American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 67 | P a g e CONSTRUCTION OF CORRELATION MODELS OF OSCILLATORY PROCESSES BY AN ALTERNATIVE METHOD Sh. B. Ochilov, O. Q. Akramova, N. N. Rasulova Bukhara State Technical University, Bukhara, Uzbekistan Abstract This article examines economic processes with correlational dependencies that change over time according to a harmonic law, and it also proposes a method for determining the period, phase, and frequency of the oscillations in these processes. The article proposes a method for determining the coefficients A and B in the harmonic function, which was obtained as a result of research. Keywords: Differential equation, finite differences, period of oscillation, frequency of oscillation, harmonic functions, cyclicity of fluctuations in crop yields, oscillation phase. Introduction Many processes in nature occur on the basis of oscillatory laws. Based on the results of experiments, the main characteristics, such as the oscillation period, frequency, oscillation phase and amplitude, are easily determined if the dependence is functional. For example, oscillations of electromagnetic waves, oscillations of a mathematical pendulum. However, when studying processes with correlational dependencies, the task becomes more complex, as it requires determining the period of oscillation, amplitude, and other parameters. Additionally, it is necessary to analyze and compare large volumes of data over an extended period. In this paper, an attempt is made to answer the following questions: 1. Is it possible to determine whether the process уi=f(xi) is harmonic if the results of experiments on the process Т at points хi and уi are known? Is there a mathematical criterion to verify the correctness of this statement? 2. How to determine the method for finding the period of oscillation, frequency, and phase of oscillation? 3. How can the coefficients A and B be determined if the harmonic function obtained from the study is 𝑦 = 𝐴𝑠𝑖𝑛(𝜔𝑥) + 𝐵𝑐𝑜𝑠(𝜔𝑥)? American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 68 | P a g e MATERIALS AND METHODS When solving practical problems, researchers often face the question of whether a given process is an oscillatory process. This question arises in several cases, such as: studying the yield of agricultural crops, examining the fluctuation in demand for goods and services depending on the season, and studying the periodicity of earthquake occurrences in seismically active zones. Oscillatory processes of agricultural crop yields have been examined in the scientific works of Russian researchers, in which the regularity of grain crop yield fluctuations has been established [1,2,3]. Based on the study of long-term time series, the authors identified the existence of a certain yield fluctuation cycle and established the regularity of this process. Additionally, several articles were published in the journal “Хлопководство” (Cotton Cultivation) where an attempt was made to prove the existence of cyclical fluctuations in crop yields across all cotton-growing republics of the former Soviet Union. It was also demonstrated that these regularities apply to other agricultural crops as well. To prove that a particular process under study is a cyclical process, it is necessary to analyze a large amount of statistical data, check their synchrony, compare the graphs of these phenomena, and much