# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
## 3 State Space Iterating (SSI) Scheme
### Preliminary Concepts
#### 3.1.1 Poincare Map and Lorenz Map  
For a given initial condition (I.C.) \(\mathbf{x}_{0}\), a start time \(t_{0}\), and a time range \(T\), Equation (1) determines a solution \(\mathbf{x}(t)\), where \(t_{0}\leq t\leq t_{0}+T\). Such a solution can be depicted as an  
Figure 1: Illustration of freeplay and three subdomains separated by two freeplay boundaries in state space. orbit \(\Gamma\) in state space, as shown in Figure 2a, where the closed orbit \(\Gamma_{C}\) indicates a periodic solution of the system. The Poincare map was defined by [PERSON] in 1881 [15]. [PERSON] provided good interpretations of the Poincare map in Section 3.4 of his book [16]. Suppose that \(\Sigma\) is a hyperplane perpendicular to a periodic orbit \(\Gamma_{C}\) at \(\mathbf{x}_{0}\); then, for any point \(\mathbf{x}\in\Sigma\) sufficiently near \(\mathbf{x}_{0}\), the orbit \(\Gamma\) through \(\mathbf{x}\) at \(t=0\) will cross \(\Sigma\) again at another point \(P(\mathbf{x})\). The mapping \(\mathbf{x}\to P(\mathbf{x})\) is called a Poincare map, or the first return map. Figure 2a illustrates the Poincare map in state space. If we could find a fixed point \(\mathbf{x}_{L}\) on the Poincare map--that is, \(\mathbf{x}_{L}-P(\mathbf{x}_{L})=0\)--then a periodic solution would be found at \(\mathbf{x}_{L}\) and an LCO could be determined. However, only a few researchers have tried to find the fixed points of a Poincare map. An example can be found in [PERSON]'s paper [17], in which he derived the expression of a Poincare map \(P(\mathbf{x})\) numerically and calculated the fixed points of \(P(\mathbf{x})\). Nevertheless, the results presented by [PERSON] were dependent on an artificial parameter \(\lambda\) that has no physical meaning, as the author stressed. The sophisticated procedures for finding fixed points on a Poincare map could be difficult to apply to a general aeroelastic system. Other applications of Poincare maps are actually based on phase diagrams, upon which a set of points at the Poincare section \(\Sigma\) are projected. This phase diagram consists of many discrete points and is usually simply called a \"Poincare section\". Figure 2b shows a Poincare section derived by [PERSON] [13], based on which the author explained the mechanism of the associated bifurcations and confirmed that the system was likely undergoing chaotic motion. Many examples can be found in Figure 12 in [10], Figure 10 in [18], Figure 13 in [19], and Figure 11 in [20]. The Poincare section provides evidence for the type of nonlinear response (e.g., LCO, quasi-periodic motion, or chaotic motion). However, it cannot aid in time integration for finding LCOs. Instead, it usually plays the role of confirming what has been derived from time integration. The Lorenz map was proposed by [PERSON] [21] while studying his famous three-DoF system from a simplified model of convection rolls in the atmosphere. An excellent introduction on [PERSON]'s system and the Lorenz map is presented by [PERSON] in Section 9 of his book [22]. In [PERSON]'s system, three system states, \(x,y,\) and \(z\), vary with time. To  
Figure 2: Illustration of a Poincaré map on (**a**) state space and (**b**) the Poincaré section. (**a**) is based on Figure 1 in Section 3.4 of [PERSON]’s book, and (**b**) is based on Figure 20 (**a**) in [PERSON] paper. (**b**) shows the normal method for a Poincaré map to be applied on the analysis of nonlinear behaviors in an aeroelastic system. grasp the features of the system, [PERSON] focused only on the connection between the \(n\)-th local maximum \(z_{n}\) and the next one, \(z_{n+1}\), as shown in Figure 3. Then, a map \(z_{n+1}=L(z_{n})\), known as the Lorenz map, was obtained through numerous combinations of \(z_{n}\) and \(z_{n+1}\), as presented in Figure 4. Both Figures 3 and 4 were originally presented by [PERSON] [22]. The Lorenz map (Figure 4) is significant because it reflects the nonlinear pattern of \(z(t)\). Note that there are almost no \"thicknesses\" to the graph in Figure 4, so the pattern of is clear.