In this article, we have tried to introduce one of the most important topics in the subject of dynamical systems, namely the MELNIKOV function, in simple language. MELNIKOV function is one of the tools that can express the effect of the small perturbation on the homoclinic orbits. This issue is also related to breaking a homoclinic orbit under the effect of some small perturbations. When a small perturbation occurs in a dynamical system, some dynamical behaviors of the system may change. Here, we try to explain how to compute the formula of this function fluently. Therefore, in the first part, we will introduce some preliminary concepts and properties, and then in the second part, we will describe the construction of the MELNIKOV function. Finally, by stating the results of the fundamental matrix solutions and then using them, we will construct the MELNIKOV function near a homoclinic or hetroclinic orbit. 1- IntroductionOne of the fundamental issues that have recently attracted the attention of researchers in the field of theoretical research is the issue of dynamical systems. The phenomenon of touching stable and unstable manifolds is an example of behavior in a dynamical system that results in the existence of special solutions. If the stable and unstable curves of an equilibrium point touch each other, then the solution is homoclinic, and if the stable curves of one equilibrium point intersect with the unstable curve of another equilibrium point, then we will have a heteroclinic solution. Investigating the existence of such solutions for a specific dynamical system has always been one of the important issues, and therefore many people have done good research in this field, for example, some examples of such research can be found in [8, 4,5] observed. In addition, many researchers have tried to solve certain cases of Hilbert's 16th problem using the MELNIKOV function. As an example, we can refer to [2, 5, 6]. The subject we will discuss in this article is related to the breaking of a homoclinic orbit. When a perturbation occurs in a dynamic system, some dynamical behaviors of the system may change; One of these changes can be the breaking of the homoclinic orbit in the phase space. One of the tools that can express the effect of small perturbations on the homoclinic orbit for us is the MELNIKOV function, which in this article we try to explain how to make fluently. To continue our discussion more easily, it is necessary to familiarize the reader with a series of basic concepts, so in the second part we will introduce some preliminary concepts, and then in the third part we will describe the construction of the MELNIKOV function. 3- Main ResultsWe consider the differential equation $ \dot{x} = f(x,\mu) $ where $ x \in \mathbb{R}^2 $, $ \mu \in \mathbb{R} $ and $f$ is at least $C^2$. Suppose there are two hyperbolic saddles $ p_{-}(\mu) $ and $ p_{+}(\mu) $. Suppose that for $ \mu = 0 $ there is a solution of this equation in the form of $ x_{\ast}(t) $ such that\begin{align*}\lim_{t\rightarrow - \infty} x_{\ast}(t) = p_{-}(0),~~~~~~ \lim_{t\rightarrow + \infty} x_{\ast}(t) = p_{+}(0) \end{align*} Our goal is to check that when $ \mu $ changes, how this connection (that is, the saddle connection between $p_{-}(t)$ and $p_{+}(t)$) is broken.\\We put $ x_{\ast}(0) = x_{0} $. The velocity vector $ x_{\ast}(t) $ at $ t = 0 $ is:\begin{align*}\dot{x}_{\ast}(0) = f(x_{0},0) = (f_{1}(x_{0},0),f_{2}(x_{0},0 )).\end{align*}If we put\begin{align*}u_{0} = \frac{1}{\Vert f(x_{0},0)\Vert^{2}}(-f_{2}(x_{0},0) , f_{1 }(x_{0},0)), \end{align*}then obviously, the vector $ u_{0} $ is perpendicular to $ f(x_{0},0) $. We consider a line segment $ \Sigma $ passing through $ x_{0} $ and in the direction of $ u_{0} $. Therefore, $ \Sigma $ with the formula $ x = x_{0} + \xi u_{0} $ where $ \vert \xi \vert < \alpha $ for some $ \alpha>0$. Suppose $ x_{-}(t,\mu) $ is a solution of $ \dot{x} = f(x,\mu) $ such that$$1)~~x_{-}(0,\mu) \in \Sigma,$$$$2)~~\lim_{t\rightarrow -\infty}x_{-}(t,\mu) = p_{-}(\mu),$$$$3)~~x_{-}(t,0) = x_{\ast}(t).