## 2 Noise Reduction Mechanism and Frequency Estimation Method Based on the Coupled Lorenz System
### Selection and Configuration of Chaotic Model  
Most of the existing studies on weak signal detection are based on the Duffing oscillator system. This kind of system contains a periodic dynamic force, whose frequency must be consistent with the tested signal, and it runs at the critical point of the state transition. These requirements greatly increase the complexity of the system and such that it cannot be applied to situations where there are several periodic signals with different frequencies or the tested signal with an unknown frequency.  
Figure 1: The principle block diagram of the detection system.  
The Lorenz system is a classic three-dimensional chaotic system that was first studied by [PERSON] in 1963 [20]. This system has the advantages of a simple structure and easy construction and has chaotic solutions under some parameter values and initial conditions. Therefore, we studied the method of detecting the frequency of weak signal using the Lorenz system. Its mathematical model is as follows:  
\[\begin{cases}\dfrac{dx}{dt}=\sigma(y-x)\\ \dfrac{dy}{dt}=x(\rho-z)-y+r(t).\\ \dfrac{dz}{dt}=xy-\beta z\end{cases} \tag{2}\]  
The \(r(t)\) is the test signal, which does not exist in the standard Lorenz system. It changes the model (Equation (2)) into a higher-dimensional dynamical system. We studied the dynamic characteristics of this system by setting its structural parameters (\(\sigma=10\), \(\beta=2.6\)) and obtained the bifurcation diagram of the parameter \(\rho\), as shown in Figure 2.  
This figure can be divided into five intervals, in which the critical points are \(\rho=24\), \(150\), \(169\) and \(215\). Therefore, the running state of the chaotic system goes through the multi-periodic state and chaotic state as the parameter \(\rho\) increases. Furthermore, we made the system run in different states by changing the value of \(\rho\), and observed the changes in the phase space trajectory with different signal inputs \(r(t)\), as shown in Figure 3.  
In Figure 3a, \(\rho=20\), and the system runs in a multi-periodic state, while in Figure 3b,c, \(\rho=50\), and the system runs in chaotic state. In the above figure, curve \(a\) is obtained when \(r\) (\(t)=0\), curve \(b\) is obtained when \(r\) (\(t)=0.001\sin\) (\(t\)), curve \(c\) is obtained when \(r(t)\) is a white noise with a variance of 0.001, and curve \(d\) is obtained when \(r(t)\) is composed of the above sinusoidal signal and the noise signal.  
It can be seen that the four curves almost coincide with each other and are hard to distinguish in Figure 3a. Correspondingly, in Figure 3b,c, curves \(a\) and \(c\) are almost coincident, and curves \(b\) and  
Figure 3: The phase space trajectory diagram of the Lorenz system running in multi-periodic state (**a**) and chaotic state (**b**) and (**c**).  
Figure 2: Bifurcation diagram of the parameter \(\rho\) of the Lorenz system.  
\(d\) are also very close to each other, but there are great differences between the two groups of curves. This figure shows that the phase space trajectory will only swing slightly under the influence of noise, but will produce a substantial offset under the influence of a periodic signal, and the offset value is far greater than the amplitude of the periodic signal when the system is running in a chaotic state. Meanwhile, the values of state variables are bounded, which is conducive to the detection of a weak signal in strong noise. Therefore, we use the chaotic synchronization system based on the Lorenz system to obtain the change value of the phase space trajectory to achieve noise reduction.