In this section, system (1)–(2) is studied for two different cases in two different spaces. One of these cases is \(r_{1} = r_{2} = r\) for which the characteristic equation becomes much simpler. In other case, \(r_{1} \ne r_{2}\) state, which constitutes the main part of our analysis.

### 3.1
\((\alpha_{1} - \alpha_{2})\) parameter space

In this subsection, we determine the conditions under which the system is unstable. Moreover, the regions on which the characteristic equation has roots with the same number of positive real parts are shown for fixed delay values.

In order to find the stability region of system (1)–(2) in the parameter space, the D-partition method is applied. If the characteristic equation

$$ g(\lambda) = \lambda^{2} - (\alpha_{1} + \beta_{2})\lambda+ (\alpha _{1} - \lambda) \theta_{2}e^{ - \lambda r_{2}} - \alpha_{2}\theta_{1}e^{ - \lambda r_{1}} + \alpha_{1}\beta_{2} - \alpha_{2} \beta_{1} = 0, $$

(8)

corresponding to system (1)–(2), has a zero root, then we have

$$ (\theta_{2} + \beta_{2})\alpha_{1} = ( \theta_{1} + \beta_{1})\alpha_{2}. $$

(9)

This straight line is one of the lines forming the boundary of the D-partition. Substituting \(\lambda= ib\) in characteristic equation (8) and equating the real and imaginary parts to zero, we have

$$\begin{aligned}& U = - b^{2} - \theta_{2}b\sin(br_{2}) + \alpha_{1}\theta_{2}\cos(br_{2}) - \alpha_{2}\theta_{1}\cos(br_{1}) + \alpha_{1}\beta_{2} - \alpha_{2} \beta_{1} = 0, \end{aligned}$$

(10)

$$\begin{aligned}& V = - \alpha_{1}b - \beta_{2}b - \theta_{2}b \cos(br_{2}) - \alpha_{1}\theta_{2} \sin(br_{2}) + \alpha_{2}\theta_{1} \sin(br_{1}) = 0. \end{aligned}$$

(11)

From equations (10)–(11), D-curves are obtained as follows:

$$\begin{aligned}& \alpha_{1}(b) = \frac{\theta_{1}\theta_{2}b\cos(b(r_{1} - r_{2})) + \theta_{1} b^{2}\sin(br_{1}) + \theta_{1}\beta_{2}b\cos(br_{1}) + \theta_{2}\beta_{1}b\cos(br_{2}) + \beta_{1}\beta_{2}b}{\theta_{1}\theta_{2}\sin(b(r_{1} - r_{2})) + \theta_{1} \beta_{2}\sin(br_{1}) - \theta_{2}\beta_{1}\sin(br_{2}) - \theta_{1}b\cos(br_{1}) - \beta_{1}b}, \end{aligned}$$

(12)

$$\begin{aligned}& \alpha_{2}(b) = \frac{b^{3} + (\theta_{2}^{2} + \beta_{2}^{2})b + 2\theta_{2}\beta_{2}b\cos(br_{2}) + 2\theta_{2}b^{2}\sin (br_{2})}{\theta_{1}\theta_{2}\sin(b(r_{1} - r_{2})) + \theta_{1} \beta_{2}\sin(br_{1}) - \theta_{2}\beta_{1}\sin(br_{2}) - \theta _{1}b\cos (br_{1}) - \beta_{1}b} \end{aligned}$$

(13)

as *b* tends to 0, we obtain the cusp point

$$\begin{aligned}& p_{1}: = \lim_{b \to0}\alpha_{1}(b) = - \frac{(\theta_{1} + \beta_{1})(\theta_{2} + \beta_{2})}{\theta_{1} + \beta_{1} - \theta_{1}r_{1}(\theta_{2} + \beta_{2}) + \theta_{2}r_{2}(\theta_{1} + \beta_{1})}, \\& p_{2}: = \lim_{b \to0}\alpha_{2}(b) = - \frac{(\theta_{2} + \beta_{2})^{2}}{\theta_{1} + \beta_{1} - \theta_{1}r_{1}(\theta_{2} + \beta_{2}) + \theta_{2}r_{2}(\theta_{1} + \beta_{1})}. \end{aligned}$$

As the next step, these results are illustrated for various values of parameters. The curves (12)–(13) and the straight line (9) form the D-partition shown in Fig. 1 for \(\beta_{1} = 1.4\), \(\beta_{2} = 0.4\), \(\theta_{1} = 1.25\), \(\theta_{2} = 1\), \(r_{1} = 1\) and \(r_{2} = 0.25, 0.5, 1, 1.5, 1.8\).

