# Synchronization of Monostatic Radar Using a Time-Delayed Chaos-Based FM Waveform
## 5 Design of Time-Delay Chaotic Systems
The equilibrium points (zeroes of PWL functions defined by Equation (3)) are stable for \(\tau<0.45\) ms, and the trajectory of the system (1) converges with it, with no PLC occurring in this case. When the time delay (\(\tau\)) is increased until it reaches \(\tau\geq 0.45\) ms, chaos occurs, because the LC is expected and begins to interact with unstable equilibrium points [30]. Notably, this time-delayed system was successfully implemented using OrCAD PSpice Designer and illustrated for different multi-scrolls by [PERSON] et.al. in [30]. Figure 5: \(F(x_{\tau})\) and its characteristic regions of chaotic 2-scroll attractor. Figure 6: The bifurcation diagram of the system (1) with nonlinearity (3) as a function of time delay \(\tau\) for system parameters \(a=1\), \(b=2.6\), and \(k=1800\). Through the graphical solution (2), the dynamic of the system is represented in terms of the conditions for the existence of chaos for \(\tau=0.6\) ms, \(a=1\), and \(k=1500\), and the Nyquist plot of \(L(j\omega)\) intersects the curve \(-1/N(A)\), as shown in Figure 7. It leads to a stable PLC and the system behaves chaotically. Except for the equilibrium points, any number in the basin of attraction can be used as the initial condition for the system (1). The parameter values of the system (1) can be selected as \(a=1\), \(b=3\), has a chaotic attractor with PWL nonlinearity (3), and \(x(0)=0.5\) and \(\tau=0.6\) ms, as shown in Figure 8. In fact, the phase trajectory over time is clearly shown in this figure, where a chaotic signal can introduce various attractors once the time delay is changed [33]. Similarly, in [34], [PERSON] et al. showed that a chaotic oscillator utilizing a flux-controlled memristor can also produce various signals with increasing amplitude and frequency for the broad requirements in engineering applications. Accordingly, system (1) with the selected parameter values \(a=1\), \(b=2\), and \(k=1800\), and \(x(0)=-1\) can also produce the delay-dependent chaotic attractor with double scrolls at the time delay \(\tau^{*}\geq 0.45\) ms. It means that \(\tau\) has a significant effect once this delay has been increased, leading to different delay-dependent double-scroll attractors. Hence, the more complex chaotic behavior of \(\tau\geq 0.6\) ms indicates the more complex attractors generated with different amplitude and frequency in order to meet more secure communication, especially in chaotic radar applications [33]. \[\dot{x}(t)=k[-ax(t)-bF(x_{\tau})] \tag{1}\]0 Chaotic attractors for \(x(0)=0.5\), \(\tau=0.6\) ms, \(a=1\), \(b=3\), and \(k=1500\). (a) Time delay \(\tau=0.6\) ms, (b) time delay \(\tau=0.5\) ms.  
\[\dot{x}(t)=k[-ax(t)-bF(x_{\tau})] \tag{1}\]1 A graph of the equation’s solution (2). The dashed line is the function \(-1/N(A)\), whereas the continuous curve is a Nyquist plot of \(L(j\omega)\) with \(\tau=0.6\) ms. Note that the output signal obtained from the dynamical Lur'e system in Figure 4 is characteristic of the broad band chaotic spectrum. The Fast Fourier Transformer (FFT) is applied on the chaotic signal. Then, the power spectrum of the output signal \(x(t)\) can be obtained, as shown in the Figure 9, for the time delay \(\tau=0.6\) ms and sampling frequency \(f_{\rm s}=4\) MHz. This base band spectrum deploys below the frequency range of 3 kHz. Thus, the chaotic signal is classified with a Very-Low Frequency (VLF) chaotic band, and such a feature allows direct long-wave chaotic radar system to be built for anti-stealth applications [2].