# Synchronization of Monostatic Radar Using a Time-Delayed Chaos-Based FM Waveform
## 5 Design of Time-Delay Chaotic Systems  
In this study, chaotic behavior in a dynamical system with a single state variable and a minimum number of elements is addressed in detail using a simple time-delay chaotic system in Lur'e form. The Lur'e system consists of three elements, namely time-delay, an integrator, and non-linearity, respectively, as shown in Figure 4. The time-delay affects the dynamic of the systems, such as single state time-delay, which makes the system infinite-dimensional, the integrator officiates the system dynamics of a single state, and the non-linearity is a necessary part and is wanted as a major, so chaos is a distinguishing characteristic of nonlinear systems [30; 31]. Consequently, the dimensionless equation of the Lur'e system can be expressed as follows  
\[\dot{x}(t)=k[-ax(t)-bF(x_{\tau})] \tag{1}\]  
where \(x(t)\in\mathbb{R}\) is the state variable, \(x_{\tau}=x(t-\tau)\), \(F(\,\cdot\,):\mathbb{R}\rightarrow\mathbb{R}\) is a non-linear function, \(\tau\in\mathbb{R}^{+}\) is the time-delay, \(k\) is the scaling factor, and \(a\) and \(b\) are system parameters. the non-linearity is multiplied by the gain \(b\), whilst the pole of the Miller integrator is \(ka\). The Lur'e form in system (1) consists of two parts: the linear part given by \(L(s)=\frac{ke^{-sr}}{s+ka}\) and the feedback nonlinear part given by \(\mathcal{N}=bh(x)\)[30; 32]. To observe the chaos from a Lur'e system, three conditions must be satisfied: (a) the existence of a stable predicted limit cycle (PLC) and a separate unstable equilibrium point, (b) the interaction between the stable predicted limit cycle and the unstable equilibrium points, and, finally, (c) the linear part of the Lur'e system must be filtering. In our system, it is always satisfied, since \(L(s)=\frac{ke^{-sr}}{s+ka}\). Due to the piecewise linear nonlinearity under consideration, the condition of the PLC is investigated. The harmonic balance approach [32] can be used to determine the existence of stable PLC. A limit cycle (LC) solution of type \(Y_{0}(t)=A\text{sin}(\omega t)\) exists if the amplitude \(A\) of \(Y_{0}\) satisfies the following equation:  
\[1+L(j\omega)N(A)=0 \tag{2}\]  
where \(N\) is the static describing the function of nonlinearity \(\mathcal{N}\). The intersection points between the curve \(L(j\omega)\) and the curve \(-1/N(A)\) in the complex plane can be used to determine the solution to the system (2). Each intersection represents LC at a specific frequency and amplitude. The LC is considered unstable if the intersection points along the curve \(-1/N(A)\) are encircled by the curve \(L(j\omega)\). Otherwise, the LC is stable. The existence of unstable equilibrium points that are separate from the PLC is required by the second condition. With a double scroll, the chaotic attractor can be generated from the system (1), and the PWL function is defined by  
\[F(x_{\tau})=x_{\tau}-\text{sgn}[x_{\tau}] \tag{3}\]  
Figure 4: The Lur’e system’s structure. Hence, a sawtooth wave can be seen in the PWL function in the system (3). Now, using systems (1) and (3), the following equation can be used to locate equilibrium points:  
\[k[-ax(t)-bF(x_{\tau})]=0 \tag{4}\]  
As shown in Figure 5, the three equilibrium points \(\pm x_{i}^{e}\)\((i=0,1)\) are located at the intersection of \(F(x_{\tau})\) and \(x\). Only the situation \(x\geq 0\) needs to be considered because of the symmetry of the function \(F(x_{\tau})\). Obviously, based on the linear regions of the characteristic equation, the system (1) with (3) can be classified into many linear regions \(\pm D_{i}(i=1)\). From the symmetry of the function \(F(x)\), one only needs to consider the case \(x\geq 0\). Obviously, the system (1) with (3) can be classified into several linear regions (\(\pm D_{i}\)\((i=1)\)) based on the linear regions of the characteristic equation. Figure 6 depicts the bifurcation diagram of this system with respect to \(\tau\).