# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
## 6 LCO Analysis of Time Integrations with the SSI Scheme  
### LCO Results under an Airspeed of 20 m/s  
In this section, we seek to find all possible LCO solutions using time integration with the proposed SSI scheme. The SSI scheme is introduced in Section 3. First, an airspeed of \(U\) = 20 m/s and the amount of freeplay 2\(\delta\) = 0.1\({}^{\circ}\) are prescribed. A Poincare section is specified as \(\Sigma\): \(x_{2}=\alpha=\delta\), and the crossing of system states \(\mathbf{x}(t)\) to \(\Sigma\) is valid only if the crossing direction follows \(x_{4}=\dot{\alpha}>0\). Second, \(N_{T}\)-time return mapping \(F\): \(\mathbf{x}_{0}^{\{n+1\}}=F(\mathbf{x}_{0}^{\{n\}})\) is introduced with \(N_{T}\) = 11, where \(\mathbf{x}_{0}^{\{n\}}\) and \(\mathbf{x}_{0}^{\{n+1\}}\) are both at \(\Sigma\). Our final goal is to find all fixed points \(\mathbf{x}_{L}\) of \(F\) that actually represent the LCO solutions of the system. To do so, a set \(\mathbf{X}_{0}^{\{1\}}\subset\mathbb{R}^{n_{x}-1}\) is specified as follows:  
\[\mathbf{x}_{0}^{\{1\}}=\{\mathbf{x}_{0}|\;\mathbf{x}_{0}\;\mathrm{s}.t.\left\{ \begin{array}{r@{\quad}l}-0.1\leq\quad&x_{0,1}=h_{0}\quad&\leq 0.1\\ \quad&x_{0,2}=\alpha_{0}\quad&\equiv\delta\\ -0.1\leq\quad&x_{0,3}=\dot{h}_{0}\quad&\leq 0.1\\ \quad&0<x_{0,4}=\dot{\alpha}_{0}\quad&\leq 30.0\\ -0.1\leq\quad&x_{0,5}=r_{1}\quad&\leq 0.1\\ -0.1\leq\quad&x_{0,6}=r_{2}\quad&\leq 0.1\\ \quad\end{array} \right\} \tag{15}\]  
Figure 17: A close-up of Figure 16 which focuses on the airspeed from 12.4 to 13.8 m/s. Random switching between the 2-D LCO and convergence of the system responses can be observed as the initial pitch angle increases. Now, we need to check if the initial set \(\mathbf{X}_{0}^{\{1\}}\) specified in Equation (15) is contained in the basins of attraction \(\overline{U}_{L}\). Note that any initial plunge \(x_{0,1}^{\{1\}}\in X_{0,1}^{\{1\}}=[-0.1,0.1]\) will still remain in \(x_{0,1}^{\{1\}}\) after the mapping--that is, \(x_{0,1}^{\{2\}}\in X_{0,1}^{\{1\}}\). Similar situations are seen in the other system states. Therefore, we confirm that for any \(\mathbf{x}_{0}^{\{1\}}\in\mathbf{X}_{0}^{\{1\}}\), \(\mathbf{x}_{0}^{\{2\}}=F(\mathbf{x}_{0}^{\{1\}})\), which means that \(\mathbf{X}_{0}^{\{1\}}\subseteq\overline{U}_{L}\). Thus, we know that there must be some fixed points of \(F\) hidden inside region \(\mathbf{X}_{0}^{\{1\}}\). To determine the fixed points, add a line \(l_{\text{equ}}\) that represents \(x_{0,1}^{\{1\}}=x_{0,1}^{\{2\}}\) in Figure 18, and find the intersection region \(L_{0}=R_{0}\cap l_{\text{equ}}\), where region \(R_{0}\) consists of \(N_{S}\) points. Then, let \(X_{0,1}^{\{2\}}\) be the range of \(L_{0}\) projected on \(x_{0,1}^{\{1\}}\), as shown in Figure 20; that is, \(X_{0,1}^{\{2\}}=[-7.34\times 10^{-4},5.42\times 10^{-3}]\subseteq X_{0,1}^{\{1\}}\). A huge contraction is seen in \(X_{0,1}^{\{1\}}\to X_{0,1}^{\{2\}}\). Similar procedures are implemented to states \(x_{0,3}\), \(x_{0,4}\), \(x_{0,5}\), and \(x_{0,6}\). Finally, we have \(\mathbf{X}_{0}^{\{2\}}\subseteq\mathbf{X}_{0}^{\{1\}}\subseteq\overline{U}_{L}\). The procedure of \(\mathbf{X}_{0}^{\{1\}}\) contracting to \(\mathbf{X}_{0}^{\{2\}}\) is called \"Iteration-1\". Then, by repeating the procedures above, we can obtain \(\mathbf{X}_{0}^{\{3\}}\) from \"Iteration-2\", \(\mathbf{X}_{0}^{\{4\}}\) from \"Iteration-3\", and \(\mathbf{X}_{0}^{\{5\}}\) from \"Iteration-4\". For the present wing section model, four iterations are sufficient to find the LCO solutions accurately because further implementation of the iterations will show that \(\mathbf{X}_{0}^{\{5\}}\approx\)\(\mathbf{X}_{0}^{\{6\}}\approx\)\(\mathbf{X}_{0}^{\{7\}}\approx\cdots\). Table 1 presents the range of \(X_{0,j}^{\{n\}}\) in each \"Iteration-\(n\)\" for each system state \(x_{j}\), while Figure 21 illustrates the first four iterations. Figure 20: Illustration of the range of \(X_{0,j}^{\{n\}}\) being contracted to \(X_{0,j}^{\{n+1\}}\) by means of the spatial pattern. The airspeed is 20 m/s.