### Lemma 1

*If*
\(0 \le2A \le\frac{1}{r\mu} \), *then*
\(x^{2} + 2Ax\sin (xr) \ge0\)
*for*
\(\forall x \in IR\), *where*
*r*
*is a positive real number and*
\(\mu= \sup(\frac{ - \sin x}{x}) \approx0.218\).

### Proof

Since \(f(x) = x^{2} + 2Ax\sin(xr)\) is an even function, it is sufficient to show only for \(\forall x \ge0\). When \(\sin(xr) \ge0\), we obtain

$$x^{2} + 2Ax\sin(xr) \ge x^{2} \ge0. $$

On the other hand, when \(\sin(xr) < 0\), we obtain the following inequality:

$$ x^{2} + 2Ax\sin(xr) \ge x^{2} + \frac{1}{r\mu} x\sin(xr). $$

(14)

Suppose that \(x^{2} + \frac{1}{r\mu} x\sin(xr) < 0\) for \(\forall x \in IR^{ +} \) when \(\sin(xr) < 0\). By taking \(x = \frac{y}{r}\) on the left-hand side of inequality (14), we have

$$\mu< - \frac{\sin(y)}{y}, $$

which contradicts the definition of *μ*.

Consequently, the desired inequality

$$x^{2} + 2Ax\sin(xr) \ge0\quad\mbox{for } \forall x \ge0 $$

is obtained. □

### Lemma 2

*Let*
*μ*
*be defined as in Lemma *1. *If*
\(r_{1} = r_{2} = r\), \(\beta_{2}\theta_{1} = \theta_{2}\beta_{1}\), \(\vert \theta_{1} \vert < \beta_{1}\), *and*
\(0 \le2\theta_{2} \le\frac{1}{r\mu} \), *then*
\(\alpha_{2}(b) \le0\)
*for*
\(\forall b \in IR\).

### Proof

Under the conditions of the theorem, for \(\forall b \in IR - \{ 0\}\), equality (13) can be rewritten as follows:

$$ \alpha_{2}(b) = - \frac{b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2}\cos(br) + 2\theta_{2}b\sin(br)}{\theta_{1}\cos(br) + \beta_{1}}. $$

(15)

(i) If \(\vert \theta_{1} \vert < \beta_{1}\) for \(\forall b \in IR\), then inequality \(\theta_{1}\cos(br) + \beta_{1} > 0\) holds.

(ii) If \(\theta_{2}\beta_{2} \ge0\), then

$$\theta_{2}^{2} + \beta_{2}^{2} + 2 \theta_{2}\beta_{2}\cos(br) \ge \theta_{2}^{2} + \beta_{2}^{2} - 2\theta_{2}\beta_{2} = (\theta_{2} - \beta_{2})^{2} \ge0 $$

holds and if \(\theta_{2}\beta_{2} < 0\), then

$$\theta_{2}^{2} + \beta_{2}^{2} + 2 \theta_{2}\beta_{2}\cos(br) > \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2} = (\theta_{2} + \beta_{2})^{2} > 0 $$

holds. As a result, \(\theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2}\cos(br) > 0\) holds for \(\forall b \in IR\).

(iii) If \(0 \le\theta_{2} \le\frac{1}{r\mu} \), then it follows from Lemma 1 that \(b^{2} + 2\theta_{2}b\sin(br) \ge0\) holds for \(\forall b \in IR\).

It follows from (i), (ii), and (iii) that \(\alpha_{2}(b) \le 0\) for all \(\forall b \in\Re- \{ 0\}\). Moreover, we have

$$\lim_{b \to0}\alpha_{2}(b) = - \frac{(\theta_{2} + \beta_{2})^{2}}{\theta_{1} + \beta_{1}}, $$

which is negative when \(\vert \theta_{1} \vert < \beta_{1}\). □

### Lemma 3

*If*
\(r_{1} = r_{2} = r\), \(\beta_{2}\theta_{1} = \theta_{2}\beta_{1}\), \(\vert \theta_{1} \vert < \beta_{1}\), *and*
\(2 \vert \theta_{2} \vert \le \vert \beta_{2} \vert \), *then*
\(\alpha_{2}(b) \le 0\)
*for*
\(\forall b \in IR\).

