as (26), (27) for *m* odd and (30), (31) for *m* even. The proof is complete. □

## 3 Asymptotic stability discussion

*k*even, the asymptotic stability region of (11) becomes $|b|+a<0$. Let us note that with the exception of the boundary, this region corresponds to (2). In the Case (II) of

*k*odd, it may be shown (by the L’Hospital rule) that the asymptotic stability conditions turn to

as $h\to 0$. These are equivalent to the conditions defining the asymptotic stability region of (1).

and then we focus on some of their monotony properties with respect to changing stepsize *h*. Finally, we compare the obtained stability intervals with the stability interval of the corresponding differential equation, as well as with the stability intervals for the forward Euler method discretization of (3).

**Corollary 4**

*Equation*(34)

*is asymptotically stable if and only if*

*Proof* The assertion is an immediate consequence of Theorem 3. Setting $a=0$, we realize that conditions (12) and (13) cannot occur. Therefore, (34) is unstable for any $b\in \mathbb{R}$ in the case of *k* even.

*k*odd, we investigate conditions (14)-(18). For $a=0$, there arises a contradiction in all conditions except for (16), which becomes

_{2}because it is imposed by the domain of the last relation (35)

_{3}. Our next aim is to simplify the delay restriction (35)

_{3}to the form more convenient for further analysis. We use the formula

*x*is decreasing for $x\in (0,\pi /2)$, we arrive at

which is the necessary and sufficient condition for asymptotic stability of (34) providing *k* is odd. □

We emphasize that the stability regions are captured just by stability intervals for values of parameter *b*. We denote ${I}_{\tau}^{M}(h)=(-\frac{1}{h}sin\frac{\pi h}{2\tau},0)$ stability intervals of (34) derived in Corollary 4. Next assertion describes the relation between stability intervals ${I}_{\tau}^{M}(h)$ with respect to stepsize *h*.

**Theorem 5**

*Let*$3\le {k}_{1}<{k}_{2}$

*be arbitrary positive odd integers and let*${h}_{1}=\tau /{k}_{1}>\tau /{k}_{2}={h}_{2}$

*be corresponding stepsizes*.

*Then*

*Proof*Let us define a function

*h*. Since the right endpoint is zero for any

*h*, we discuss only the monotony of $f(h)$. Doing this, we drop the constraint $h=\tau /k$ and we consider $f(h)$ as a function of a continuous argument

*h*. Then

Obviously, $tan(0)=0$ and ${(tan(x))}^{\prime}={cos}^{-2}(x)>1={x}^{\prime}$ for $x\in (0,\pi /6\u3009$. Therefore (36) holds for any $x\in (0,\pi /6\u3009$. Thus, we have proved that ${f}^{\prime}(h)>0$ for $h\in (0,\tau /3\u3009$ and consequently ${I}_{\tau}^{M}({h}_{2})\supset {I}_{\tau}^{M}({h}_{1})$. □

Next, we compare stability intervals ${I}_{\tau}^{M}(h)$ with the stability interval of (3), which we denote ${I}_{\tau}^{\ast}=(-\pi /(2\tau ),0)$.

**Remark 6**An important property is the behavior of ${I}_{\tau}^{M}(h)$ as $h\to 0$. Using the L’Hospital rule, we may see that

Therefore, ${I}_{\tau}^{M}(h)$ is approaching ${I}_{\tau}^{\ast}$ as $h\to 0$.

**Remark 7** In the proof of Theorem 5 we have shown that $-\frac{1}{h}sin\frac{\pi h}{2\tau}$ is an increasing function on $h\in (0,\tau /3\u3009$. Considering also Remark 6, we conclude that ${I}_{\tau}^{\ast}\supset {I}_{\tau}^{M}(h)$ for any $h=\tau /k$, where *k* is odd. Note that the midpoint method discretization of (3) is not asymptotically stable.

