# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
## 7 Conclusions  
This paper proposed a state space iterating scheme (SSI) to find LCO solutions via a visualization procedure. Inspired by the Poincare map and the Lorenz map, an \(N_{T}\)-time return mapping \(F\) was defined and visualized as a set of iteration plots, in which the LCO solutions were represented as the fixed points \(\mathbf{x}_{L}\) of \(F\) and were found within a range \(\mathbf{X}_{0}\) that continuously contracts through a series of iterations: \(\mathbf{X}_{0}^{\{1\}}\supseteq\mathbf{X}_{0}^{\{2\}}\supseteq\cdots\supseteq \mathbf{X}_{0}^{\{n\}}\). When the range \(\mathbf{X}_{0}^{\{n\}}\) is sufficiently contracted and the spatial patterns shown in the iteration  
Figure 24: Comparison of LCO non-dimensional amplitudes varying with airspeed between the state space iterating (SSI) scheme and the time integrations with the Hénon–RK45 method.  
Figure 25: Comparison of the LCO dominant frequencies varying with airspeed between the SSI scheme and the time integrations with the Hénon–RK45 method.  
plots are clear enough, the fixed points can be easily distinguished via a visual inspection, and thus, LCOs can be found.  
To verify the SSI scheme, a typical plunge-pitch wing section with pitching freeplay was established. Time integrations with a combined fourth-fifth-order Rung-Kutta method and the Henon method (Henon-RK45), as well as the proposed SSI scheme, were applied to the model. Both methods determined a three-domain LCO (3-D LCO) and a two-domain LCO (2-D LCO) under a wide range of flow velocities. The SSI scheme confirmed the minimal \(\mathbf{X}_{0}^{\{n\}}\) at the fourth iteration and ultimately found the two LCOs \(\mathbf{x}_{L}\in\mathbf{X}_{0}^{\{4\}}\) simply through a visual inspection of the iteration plots. The LCO results obtained by SSI were found to be well aligned with those of the Henon-RK45 method, as shown in Figures 24 and 25.  
The highlights of the study are as follows: (1) To obtain the LCO results, Henon-RK45 checked 100 initial pitch angles for each of the 73 airspeed cases, without guaranteeing that any other arbitrary initial conditions (I.C.s) would lead to the same LCO results, while SSI used 1000 arbitrary I.C.s for each of the 18 airspeed cases, uncovered clear spatial patterns of the system responses, and determined the basins of attraction of the system, which makes the LCO results suitable for any I.C.; (2) the spatial patterns depicted by the SSI scheme could be easily utilized in future studies on other nonlinear responses, such as quasi-periodic motions and chaotic motions, which are usually analyzed via Poincare sections; (3) the SSI scheme can be easily extended to any other kind of structural nonlinearities and applied to an aeroelastic system with higher dimensions.  
However, further studies on SSI are needed, including its application to points (2) and (3) above. We must also determine a better method for selecting arbitrary I.C.s and the parameter \(N_{T}\), which is currently part of our ongoing research.  
Conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization and writing (original draft) were conducted by [PERSON] Project administration was conducted by [PERSON] Funding acquisition was conducted by [PERSON] Writing (review & editing) was conducted by both [PERSON] and [PERSON]. Resources and supervision were conducted by both [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
Both this research and the APC was funded by the National Key Research and Development Program of China (grant number 2017 YFB0503002).  
Not applicable.  
Data Availability Statement: Data is contained within the article or Appendix A.  
This work was supported in part by the National Key Research and Development Program of China under Grant 2017 YFB0503002.  
The authors declare no conflict of interest.  
Parameters of the Wing Section Model  
This appendix provides the parameters and matrices of the numeric model in Section 3.1.