### Proof

It follows from the proof of Lemma 2 that the denominator of equality (15) is positive. Now let us investigate the following cases for the numerator of equality (15). Since the numerator of equality (15) is an even function of *b*, it is sufficient to show for \(\forall b \ge0\).

(i) If \(0 \le2\theta_{2} \le\beta_{2}\), then

$$\begin{aligned} \begin{aligned} b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2}\cos(br) + 2\theta_{2}b \sin(br) &\ge b^{2} + \theta_{2}^{2} + \beta_{2}^{2} - 2\theta_{2}\beta_{2} - 2 \theta_{2}b \\ &= (b + \theta)^{2} + \beta_{2}^{2} - 2 \theta_{2}\beta_{2} \ge0 \end{aligned} \end{aligned}$$

holds.

(ii) If \(0 \le2\theta_{2} \le- \beta_{2}\), then

$$\begin{aligned} b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2}\cos(br) + 2\theta_{2}b \sin(br) \ge& b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2} - 2 \theta_{2}b \\ =& (b - \theta)^{2} + \beta_{2}^{2} + 2 \theta_{2}\beta_{2} \ge0 \end{aligned}$$

holds.

(iii) If \(0 \le- 2\theta_{2} \le\beta_{2}\), then

$$\begin{aligned} b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2}\cos(br) + 2\theta_{2}b \sin(br) \ge& b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2} + 2 \theta_{2}b \\ =& (b + \theta)^{2} + \beta_{2}^{2} + 2 \theta_{2}\beta_{2} \ge0 \end{aligned}$$

holds.

(iv) If \(\beta_{2} \le2\theta_{2} \le0\), then

$$\begin{aligned} b^{2} + \theta_{2}^{2} + \beta_{2}^{2} + 2\theta_{2}\beta_{2}\cos(br) + 2\theta_{2}b \sin(br) \ge& b^{2} + \theta_{2}^{2} + \beta_{2}^{2} - 2\theta_{2}\beta_{2} + 2 \theta_{2}b \\ =& (b + \theta)^{2} + \beta_{2}^{2} - 2 \theta_{2}\beta_{2} \ge0 \end{aligned}$$

holds. □

### Theorem 5

*We suppose that the conditions of Lemma *2
*or Lemma *3
*hold and if*

$$ \left . \textstyle\begin{array}{l} (\theta_{2} + \beta_{2})\alpha_{1} < (\theta_{1} + \beta_{1})\alpha_{2}, \\ \alpha_{2} > 0 \end{array}\displaystyle \right \} $$

(16)

*are satisfied*, *then the characteristic equation of system* (1)*–*(2) *has only one root with positive real part*.

### Proof

It follows from Lemma 2 or Lemma 3 that D-curves of system (1)–(2) are located outside of the region which is determined by inequality system (16) in \((\alpha_{1} - \alpha_{2})\) parameter space. Suppose that there were two points within the region (16) with different numbers of roots with positive real parts. Then along any arc within that region connecting the points there must be a point where some of the roots of the characteristic equation lie on the imaginary axis. This point must lie on the D-curves, giving a contradiction. As a result, the number of roots of the characteristic equation with positive real parts does not change in the region (16). In other words, if the system is unstable for specific values of the parameters which satisfy the conditions of the theorem, the instability of the system has been shown. Using Stepan’s formula (6) with \(\beta_{1} = \beta_{2} = 1.2\), \(\theta_{1} = \theta_{2} = 1\), \(r_{1} = r_{2} = 1\), \(\alpha_{1} = 0.1\), \(\alpha_{2} = 1\) values, it is obtained that the characteristic equation of the system has one root with a positive real part. □

### Theorem 6

*We suppose that the conditions of Lemma *2
*or Lemma *3
*hold and if*

$$ \left . \textstyle\begin{array}{l} (\theta_{2} + \beta_{2})\alpha_{1} > (\theta_{1} + \beta_{1})\alpha_{2}, \\ \alpha_{2} > 0 \end{array}\displaystyle \right \} $$

(17)

*are satisfied*, *then the characteristic equation of system* (1)*–*(2) *has two roots with positive real parts*.

### Proof

The proof follows the lines of the proof of Theorem 5. □

It follows from Theorems 5 and 6 that since the region on which \(\alpha_{2} > 0\) has no curves but straight line (9), the stability of the system does not change. Moreover, the straight line (9) splits this region into two subregions on which the characteristic equation of the system has either one or two roots with positive real parts. As a result, the system is unstable under the assumption of Theorems 5 and 6.