$$This answer is on the unstable manifold $ p_{-}(\mu) $.Similarly, suppose that $ x_{+}(t,\mu) $ is a solution of $ \dot{x} = f(x,\mu) $ such that$$1)~~ x_{+}(0,\mu) \in \Sigma,$$$$2)~~\lim_{t\rightarrow +\infty}x_{+}(t,\mu) = p_{+}(\mu),$$$$3)~~x_{+}(t,0) = x_{\ast}(t).$$This solution is also placed inside the stable manifold $ p_{+}(\mu) $ (pay attention to the Figures 1 and 2).  We define the separation function as follows:\begin{align*}s(\mu) = \xi_{-}(\mu) - \xi_{+}(\mu). \end{align*}We put $ \psi_{0} = (-f_{2}(x_{0},0),f_{1}(x_{0},0)) $ then $\dot{x}_{*}(t)=(\dot{x}_{*1}(t),\dot{x}_{*2}(t))$ is a solution of the equation $\dot{v}(t)=A(t)v$ and so,$$ \psi(t) = exp\left(-\int_{0}^{t} div f(x_{\ast}(s),0)ds\right)\left(-\dot{x}_{\ast2}(t), \dot{x}_{\ast1}(t)\right)$$is a solution of the adjoint equation $\dot{w}=-wA(t)$, with the initial condition$$\psi(0)=(-\dot{x}_{*2}(0),\dot{x}_{*1}(0))=(-f_2(x_0,0),f_1(x_0,0))=\psi_0,$$Since we have $\psi_0u_0=1,$ it follows that:\begin{align*}\frac{d \xi_{\pm}}{d\mu}(0) = \psi_{0}\frac{d \xi_{\pm}}{d\mu}(0)u_{0} = \psi_{0}\frac{\partial x_{\pm}}{\partial \mu}(0,0).\end{align*}Also $ \frac{\partial x_{\pm}}{\partial \mu}(t,0) $ is an answer of the following equation:\begin{align}\label{b50}\dot{v} = D_{x}f(x_{\ast}(t),0)v + \frac{\partial f}{\partial \mu}(x_{\ast}(t) ,0).\end{align}Let's assume that the state transition matrix of the homogeneous linear system $ \dot{v} = D_{x}f(x_{\ast}(t),0) v $ is $ \phi(t,s) $. Now, according to the parameter change formula, we have:\begin{align*}\frac{\partial x_{+}}{\partial\mu}(0,0)=\phi(0,T)\frac{\partial x_{+}}{\partial\mu}(T, 0)-\int_{0}^{T}\phi(0,s)\frac{\partial f}{\partial\mu}(x_{\ast}(s) ds, 0)\end{align*}And\begin{align*}\frac{\partial x_{-}}{\partial\mu}(0,0)=\phi(0,-T)\frac{\partial x_{-}}{ \partial\mu}(-T, 0)-\int_{-T}^{0}\phi(0,s)\frac{\partial f}{\partial\mu}(x_{\ast} (s), 0) ds\end{align*}So\begin{eqnarray}\label{b51}\frac{d \xi_{-}}{d \mu}(0) = \psi_0 \frac{\partial x_{-}}{\partial\mu}(0,0) &=& \psi (0)(\phi(0,-T)\frac{\partial x_{-}}{\partial \mu}(-T,0) + \int_{-T}^{0}\phi( 0,s)\frac{\partial f}{\partial\mu}(x_{\ast}(s),0)ds)\\&=& \psi(-T)\frac{\partial x_{-}}{\partial \mu}(-T,0) + \int_{-T}^{0}\psi(s) \frac{\partial f}{\partial \mu}(x_{\ast}(s),0)ds. \end{eqnarray}Now, if $t\rightarrow\infty$ then:\begin{align*}\lim_{T\rightarrow +\infty}\psi(-T)\frac{\partial x_{-}}{\partial\mu}(-T,0) = 0. \end{align*}So\begin{align*}\frac{d \xi_{-}}{d \mu}(0) = \int_{-\infty}^{0} \psi(s)\frac{\partial f}{\partial \mu}( x_{\ast}(s),0)ds. \end{align*}Similarly\begin{align*}\frac{d \xi_{+}}{d \mu}(0) = \int_{0}^{+\infty} \psi(s)\frac{\partial f}{\partial \mu }(x_{\ast}(s),0)ds. \end{align*}So\begin{align*}s'(0) = \xi'_{-}(0) - \xi'_{+}(0) = \int_{-\infty}^{\infty} \psi(s)\frac {\partial f}{\partial \mu}(x_{\ast}(s),0).ds \end{align*} The recent integral is called the MELNIKOV integral if the function $f$ is dependent on $t$, that is, the right side of the differential equation is $f(x,\mu, t)$, then the value of MELNIKOV integral will also depend on $t$. 3- ConclusionsThis article presents a unique perspective on constructing the MELNIKOV integral, different from what is typically found in textbooks. The problem's assumptions are presented in a way that accounts for both homoclinic and heteroclinic states. By following the method outlined in the article, interested readers can calculate MELNIKOV's integral formula in the time-dependent mode by considering the time-dependent perturbation on the system.