Finally, we discuss a relation between ${I}_{\tau}^{M}(h)$ and asymptotic stability intervals for the forward Euler discretization of (3). They are derived in [12], and we denote them as ${I}_{\tau}^{E}(h)=(-\frac{2}{h}cos\frac{\pi \tau}{2\tau +h},0)$.

**Theorem 8**

*Let*$k\ge 3$

*be an arbitrary positive odd integer and let*$h=\tau /k$

*be the corresponding stepsize*.

*Then*

*Proof*Since the right endpoints of ${I}_{\tau}^{M}(h)$ and ${I}_{\tau}^{E}(h)$ are zero for any

*h*, we investigate only the behavior of the left endpoints with respect to changing stepsize

*h*. We define a function

*h*. In the further analysis, we drop the constraint $h=\tau /k$ and consider both functions $f(h)$ and $g(h)$ to be functions with a continuous argument for $h\in (0,\tau \u3009$ (we extend the domain of $f(h)$ to simplify the proof). Thus our aim is to show that $f(h)-g(h)<0$ for any $h\in (0,\tau )$,

*i.e.*,

To do this, we introduce the following proposition.

**Lemma** *Let* $F\in {C}^{3}\u3008a,b\u3009$ *be a function such that* $F(a)=F(b)=0$, ${F}^{\prime}(a)\le 0$, ${F}^{\prime}(b)>0$, ${F}^{\u2033}(a)<0$, ${F}^{\u2033}(b)>0$ *and* ${F}^{\u2034}(t)>0$ *for all* $t\in (a,b)$. *Then* $F(t)<0$ *for all* $t\in (a,b)$.

*Proof* Since ${F}^{\u2034}(t)>0$ for all $t\in (a,b)$, the function ${F}^{\u2033}(t)$ is increasing. Since ${F}^{\u2033}(a)<0<{F}^{\u2033}(b)$, there is a unique point ${t}_{1}\in (a,b)$ such that ${F}^{\u2033}({t}_{1})=0$. Thus, the function ${F}^{\prime}(t)$ is decreasing in $(a,{t}_{1})$ and increasing in $({t}_{1},b)$. Further, since ${F}^{\prime}({t}_{1})<{F}^{\prime}(a)\le 0$ and ${F}^{\prime}(b)>0$, there is a unique point ${t}_{2}\in (a,b)$ such that ${F}^{\prime}({t}_{2})=0$. Therefore, $F(t)$ is decreasing in $(a,{t}_{2})$ and increasing in $({t}_{2},b)$. Taking into account $F(a)=F(b)=0$, we obtain that $F(t)<0$ for $t\in (a,b)$. □

since each term in the sum is positive for all $s\in (2,3)$. Then by the previous lemma, we have that $G(s)<0$ for all $s\in (2,3)$ and consequently $f(h)<g(h)$ for $h\in (0,\tau )$, which concludes the proof. □

## 4 Conclusions

To summarize the previous, the main result formulated in Theorem 3 describes the asymptotic stability regions of difference equation (11). This equation actually represents a discretization of delay differential equation (1) by a modified midpoint rule. It was shown that the asymptotic stability regions depend not only on the value of stepsize *h*, but also on the parity of *k*. In the case $a=0$, the obtained result was given to the connection with the results known for the Euler discretization of (3). Moreover, the connection with asymptotic stability properties of delay differential equation (3) was also mentioned. This discussion points out some interesting properties of the stability regions for the discrete form of the delay differential equation (1). The authors believe that analogous investigation is possible also for more complicated numerical formulae (applied to (1)) as far as there are known stability criteria for corresponding difference equations. Such analysis may be done, *e.g.*, for the Θ-method.

## Notes

### Acknowledgements

The first author was supported by the project FSI-S-11-3 of Brno University of Technology. The second author was supported by the grant P201/11/0768 *Qualitative properties of solutions of differential equations and their applications* of the Czech Science Foundation.

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