## Appendix A Proofs of Theorems and Corollaries
Therefore, there exists interval \(I_{k\psi}=[h_{k}(\psi_{k2}),h_{k}(\psi_{k1})]\supset[\psi_{k1},\psi_{k2}]\) such that \(h_{k}(\psi_{k2})<\psi_{k1}<\psi_{k2}<h_{k}(\psi_{k1})\). Furthermore, in the interval \([h_{k}(\psi_{k2}),\psi_{k1}]\subset I_{k\psi}\), \(h_{k}\) has a positive slope \(Dh_{k1}\) such as \(0<Dh_{k1}<1\). In the interval \([\psi_{k1},\psi_{k2}]\subset I_{k\psi}\), \(h_{k}\) has negative slope \(Dh_{k2}\) such as \(Dh_{k2}<-1\). In the interval \([\psi_{k2},h_{T}(\psi_{k1})]\subset I_{k\psi}\), \(h_{k}\) has positive slope \(Dh_{k3}\) such as \(0<Dh_{k3}<1\). Then, \(h_{k}(\psi_{k1})-h_{k}(h_{k}(\psi_{k1}))=h_{k}(\psi_{k1})-(h_{k}(\psi_{k2})+Dh _{k3}(h_{k}(\psi_{k1})-\psi_{k2}))>h_{k}(\psi_{k1})-(h_{k}(\psi_{k2})+(h_{k}( \psi_{k1})-\psi_{k2}))=\psi_{k2}-h_{k}(\psi_{k2})>0\) holds because \(h_{k}(h_{k}(\psi_{k1}))=h_{k}(\psi_{k2})+Dh_{k3}(h_{k}(\psi_{k1})-\psi_{k2})< h_{k}(\psi_{k2})+(h_{k}(\psi_{k1})-\psi_{k2})\) holds. In the similar way above, \(h_{k}(h_{k}(\psi_{k2}))-h_{k}(\psi_{k2})=h_{k}(\psi_{k1})-Dh_{k1}(\psi_{k1}-h _{k}(\psi_{k2}))-h_{k}(\psi_{k2})>h_{k}(\psi_{k1})-(\psi_{k1}-h_{k}(\psi_{k2}) )-h_{k}(\psi_{k2})=h_{T}(\psi_{k1})-y_{k1}>0\) holds because \(h_{k}(h_{k}(\psi_{k2}))=h_{k}(\psi_{k1})-Dh_{k1}(\psi_{k1}-h_{k}(\psi_{k2}))>h_{ k}(\psi_{k1})-(\psi_{k1}-h_{k}(\psi_{k2}))\) holds. Therefore, it is confirmed that \(h_{k}(\psi_{k1})=\max_{x\in I_{k\psi}}h_{k}(x)\) and \(h_{k}(\psi_{k2})=\min_{x\in I_{k\psi}}h_{k}(x)\). Thus \(h_{T}(I_{k\psi})\subset I_{k\psi}\) holds. Item 2 is proved. Finally for any \(\psi_{0}\in I_{k\psi}\), clearly \(h_{k}(\psi_{0})\in I_{k\psi}\). for any \(\psi_{0}\in(-\infty,\psi_{k1})\), \(h_{k}(\psi_{0})>\psi_{0}\) from Equation (A1) or Equation (A3). \(h_{k}^{j}(\psi_{0})\in(-\infty,\psi_{k1}),j(>0)\in\mathcal{Z}\) is a monotonically increasing sequence. Therefore, for any \(\psi_{0}\in(-\infty,\psi_{k1})\), there exists \(j\)th iterate of \(h_{k}\) such that \(h_{k}^{j}(\psi_{0})\in I_{k\psi},j(>0)\in\mathcal{Z}\). In the similar way for any \(\psi_{0}\in(\psi_{k2},\infty)\), \(h_{k}(\psi_{0})<\psi_{0}\) from Equation (A2) or Equation (A4). \(h_{k}^{j}(\psi_{0})\in(\psi_{k2},\infty),j(>0)\in\mathcal{Z}\) is a monotonically decreasing sequence. Therefore, for any \(\psi_{0}\in(\psi_{k2},\infty)\), there exists \(j\)th iterate of \(h_{k}\) such that \(h_{k}^{j}(\psi_{0})\in I_{k\psi},j(>0)\in\mathcal{Z}\). Item 3 is proved. The proof is now complete. Proof of Theorem 3.: To prove theorem 3, we follow [PERSON] or [PERSON]'s line for \"\(\varphi\)_has a homoclinic point_\" implying Definition 1 in [15; 21]. First of all, we introduce the following definitions, propositions, and notations that are used throughout the proof:  
Definition A1 (Devaney [15]). : \(\Sigma_{2}=\{s=(s_{0}s_{1}s_{2}\ldots)|\,s_{j}=0\,\,or\,1\}\). \(\Sigma_{2}\) is called the _sequence space_ on the two symbols \(0\) and \(1\). Elements of \(\Sigma_{2}\) are infinite sequences of integers, such as \((000\ldots)\) or \((0101\ldots)\). We may make \(\Sigma_{2}\) into a metric space as follows. For two sequences \(s=(s_{0}s_{1}s_{2}\ldots)\) and \(t=(t_{0}t_{1}t_{2}\ldots)\), define the distance between them by \(d(s,t)=\sum_{i=0}^{\infty}\frac{|s_{i}-t_{i}|}{2}\). Since \(|s_{i}-t_{i}|\) is either \(0\) or \(1\), this infinite series is dominated by the geometric series \(\sum_{i=0}^{\infty}\frac{1}{2^{i}}=2\); therefore, it converges. Definition A2 (Devaney [15]). : The shift map \(\sigma:\Sigma_{2}\rightarrow\Sigma_{2}\) is given by \(\sigma(s_{0}s_{1}s_{2}\ldots)=(s_{1}s_{2}s_{3}\ldots)\). The shift map simply \"forgets\" the first entry in a sequence, and shifts all other entries one place to the left. Clearly, \(\sigma\) is a two-to-one map of \(\Sigma_{2}\), as \(s_{0}\) may be either \(0\) or \(1\). Moreover, in the metric defined above, \(\sigma\) is a continuous map. **Proposition A1** ([PERSON] [15]). : \(d(s,t)\) _is a metric on \(\Sigma_{2}\)._  
**Proposition A2** ([PERSON] [15]). : _Let \(s,t\in\Sigma_{2}\) and suppose \(s_{i}=t_{i}\) for \(i=0,1,\ldots,n\).