## 3 Effect of Linear Damping on System Stability
### No Dimensional Damping Coefficient Changes  
Equation (2) is an iterative equation. We introduce the concept of the number of iteration cycles \(M\) here, which represents the time it takes for the system to transition from a resting state to an investigative state. When given that \(M=500\) iteration cycles and the initial values \((x_{0},y_{0})=(0,0)\), with \(r\) equal to \(0.001\), \(0.003\), and \(0.005\), respectively, Figure 2 shows the phase trajectory diagrams and Poincare cross-sections of system (2).  
After analyzing Figure 3, it can be observed that for a given cycle number \(M=500\), the motion stability of system (2) gradually increases with an increase in the damping coefficient. It is worth noting that even though the error during the cycle in the phase trajectory diagram of system (2) is already tiny when \(r\) equals \(0.003\), which cannot be identified solely based on the phase trajectory diagram and requirescross-section, for micro- and nano-measurements using TM-AFM, this tiny error will be amplified by the optical lever principle and detected by the photoelectric displacement sensor, resulting in increased measurement errors.  
To further analyze the evolution law of system (2)'s stability as the damping coefficient changes, with \(r\) being taken as the independent variable, Figure 4 shows the bifurcation diagram of system (2) when \(r\in(0,\,0.01)\).  
It can also be observed from Figure 4 that as \(r\) gradually increases, the number of bifurcations in the bifurcation diagram decreases gradually and consequently enhances the motion stability of system (2).  
Figure 4: \(r\in(0,\,0.01)\), bifurcation diagram of system (2).  
Figure 3: Phase trajectories and Poincaré cross sections of system (2) with different r values: (**a,d**) \(r=0.001\), system (2) exhibits an extremely unstable state of motion; (**b,e**) \(r=0.003\), system (2) shows a motion state of period 19, and its stability is improved; (**c,f**) \(r=0.005\), the motion state of system (2) is stable and the motion is period 1.  
The iterative cycle is repeated 500 times, with the initial value point set as \((x_{0},\,y_{0})=(0.0)\), \(r\) ranging from 0.001 to 0.008 with a step size of 0.001. The attractive domains of system (2) in the \(x\in(-0.5,\,0.5),\,y\in(-0.5,\,0.5)\) space as shown in Figure 5.  
When \(r=0.005\), although the initial value point is located in the suction basin of period 1 as shown in Figure 5e, the space occupied by the suction basin of period 1 is not large, and there are several ribbon-like quasi-periodic suction basins outside the suction basin of period 1; therefore, system (2) has relatively poor anti-interference ability. Additionally, considering that the variation in \(r\) between Figure 5a-h is 0.001, there is a very fast evolution speed of the attraction domain in space from Figure 5a-e, resulting in obvious changes in the image of the attraction domain; however, from Figure 5f-h, there is a slow evolution speed for the attraction domain in space and almost no change in shape or area for the central part of the attraction basin during period 1. The speed at which the suction basin nears its initial value point evolves and can also be used to evaluate system stability: faster evolution indicates a higher likelihood for changes in where this initial value point lies within its corresponding suction basin and thus suggests an unstable motion state for this system; conversely, slower evolution rates indicate greater stability for both their respective suction basins and the corresponding system motion states.