more. RESULTS AND DISCUSSION The question arises: is it possible to determine if the process under study is an oscillatory (harmonic) process if the observation results are known at limited points? For this, there is a finite number of points хi and yi, where 𝑖 = 1, 𝑁̅̅ ̅̅ ̅. A mathematical criterion also needs to be developed, such that when it is satisfied, the function 𝑦𝑖 = 𝑓(𝑥𝑖) will be exactly harmonic, and not some other function. Let us assume that we have a function 𝑦 = 𝑓(𝑥) and that this function is harmonic:: 𝑦 = 𝐴 sin 2𝜋𝑥 𝑇 + 𝐵 cos 2𝜋𝑥 𝑇 To determine the criterion, we calculate the derivatives 𝑑𝑦 𝑑𝑥 and 𝑑2𝑦 𝑑𝑥2 : 𝑑𝑦 𝑑𝑥 =𝐴 2𝜋 Т cos 2𝜋𝑥 Т − B 2𝜋 Т 𝑠𝑖𝑛 2𝜋𝑥 Т and 𝑑2𝑦 𝑑𝑥2 = −А ( 2𝜋 Т ) 2 𝑠𝑖𝑛 2𝜋𝑥 Т − В ( 2𝜋 Т ) 2 cos 2𝜋𝑥 Т . Taking into account that у=𝐴 sin 2𝜋𝑥 Т +B𝑐𝑜𝑠 2𝜋𝑥 Т , we have 𝑑2𝑦 𝑑𝑥2 = − ( 2𝜋 Т ) 2 (А𝑠𝑖𝑛 2𝜋𝑥 Т + В cos 2𝜋𝑥 Т ) or 𝑑2𝑦 𝑑𝑡2 = − ( 2𝜋 Т ) 2 у. In other words, 𝑑2𝑦 𝑑𝑥2 = 𝑦′′ = −𝜔2y, where 𝜔 = 2𝜋 Т . Here, T is the period of oscillation, and 1 Т is the frequency of the oscillatory process under consideration. From this, it follows that: у′′ 𝑦 = −𝜔2 = 𝑐𝑜𝑛𝑠𝑡. (1) Condition (1) is the criterion for the harmonicity of the process 𝑦 = 𝑓(𝑥). American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 69 | P a g e Thus, if for the process 𝑦 = 𝑓(𝑥), the ratio у′′ 𝑦 remains a constant, then this process is a harmonic process. Since we do not have the specific function of the process 𝑦𝑖 = 𝑓(𝑥𝑖), but only the observation results at certain points хi and yi , where 𝑖 = 1, 𝑁̅̅ ̅̅ ̅ , we will try to replace 𝑑𝑦 𝑑𝑥 и 𝑑2𝑦 𝑑𝑥2 with finite differences, which can be expressed in terms of хi and yi , where 𝑖 = 1, 𝑁̅̅ ̅̅ ̅. If 𝑦 = 𝑓(𝑥) is a continuous function and has continuous first and second derivatives, then from the definition of the derivative 𝑦′ = 𝑓′(𝑥0) lim ∆х→0 𝑦(𝑥0 + ∆𝑥) − 𝑦(𝑥0) ∆х = lim ∆х→0 ∆𝑦 ∆х = 𝑓′(𝑥0) = 𝑦′ we have: 𝑦(𝑥0+∆𝑥)−𝑦(𝑥0) ∆𝑥 ≈ 𝑓′(𝑥0) = 𝑦′ or ∆у ∆𝑥 ≈ 𝑓′(𝑥0) = 𝑦′; By the definition of the second derivative 𝑦′′ = 𝑓′′(𝑥0) lim ∆х→0 𝑦′(𝑥0 + ∆𝑥) − 𝑦′(𝑥0) ∆𝑥 = lim ∆х→0 ∆2𝑦 ∆𝑥2 = 𝑓′′(𝑥0) = 𝑦′′ we have: ∆2𝑦 ∆𝑥2 ≈ 𝑓′′(𝑥0) = 𝑦′′. Therefore, the derivatives can be approximately replaced by finite differences: 𝑦′ = 𝑑𝑦 𝑑𝑥 ≈ ∆𝑦 ∆𝑥 , 𝑦′′ = 𝑑2𝑦 𝑑𝑥2 ≈ ∆2𝑦 ∆𝑥2 . Based on this, our criterion for the cyclicity of the function takes the following form: if the results of the experiment хi and yi, are known, then a necessary condition for the cyclicity of the process is the fulfillment of the condition: ∆2𝑦 ∆𝑥2 /у = const. The theory outlined above can be examined using the following specific example. In this case, the condition ∆2𝑦 ∆𝑥2 /у =const plays an important role in the correct selection of the function 𝑦 = 𝑓(𝑥). Table №1. The coincidence of the values of the harmonic function y1 and the straight line y2 within specific intervals of the values of «х» х у1 = 3sin(2*3,14*х/5) у2 = 2,8591x + 0,2151 у1- у2 0,035 0,131837528 0,3151685 0,183331 0,07 0,263420324 0,415237 0,151817 0,105 0,394494146 0,5153055 0,120811 0,14 0,524805739 0,615374 0,090568 0,175 0,654103318 0,7154425 0,061339 0,21 0,782137057 0,815511 0,033374 American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 70 | P a g e 0,245 0,908659575 0,9155795 0,00692 0,28 1,033426407 1,015648 -0,01778 0,315 1,156196483 1,1157165 -0,04048 0,35 1,276732591 1,215785 -0,06095 0,385 1,394801834 1,3158535 -0,07895 0,42 1,510176082 1,415922 -0,09425 0,455 1,622632412 1,5159905 -0,10664 0,49 1,73195354 1,616059 -0,11589 0,525 1,837928239 1,7161275 -0,1218 0,56 1,940351747 1,816196 -0,12416 0,595 2,039026165 1,9162645 -0,12276 0,63 2,133760837 2,016333 -0,11743 0,665 2,224372719 2,1164015 -0,10797 0,7 2,310686734 2,21647 -0,09422 0,735 2,392536109 2,3165385 -0,076 0,77 2,469762696 2,416607 -0,05316 0,805 2,542217281 2,5166755 -0,02554 0,84 2,609759868 2,616744 0,006984 0,875 2,672259955 2,7168125 0,044553 0,91 2,729596781 2,816881 0,087284 0,945 2,78165956 2,9169495 0,13529 0,98 2,828347698 3,017018 0,18867 1,015 2,869570988 3,1170865 0,247516 Table №1 presents the experimental data connecting хi and yi. At first glance, it appears to be a typical linear relationship у=а0+bх between х and у. The data was processed using traditional methods, and the following result was obtained: у2 =2,8591х+1,2151. The correlation coefficient is r=0,98, and the difference between the actual and calculated values |у𝑎 − у𝑐| ≈ 0. But in reality, this is not the case. The given data represents the values of the function у1=3sin(1,256x). If we examine the values of the function у2=2,8591х+0,2151 and the values of у1=3sin(1,256x) on the graph (Fig. 1), we can see that they are virtually identical. American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 71 | P a g e Figure 1. Comparison of real and modeled values. This is the main reason for our research. In other words, it is necessary to find a mathematical statement that provides a clear answer to the question about the form of the relationship between the data хi and yi : linear у=а0+bх or harmonic у=𝐴 sin 2𝜋𝑥 Т +B cos 2𝜋𝑥 Т . The necessary condition for the existence of a harmonic dependence between хi and yi was stated in formula (1). To verify the correctness of our theory, let's consider an example presented in Table №2. In the seventh column of this table, the values of the second-order finite difference ∆2у𝑖 ∆𝑥2 /у are calculated, and their average value is computed as −1,13= −𝜔2, from which 𝜔 = √1,13 = 1,06. Since 𝜔2 = ( 2𝜋 Т ) 2 = ∆2𝑦𝑖 ∆𝑥𝑖 2 /𝑦𝑖, the period of oscillation Т can be determined in this case as: Т = 2𝜋 𝜔 = 6,28 1,1 =5,7 seconds; and 𝜔 = 1,1𝑟𝑎𝑑/𝑠. The amplitude of the oscillation process А is determined as follows: уi=A*sin(1,1х). From Table №2 when х=0,63, we have 2,1337=Asin(1,1*0,63), and from this, we find that A=3,5. Therefore, our function looks as follows: y=3,5sin(1,1х)=3,5sin( 2𝜋 5.7 х). To verify the suitability of the model and demonstrate the consistency of the proposed method, the calculated and real data were compared. y = 2.8591x + 0.2151 R² = 0.9834 0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 72 | P a g e Тable №2. Comparison of real and modeled data for the function y=3,5sin( 2𝜋 5.7 х) х Y ∆y ∆y/ω∆x ∆2y ∆2y/ω∆x2 (∆2y/ω ∆x2)/y Ya Ya-Yc 0,0350 0,1318 0,1316 2,9932 -0,0116 -0,2634 -1,9977 0,1347 0,0029 0,0700 0,2634 0,1311 2,9817 -0,0173 -0,3944 -1,4973 0,2692 0,0058 0,1050 0,3945 0,1303 2,9643 -0,0231 -0,5247 -1,3301 0,4034 0,0089 0,1400 0,5248 0,1293 2,9413 -0,0287 -0,6540 -1,2462 0,5369 0,0121 0,1750 0,6541 0,1280 2,9125 -0,0344 -0,7820 -1,1955 0,6696 0,0155 0,2100 0,7821 0,1265 2,8781 -0,0399 -0,9085 -1,1616 0,8013 0,0192 0,2450 0,9087 0,1248 2,8382 -0,0454 -1,0333 -1,1371 0,9319 0,0232 0,2800 1,0334 0,1228 2,7928 -0,0508 -1,1560 -1,1186 1,0610 0,0276 0,3150 1,1562 0,1205 2,7419 -0,0561 -1,2765 -1,1041 1,1886 0,0324 0,3500 1,2767 0,1181 2,6858 -0,0613 -1,3946 -1,0923 1,3145 0,0377 