Interactions of the translational motion of a gyrostat satellite on its attitude dynamics is considered in this paper. The mathematical model is first derived using the Hamiltonian method for the rotation-Translation motion of the gyrostat satellite followed by the reduction of the coupled equations of motion using the extended Deprit canonical transformation. The analytical MELNIKOV method along with the numerical Lyapunov exponent criterion, Poincare' section, trajectories of phase portrait, and the time history responses are used to study the heteroclinic bifurcation and chaos phenomena on the reduced model. On the basis of the results obtained from MELNIKOV integral, the parameters of the gyrostat satellite can be designed to prevent chaos in the system in the absence of a control system.

In this paper, the nonlinear vibration of a Euler–Bernoulli nanobeam resting on a non-linear viscoelastic foundation is investigated. It is assumed that the nanobeam is subjected to a harmonic excitation that can be representative of an electrostatic field. The non-linear viscoelastic foundation is considered for both hardening and softening cases. By neglecting of the in-plane inertia, Eringen's nonlocal elasticity theory is used to model and derive the equation of motion of the nanobeam. Using the Galerkin method and the first mode shape, the obtained partial differential equation is reduced to the ordinary differential equation. Calculating the system's equilibrium points lead to heteroclinic bifurcation and the heteroclinic orbits are obtained. Then, using the MELNIKOV integral method, the chaotic motion of the system is studied analytically, and the safe region of the system is determined respect to the parametric space of the problem. When the viscoelastic foundation has a hardening characteristic, the chaotic behavior in the system does not occur. It has been observed that the use of nonlocal elasticity theory is necessary to investigate the chaotic behavior of nanobeam, and using the classical theory of elasticity may place the system in the chaotic region.

Purpose: The flourishing of political parties and currents is one of the signs of the degree of development in societies. In contemporary Iran and since the formation of the constitutional movement, political currents have gone through many ups and downs and it can be said that political currents in Iran after the Islamic Revolution have not yet reached the stage of institutionalization and stability. A characteristic feature of political currents in Iran is the divergence and division among political currents in recent decades, and this can be one of the reasons for the instability and cross-sectional and seasonal activity of political parties in Iran. Therefore, the necessity of leading research seeks to answer the question of how factors and divergence among political currents in Iran after the victory of the Islamic Revolution can be analyzed? And what are the scenarios for the advancement of political currents in Iran?Method: To answer this question, the method of causal-layer ANALYSIS, which is one of the qualitative methods in futures research, has been used.Findings: The research findings indicate that this divergence is due to a wide range of reasons from the level of causal systems (from the institutionalization of power to the formation of parties as elitist initiatives), worldview and discourse (from charismatic political authority to culture). Subsidiary-follower politics to myth-metaphor (Iranian individualism to belief in a strong state-weak society) can be analyzed.Conclusions: Three scenarios for the future of Iranian political currents can be considered: integration of currents as the security valve of the political system, the collapse of political currents in the traditional form, integration and consolidation in new social movements (virtualized parties).

