# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
## 3 State Space Iterating (SSI) Scheme
### LCO Analysis Using the SSI Scheme  
As explained in Section 3.2, an LCO is a fixed point \(\mathbf{x}_{L}\) of the mapping \(F^{N_{T}}\): \(\mathbf{x}_{L}=F^{N_{T}}(\mathbf{x}_{L})\), from which we can find the fixed point simply by visual inspection. A line \(l_{\text{equ}}\) in the iteration plot for \(x_{j}\), which has a unit slope and indicates that \(l_{\text{equ}}\): \(x_{0,j}^{*}=x_{0,j}\), as shown in Figure 7a,b. Assume that \(N_{S}\) points on the iteration plot form a perfect line \(l_{0}\), which intersects \(l_{\text{equ}}\) at Point \(A\); then, \(A\) provides the \(j\)-th component of \(\mathbf{x}_{L}\): \(x_{L,j}=x_{0,j}\big{|}A=x_{0,j}^{*}\big{|}_{A}\). If a fixed point can be found at all iteration plots, then at those fixed points, all system states will behave as periodic motions, and an LCO is determined. Stability can also be determined visually. For simplicity, we use \(F\) to represent the mapping instead of \(F^{N_{T}}\). We define the slope \(F^{\prime}_{j}\) as \(F^{\prime}_{j}=\text{d}F_{j}/\text{d}x_{0,j}\), where \(F_{j}\) is the \(j\)-th  
Figure 6: Illustration of an iteration plot, in which point \(A\) represents how a state behaves when the system evolves from the upper freeplay boundary and returns to that boundary after crossing it several times. Figure 7: Illustration of the \(j\)-th iteration plot, where the spatial pattern indicates the occurrence of a limit cycle oscillation (LCO). The LCO is (**a**) stable or (**b**) unstable with respect to any perturbation in the \(j\)-th state. component of the mapping \(x_{0,j}^{*}=F_{j}(x_{0,j})\). An LCO is unstable if \(\left|F^{\prime}_{j}\right|>1\). We can prove this by linearizing \(F_{j}\) near a fixed point. Suppose that the fixed point is at \(x_{0,j}=x_{0,j}^{*}=\overline{x}\). A small perturbation \(\delta x_{0,j}=x_{0,j}-\overline{x}\) is given as the input of the mapping. Then,  
\[x_{0,j}^{*}=F_{j}(x_{0,j})\approx F_{j}(\overline{x})+F^{\prime}_{j}(\overline {x})\cdot(\overline{x}-x_{0,j}) \tag{9}\]  
which means that  
\[F_{j}(x_{0,j})-F_{j}(\overline{x})\approx F^{\prime}_{j}(\overline{x})\cdot( \overline{x}-x_{0,j}) \tag{10}\]  
or  
\[\left|\delta F_{j}\right|\approx\left|F^{\prime}_{j}(x_{0,j})\right|\cdot \left|\delta x_{0,j}\right| \tag{11}\]  
As time progresses, the mapping will be repeated over and over again a \(\mathrm{s}\ x_{0,j}^{*}= F_{j}(F_{j}(\ldots F_{j}(x_{0,j})\ldots))\), finally yielding:  
\[\lim_{t\to\infty}\left|\delta F_{j}\right|=\lim_{N\to\infty}\left|F^{\prime} _{j}(x_{0,j})\right|^{N}\cdot\left|\delta x_{0,j}\right|\to+\infty \tag{12}\]  
which indicates that the LCO is unstable. Conversely, if \(\left|F^{\prime}_{j}\right|<1\), the LCO is stable. Figure 7 illustrates situations of both a stable and an unstable LCO. For simplicity, only one \(l_{\mathrm{equ}}\) representing \(F^{\prime}_{j}=1\) is plotted. However, to find fixed points may not be easy, since the patterns shown in the iteration plots may be complicated. As shown in Figure 8, the \(N_{S}\) points form a range \(R_{0}\) instead of the \(l_{0}\) in Figure 7; therefore, there is no obvious fixed point that can be found. Presumably, a fixed point is hidden inside of the intersection line \(L_{0}\): \(L_{0}=R_{0}\cap l_{\mathrm{equ}}\). Here, we introduce an iterating procedure to gradually find the fixed point hidden in \(L_{0}\), which is the SSI scheme proposed in this paper. Figure 8: Illustration of the \(j\)-th iteration plot, where the spatial pattern indicates the occurrence of an LCO but the effects of transient phases and noises are also taken into consideration, so that the distributing region of all points is a belt region \(R_{0}\) rather than a line, as shown in Figure 3.  
First, for each state \(x_{j}\), specify a range \(X_{0,j}^{\{n\}}\) and select \(N_{S}\) samples \(x_{0,j}\) randomly from that range (\(x_{0,j}\in X_{0,j}^{\{n\}}\)). By combining all \(X_{0,j}^{\{n\}}\) values, we obtain a set \(\mathbf{X}_{0}^{\{n\}}\in\mathbb{R}^{n_{x}-1}\) that contains all \(N_{S}\) I.C.