0,3850 1,3948 0,1154 2,6245 -0,0664 -1,5099 -1,0825 1,4383 0,0435 0,4200 1,5102 0,1125 2,5582 -0,0713 -1,6224 -1,0743 1,5601 0,0499 0,4550 1,6226 0,1093 2,4868 -0,0761 -1,7317 -1,0672 1,6795 0,0569 0,4900 1,7320 0,1060 2,4107 -0,0808 -1,8376 -1,0610 1,7965 0,0645 0,5250 1,8379 0,1024 2,3299 -0,0853 -1,9400 -1,0556 1,9108 0,0728 0,5600 1,9404 0,0987 2,2446 -0,0896 -2,0387 -1,0507 2,0222 0,0819 0,5950 2,0390 0,0947 2,1550 -0,0938 -2,1334 -1,0463 2,1307 0,0916 0,6300 2,1338 0,0906 2,0612 -0,0978 -2,2240 -1,0423 2,2360 0,1022 0,6650 2,2244 0,0863 1,9635 -0,1016 -2,3103 -1,0386 2,3380 0,1136 0,7000 2,3107 0,0818 1,8619 -0,1052 -2,3922 -1,0353 2,4365 0,1258 0,7350 2,3925 0,0772 1,7567 -0,1086 -2,4694 -1,0321 2,5314 0,1388 0,7700 2,4698 0,0725 1,6482 -0,1117 -2,5418 -1,0292 2,6225 0,1528 0,8050 2,5422 0,0675 1,5365 -0,1147 -2,6093 -1,0264 2,7098 0,1676 0,8400 2,6098 0,0625 1,4217 -0,1175 -2,6718 -1,0238 2,7931 0,1833 0,8750 2,6723 0,0573 1,3043 -0,1200 -2,7292 -1,0213 2,8722 0,1999 0,9100 2,7296 0,0521 1,1843 -0,1223 -2,7812 -1,0189 2,9470 0,2174 0,9450 2,7817 0,0467 1,0621 -0,1243 -2,8279 -1,0166 3,0175 0,2359 0,9800 2,8283 0,0412 0,9377 1,0150 2,8696 In the table, the values of хi are spaced at equal intervals of 0.035, so ∆хi=∆x=0,035, and ∆𝑥𝑖 2 = ∆𝑥2 = 0,01225. Table №2 presents the results of the calculation of the average values, |у𝑐 − у𝑎| = 0,0845. Thus, we can draw the following conclusions: American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 73 | P a g e 1. If the results of the experiments are known, i.e., the values of хi and yi , 𝑖 = 1, 𝑁̅̅ ̅̅ ̅, then it is possible to determine whether the dependence 𝑦 = 𝑓(𝑥) is harmonic or not. 2. To determine the harmonicity of the function 𝑦 = 𝑓(𝑥), we calculate the finite difference ∆2𝑦 ∆𝑥2 and its ratio to у: if ∆2𝑦 ∆𝑥2 /𝑦 = 𝑐𝑜𝑛𝑡 = 𝜔2, then the studied dependence is harmonic, i.e., it has the following form: 𝑦 = 𝐴sin 2𝜋𝑥 Т + 𝐵с𝑜𝑠 2𝜋𝑥 Т . 3. One of the advantages of this method is that, knowing only the values of хi and yi , where 𝑖 = 1, 𝑁̅̅ ̅̅ ̅ , we can determine both the oscillation period Т and the oscillation amplitudes А and В; 4. The amplitudes А and В are determined based on the solution of the second-order differential equation, which has the form: 𝑦′′ = −𝜔2у. In this case, the characteristic equation к2= - ω has an imaginary solution. Therefore, the general solution of the differential equation is: у =Аsin√𝜔𝑡 + 𝐵𝑐𝑜𝑠√𝜔𝑡. The coefficients of this function are determined based on the initial conditions 𝑦(𝑥0) = 𝑦0 and 𝑦′(𝑥0) = 𝑦0 ′ . From these initial conditions, we obtain a system of two equations with two unknowns, and by solving this system, we find the coefficients А and В. To verify this theory in practice, let's consider specific data on cotton yield in the period from 1991 to 2023 in the Bukhara region of the Republic of Uzbekistan. Back in the late 20th century, M.A. Abdullayeva proved the existence of cyclicality in cotton yields, the results of which were published in the 1982-1984 period in the journal “Хлопководство” (Cotton Cultivation). The author proposed a yield graph over an extended period based on a specific cyclicity. Based on this evidence, we propose the most advanced methodology for determining these cycles. Here, ∆𝑡𝑖 = 1. Table №3 Table of calculations for determining the oscillation cycles of cotton yield in the Bukhara region of the Republic of Uzbekistan. t уi ki=yi-25,8 ∆к𝒊 ∆𝒕𝒊 ∆𝟐к𝒊 ∆𝒕𝒊 𝟐 ∆𝟐к𝒊 ∆𝒕𝒊 𝟐 /𝐤𝐢 = 𝝎𝒊 𝟐 1. 