Recognition and understanding the genetic control of traits, combining ability and genetic structure are directly related to the success of breeding programs. For this purpose, a 7 × 7 one-way diallel design was conducted in a randomized complete block design with three replications. The measured traits were included plant height, height to the first capsule, number of days to 50% and 90% of flowering, number of capsules per plant, number of seeds per capsule, number of days to physiological ripening, number of branches, leaves number and length, 1000-seeds weight, capsule weight, length and width, chlorophyll a, b and total chlorophyll, biological andeconomic yields, harvest index, oil and protein percentage. ANALYSIS of variance showed that there was a significant difference between genotypes and diallel ANALYSIS showed that the additive variance of all traits and dominant variance of all traits except height to the first fruit-bearing capsule were significant. The oltan cultivar was the best and Ardestan genotype was the worst genotype in terms of general combining ability. Sabzevar×TS-3 and Sirjan×Fars were the best hybrids in most traits. The general heritability was between 0.90 to 0.96 for biologic yield and number of branches, respectively and narrow heritability was between 0.36 to 0.91 for the number of branches and harvest index, respectively. The ANALYSIS of variance by Hayman method confirmed the results of Griffing ANALYSIS.

As the most complex manufactured structures, cities face excessive population growth. Their expansion has intensified on high-risk sites, and the available evidence also indicates the continuous increase of all types of natural crises in terms of intensity and frequency. Scientific and experimental findings show that the best way to deal with danger is to promote the resilience of settlements in different dimensions (social, economic-livelihood, physical-spatial and institutional); in other words, resilience in both human and environmental dimensions comprehensively. It decreases and increases. This research has evaluated and analyzed the components of resilience in Sari. The method of the present study is applied in terms of purpose and descriptive-analytical and field in nature. The statistical population in this research includes citizens living in the four districts of Sari, and the sample size was determined based on Cochran's formula of 383 people, who were selected from among the statistical population by stratified sampling. The questionnaire is the method of collecting library and field information and its most important tool. For data ANALYSIS, descriptive and inferential statistics (one-sample t-test and structural equation modeling) were used by SPSS and Smart PLS software, and entropy and SAW models were exerted. The research results indicate that the situation of the four regions of Sari regarding social components has better conditions than other dimensions of resilience. In terms of institutional components, they have a vulnerable state. According to the entropy model, among the components of resilience, the institutional dimension has the most weight, and the economic dimension has the least weight. Moreover, according to the SAV model, Region 1 ranks first, and Region 3 of Sari ranks last in having the components of resilience dimensions.

Purpose: Food security is a critical global challenge that is influenced by research and innovation in the field. Therefore, the objective of this study is to analyze the scientific output of developing countries in food security and examine its relationship with patents and Gross Domestic Product (GDP).Methodology: This applied research utilized the Scientometric approach. A total of 8,416 papers published between 1992-2023 in the field of food security by developing countries were included in the study using citation databases from Clarivate Analytics. Additionally, patent registrations from the WIPO database and GDP data from the World Bank were analyzed. Information was collected through note-taking, and the data was analyzed using Pearson's correlation coefficient.Findings: The findings reveal an upward trend in the publication and citation of scientific outputs related to food security in developing countries. China has higher numbers of papers, patents, GDP, and food production index compared to Iran, Japan, and South Korea. There is also a positive correlation observed between population and the number of papers, gross production and the number of papers, food production and the number of published papers, as well as the number of patents and papers citing scientific outputs of countries.Conclusion: These results highlight the significant relationship between increasing scientific output, GDP, the number of patents, and food security. Greater emphasis on food security contributes to enhanced scientific output, GDP, and innovation. Similarly, increasing scientific output, GDP, and innovation positively impact food security in countries.Value: This study emphasizes the importance of scientific outputs in driving technological advancements, innovations, and ultimately, ensuring food security in developing countries.

The linkage between sustainable agriculture, poverty and agricultural extension efforts and their impacts on rural centers in Behbahan Shahrestan has been discussed in this paper. Data were collected from 200 farmers in 40 villages of this Shahrestan. A multi-stage stratified random sampling technique was used for selecting villages and farmers. The findings of path ANALYSIS in three different causal models provide the complexity of relationships between variables and environmental degradation so that there is a causal relationship between poverty and unsustainability. Lack of direct causal effect of use of technology and extension efforts on sustainability in three models indicated the structural and institutional limitations of extension in diffusion of appropriate technologies. Finally, recommendations regarding regional planning with respect to socio-economic characteristics and changing from TOT approach to other alternatives and revising the education programs of extension agents are provided.