### 3.2
\((\beta_{1} - \beta_{2})\) parameter space

From equation (10)–(11), D-curves are obtained in \((\beta_{1} - \beta_{2})\) parameter space as follows:

$$\begin{aligned}& \beta_{1}(b) = - \frac{b^{3} + \theta_{2} b^{2}\sin(br_{2}) + \alpha_{2}\theta_{1}(b\cos(br_{1}) - \alpha_{1}\sin(br_{1})) + \alpha_{1}^{2}(b + \theta_{2}\sin(br_{2}))}{b\alpha_{2}}, \end{aligned}$$

(18)

$$\begin{aligned}& \beta_{2}(b) = - \frac{\alpha_{1}b + \theta_{2}b\cos(br_{2}) + \alpha_{1}\theta_{2}\sin(br_{2}) - \alpha_{2}\theta_{1}\sin (br_{1})}{b} \end{aligned}$$

(19)

as *b* tends to 0, we obtain the cusp point

$$\begin{aligned}& p_{1}: = \lim_{b \to0}\beta_{1}(b) = - \frac{\alpha_{1}^{2}(1 + \theta_{2}r_{2}) + \alpha_{2}\theta_{1}(1 - \alpha_{1}r_{1})}{\alpha_{2}}, \\& p_{2}: = \lim_{b \to0}\beta_{2}(b) = - \alpha_{1}(1 + \theta_{2}r_{2}) - \theta_{2} + \alpha_{2}\theta_{1}r_{1}. \end{aligned}$$

The curves (18)–(19) and the straight line (9) that form the D-partition are shown in Fig. 2 for \(\alpha_{1} = 0.2\), \(\alpha_{2} = 0.2\), \(\theta_{1} = 1.25\), \(\theta_{2} = 1\), \(r_{1} = 1\), \(r_{2} = 0.25, 0.5, 1, 1.5, 2\) and for \(r_{1} = 0.25, 1, 1.5, 2, 3\), \(r_{2} = 1\).

### Proposition 1

*There exist real numbers*
*m*
*and*
*M*
*such that*
\(m < \beta_{2}(b) < M\)
*for*
\(\forall b \in IR\).

### Proof

Let \(\mu= \sup(\frac{ - \sin x}{x})\). If *m* and *M* are defined as follows:

$$\begin{aligned}& m = \textstyle\begin{cases} - \alpha_{1} - \vert \theta_{2} \vert + \mu\alpha_{1}\theta_{2}r_{2} + \alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} < 0, \alpha_{2}\theta_{1} < 0, \\ - \alpha_{1} - \vert \theta_{2} \vert + \mu\alpha_{1}\theta _{2}r_{2} - \mu\alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} < 0, \alpha_{2}\theta_{1} > 0, \\ - \alpha_{1} - \vert \theta_{2} \vert - \alpha_{1}\theta_{2}r_{2} + \alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} > 0, \alpha_{2}\theta_{1} < 0, \\ - \alpha_{1} - \vert \theta_{2} \vert - \alpha_{1}\theta_{2}r_{2} - \mu \alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} > 0, \alpha_{2}\theta_{1} > 0, \end{cases}\displaystyle \end{aligned}$$

(20)

$$\begin{aligned}& M = \textstyle\begin{cases} - \alpha_{1} + \vert \theta_{2} \vert - \alpha_{1}\theta_{2}r_{2} - \mu\alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} < 0, \alpha_{2}\theta_{1} < 0, \\ - \alpha_{1} + \vert \theta_{2} \vert - \alpha_{1}\theta_{2}r_{2} + \alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} < 0, \alpha_{2}\theta_{1} > 0, \\ - \alpha_{1} + \vert \theta_{2} \vert + \mu\alpha_{1}\theta _{2}r_{2} - \mu\alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} > 0, \alpha_{2}\theta_{1} < 0, \\ - \alpha_{1} + \vert \theta_{2} \vert + \mu\alpha_{1}\theta _{2}r_{2} + \alpha_{2}\theta_{1}r_{1} & \mbox{if } \alpha_{1}\theta_{2} > 0, \alpha_{2}\theta_{1} > 0. \end{cases}\displaystyle \end{aligned}$$

(21)

Then it is clearly obtained that \(m < \beta_{2}(b) < M\) for \(\forall b \in IR\). □

### Theorem 7

*If*
\(\frac{\alpha_{2}\theta_{1} - \alpha_{1}\theta _{2} + \alpha_{2}\beta_{1}}{\alpha_{1}} < \beta_{2} < m\)
*is satisfied*, *then system* (1)*–*(2) *is stable*.

