## 2 The Models
### A Modified Model
#### 3.1.1 Melnikov's Approach  
By definition, the Melnikov integral is presented by  
\[M(t_{0})=\int_{-\infty}^{\infty}\!\!y_{0}(t)\left(Ay_{0}(t)|y_{0}(t)|^{p-1}-\sum_ {j=1}^{N}\!g_{i}\sin(j\omega(t+t_{0}))\right)dt, \tag{19}\]  
where the functions \(x_{0}(t)\) and \(y_{0}(t)\) are defined by Equation (18). Let us indicate the next statement for fixed \(p=2\) and \(N=1\). **Proposition 4**. : _If \(p=2\) and \(N=1\), then the zeroes of the Melnikov function \(M(t_{0})\) are received as roots of the following equation:_  
Figure 15: (**a**) The solutions of the system; (**b**) \(x\)–time series; (**c**) \(y\)–time series; (**d**) phase space (Example 8). \[-39,440 ie^{2 it\omega\omega}g1\omega^{4}P[0,\tfrac{3}{2}+\tfrac{i \omega}{2}]+7014 ig1\omega^{6}P[0,\tfrac{3}{2}+\tfrac{i\omega}{2}]+7014 ie^{2 it \omega\omega}g1\omega^{6}P[0,\tfrac{3}{2}+\tfrac{i\omega}{2}]\] \[-60 ig1\omega^{8}P[0,\tfrac{3}{2}+\tfrac{i\omega}{2}]-60 ie^{2 it \omega\omega}g1\omega^{8}P[0,\tfrac{3}{2}+\tfrac{i\omega}{2}]-2 ig1\omega^{10}P [0,\tfrac{3}{2}+\tfrac{i\omega}{2}]\] \[-2 ie^{2 it\omega\omega}g1\omega^{10}P[0,\tfrac{3}{2}+\tfrac{i\omega }{2}]+18,432 ig1\omega^{2}P[0,2+\tfrac{i\omega}{2}]+18,432 ie^{2 it\omega\omega}g1 \omega^{2}P[0,2+\tfrac{i\omega}{2}]\] \[+21,800 ig1\omega^{4}P[0,2+\tfrac{i\omega}{2}+21,800 ie^{2 it\omega \omega}g1\omega^{4}P[0,2+\tfrac{i\omega}{2}]+3066 ig1\omega^{6}P[0,2+\tfrac{i \omega}{2}]\] \[+3066 ie^{2 it\omega\omega}g1\omega^{6}P[0,2+\tfrac{i\omega}{2}]-300 i\omega^{2 it\omega\omega}g1\omega^{8}P[0,2+\tfrac{i\omega}{2}]\] \[+2 ig1\omega^{10}P[0,2+\tfrac{i\omega}{2}]+2 ie^{2 it\omega\omega}g1 \omega^{10}P[0,2+\tfrac{i\omega}{2}]-4752 ig1\omega^{2}P[0,\tfrac{5}{2}+\tfrac {i\omega}{2}]\] \[-4752 ie^{2 it\omega}g1\omega^{2}P[0,\tfrac{5}{2}+\tfrac{i\omega}{ 2}]-6360 ig1\omega^{4}P[0,\tfrac{5}{2}+\tfrac{i\omega}{2}]-6360 ie^{2 it\omega \omega}g1\omega^{4}P[0,\tfrac{5}{2}+\tfrac{i\omega}{2}]\] \[-1701 ig1\omega^{6}P[0,\tfrac{5}{2}+\tfrac{i\omega}{2}]-1701i2i^{ 2 it\omega\omega}g1\omega^{6}P[0,\tfrac{5}{2}+\tfrac{i\omega}{2}]-90 ig1\omega ^{8}P[0,\tfrac{5}{2}+\tfrac{i\omega}{2}]\] Figure 15:0 \[+576 ig1\omega^{2}P[0,3+\tfrac{i\omega}{2}]+576 ie^{2 it\omega\omegagenerating functions. Let \(\xi_{a}\) and \(\xi_{b}\) be some random variables exhibiting the corresponding laws. We shall use the symbol \(\mathbb{E}\) for the related expectations. We can rewrite the second part of system (8) as  
\[ \begin{split}&\frac{dy}{dt}=bx-\sum_{i=0}^{\left[\frac{a}{2}\right] -1}b_{i}x^{n-2i}-\epsilon\Bigg{(}A\sum_{p=1}^{k}y|y|^{2p-1}+(1+\cos(\omega t)) \left(ax+\sum_{i=0}^{\left[\frac{a}{2}\right]-1}a_{i}x^{m-2i}\right)\Bigg{)}\\ &=bx-x^{n}\sum_{i=0}^{\left[\frac{a}{2}\right]-1}b_{i}e^{-2i\ln|x |}-\epsilon\Bigg{(}A\sum_{p=1}^{k}y|y|^{2p-1}+(1+\cos(\omega t))\left(ax+x^{m} \sum_{i=0}^{\left[\frac{a}{2}\right]-1}a_{i}e^{-2i\ln|x|}\right)\Bigg{)}\\ &=bx-x^{n}\mathbb{E}\Big{[}e^{-2\xi_{b}\ln|x|}\Big{]}-\epsilon \Bigg{(}A\sum_{p=1}^{k}y|y|^{2p-1}+(1+\cos(\omega t))\Big{(}ax+\mathbb{E} \Big{[}e^{-2\xi_{a}\ln|x|}\Big{]}\Big{)}\Bigg{)}\\ &=bx-x^{n}\psi_{b}(-2\ln|x|)-\epsilon\Bigg{(}A\sum_{p=1}^{k}y|y| ^{2p-1}+(1+\cos(\omega t))(ax+\psi_{a}(-2\ln|x|))\Bigg{)}.\end{split}  \tag{21}\]  
Let us discuss some examples--discrete uniform, binomial, \(\beta\)-binomial, and hypergeometric distributions. We state the domain at the set \(\{0,1,2,\ldots,K\}\). We have \(K=\left[\frac{n}{2}\right]\) for \(\xi_{a}\) and \(K=\left[\frac{n}{2}\right]\) for \(\xi_{b}\), respectively, for \(a_{i}\) and \(b_{i}\). The coefficients \(a_{i}\) and \(b_{i}\) are defined as the probabilities:  
\[ \begin{split} p_{i}^{\text{uniform}}&=\frac{1}{K} \\ p_{i}^{\text{binomial}}&=\binom{k}{i}p^{i}(1-p)^{K-i }\\ p_{i}^{\beta-\text{binomial}}&=\binom{K}{i}\frac{B (i+\alpha,k-i+\beta)}{B(\alpha,\beta)}\\ p_{i}^{\text{hypergeometric}}&=\frac{\binom{K}{i} \binom{N-K}{M-i}}{\binom{N}{M}}.\end{split}  \tag{22}\]  
The necessary conditions are \(0<p<1\), \(\alpha>0\), \(\beta>0\), \(K\leq M\leq\left[\frac{N}{2}\right]\), and \(M\) and \(N\) are integers.