### Proof

From equation (9) and Proposition 1, the region which is determined by \(\frac{\alpha_{2}\theta_{1} - \alpha_{1}\theta_{2} + \alpha_{2}\beta_{1}}{\alpha_{1}} < \beta_{2} < m\) does not include D-curves in (\(\beta_{1} - \beta_{2}\)) parameter space.

In order to prove that on each region the characteristic equation of the system has the same number of roots with positive real parts, let us suppose that in a region there were two points for which the numbers of roots with positive real parts are different. Then along any arc within that region connecting these points there must be a point where some of the roots of the characteristic equation lie on the imaginary axis. This point must lie on the D-curves, giving a contradiction. As a result, the number of roots of the characteristic equation with positive real part does not change in the region. Thus, it is sufficient to show that it is stable for specific values which satisfy the condition of the theorem. Using Stepan’s formula (6) with \(\alpha_{1} = 0.2\), \(\alpha_{2} = 0.2\), \(\beta_{1} = - 5\), \(\beta_{2} = - 2\), \(\theta_{1} = 1.25\), \(\theta_{2} = 1\), \(r_{1} = 1\), \(r_{2} = 2\), it is shown that the characteristic equation of the system has no root with apositive real part, which implies the stability of the system (1)–(2). □

### Theorem 8

*Suppose that one of the following conditions holds*:

$$\begin{aligned} (\mathrm{A}1) &\quad M < \beta_{2} < \frac{\alpha_{2}\theta_{1} - \alpha_{1}\theta_{2} + \alpha_{2}\beta_{1}}{\alpha_{1}}; \\ (\mathrm{A}2) &\quad M < \beta_{2}\quad\textit{and}\quad \frac{\alpha_{2}\theta_{1} - \alpha_{1}\theta_{2} + \alpha_{2}\beta_{1}}{\alpha_{1}} < \beta_{2}; \\ (\mathrm{A}3) &\quad \beta_{2} < m\quad\textit{and}\quad \beta_{2} < \frac{\alpha_{2}\theta _{1} - \alpha_{1}\theta_{2} + \alpha_{2}\beta_{1}}{\alpha_{1}}; \end{aligned}$$

*then system* (1)*–*(2) *is unstable*.

### Proof

The proof is similar to the proof of Theorem 7. Figure 2 can be used to find the specific values that satisfy the conditions. □

### Proposition 2

*Let*
\(\mu= \sup(\frac{ - \sin x}{x})\). *If*
\(0 \le \theta_{2} \le\frac{1}{r_{2}\mu} \)
*and*
\(\alpha_{2} > 0\), *then there exists a real number*
*N*
*such that*
\(\beta_{1}(b) < N\)
*for*
\(\forall b \in IR\).

### Proof

It follows from Lemma 1 and equation (18) that

$$\beta_{1}(b) < - \frac{\alpha_{2}\theta_{1}\cos(br_{1}) + \alpha_{1}^{2}}{\alpha_{2}} - \frac{\alpha_{1}^{2}\theta_{2}\sin (br_{2})}{b\alpha_{2}} + \frac{\theta_{1}\alpha_{1}\sin(br_{1})}{b}, $$

holds for \(\forall b \in IR\). If *N* is defined as follows:

$$N = \textstyle\begin{cases} \frac{\alpha_{2} \vert \theta_{1} \vert + \alpha_{1}^{2} + \mu\alpha_{1}^{2}\theta_{2}r_{2}}{\alpha_{2}} + \theta_{1}\alpha_{1}r_{1} & \mbox{if }\theta_{1}\alpha_{1} > 0, \\ \frac{\alpha_{2} \vert \theta_{1} \vert + \alpha_{1}^{2} + \mu \alpha_{1}^{2}\theta_{2}r_{2}}{\alpha_{2}} - \mu\theta_{1}\alpha_{1}r_{1} & \mbox{if }\theta_{1}\alpha_{1} < 0, \end{cases} $$

then it is obtained that \(\beta_{1}(b) < N\) for \(\forall b \in IR\). □

### Theorem 9

*System* (1)*–*(2) *is unstable if*
\(N < \beta_{1}\)
*holds*.

### Proof

The proof is similar to the proof of Theorem 7. Figure 2 can be used to find the specific values that satisfy the conditions. □