34,00 8,20 -2,00000 4,90000 0,59756 2. 32,00 6,20 2,90000 -5,60000 -0,90323 3. 34,90 9,10 -2,70000 3,90000 0,42857 4. 32,20 6,40 1,20000 -6,00000 -0,93750 5. 33,40 7,60 -4,80000 5,50000 0,72368 6. 28,60 2,80 0,70000 -0,50000 -0,17857 7. 29,30 3,50 0,20000 1,60000 0,45714 8. 29,50 3,70 1,80000 -5,80000 -1,56757 9. 31,30 5,50 -4,00000 3,80000 0,69091 American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 74 | P a g e 10. 27,30 1,50 -0,20000 1,30000 0,86667 11. 27,10 1,30 1,10000 0,00000 0,00000 12. 28,20 2,40 1,10000 0,20000 0,08333 13. 29,30 3,50 1,30000 -1,00226 -0,28636 14. 30,60 4,80 0,29774 -1,77444 -0,36968 15. 30,90 5,10 -1,47670 2,34451 0,45991 16. 29,42 3,62 0,86780 -5,35104 -1,47777 17. 30,29 4,49 -4,48324 7,27765 1,62128 18. 25,81 0,01 2,79441 -0,40895 0,00000 19. 28,60 2,80 2,38546 -0,77092 -0,27533 20. 30,99 5,19 1,61454 -2,71454 -0,52349 21. 32,60 6,80 -1,10000 1,00000 0,14706 22. 31,50 5,70 -0,10000 0,00000 0,00000 23. 31,40 5,60 -0,10000 1,40000 0,25000 24. 31,30 5,50 1,30000 -3,70000 -0,67273 25. 32,60 6,80 -2,40000 1,50000 0,22059 26. 30,20 4,40 -0,90000 -0,10000 -0,02273 27. 29,30 3,50 -1,00000 2,80000 0,80000 28. 28,30 2,50 1,80000 -4,20000 -1,68000 29. 30,10 4,30 -2,40000 11,30000 2,62791 30. 27,70 1,90 8,90000 -17,90000 -9,42105 31. 36,60 10,80 -9,00000 32. 27,60 1,80 аverage -0,27805 ω 0,52730 Т 11,90963 In Table №3, the necessary calculations and data are provided, based on which the existence of oscillation cycles of cotton yield is determined for the Bukhara region of the Republic of Uzbekistan. The second column contains data on cotton yield for the period from 1991 to 2023. In the third column, the data is normalized as ki=yi ̶ ymin= yi ̶ 25,8. The fourth and fifth columns show the data ∆к𝑖 ∆𝑡𝑖 and ∆2к𝑖 ∆𝑡𝑖 2 , respectively. And the last column presents the value 𝜔2 = ∆к2 ∆𝑡2 /𝑦. In Table №3, 85% of the data are always less than one in absolute value. Therefore, the average value of |𝜔𝑖 2| is taken as |𝜔𝑖 2| = 0,27805 and ω=0,5273. Since 2𝜋 Т = 𝜔, in this case, we find that Т = 2𝜋 𝜔 = 6,28 0,5273 = 11,9. In works dedicated to determining the oscillation cycles of agricultural crop yields, cycles are determined by comparing a large amount of data over an extended period of time. However, using the proposed method, it is possible to quickly determine the cyclicity of the process, even with minimal observations. The period Т and the amplitude of oscillation А can also be determined. American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 75 | P a g e Based on simple mathematical calculations, the amplitudes of oscillation can be determined. As is known, А and В are determined taking into account the initial conditions. Therefore, we have: { 𝑦 = 𝐴𝑠𝑖𝑛𝜔𝑡 + 𝐵𝑐𝑜𝑠𝜔𝑡 𝑦′ = 𝐴𝜔𝑠𝑖𝑛𝜔𝑡 − 𝐵𝜔𝑐𝑜𝑠𝜔𝑡 Based on the initial conditions, we have: { 8 = 0,48А + 0,88 𝐵 −2 = 0,52 ∗ 0,88𝐴 − 0,52 ∗ 0,48 𝐵 { −2 = 0,45𝐴 − 0,23𝐵 8 = 0,48𝐴 + 0,88𝐵 { −4,44 = 𝐴 − 0,5𝐵 16,6 = 𝐴 + 1,83 𝐵 21,04 = 0 + 2,33B B=9 A=-4,44+0,5B=-4,44+4,5≈0, so B=9; А=0. Therefore, y=9cos 0,51t Therefore, the oscillatory process has the following form: 𝑦 = 𝑓(𝑡) + 𝐸(𝑡), where f(t) shows the general trend of yield growth, and E(t represents the influence of the external environment (noise). In our case, 𝐸(𝑡) = 𝐴𝑐𝑜𝑠 2𝜋𝑡 𝑇 = 4,4cos 2𝜋𝑡 12 ; f(t)=0.0122t2 ̶ 0.4328t+33,246, so y=0.0122t2 ̶ 0.4328t+33,246+9cos 2𝜋𝑡 12 . Тable №4 Comparison of actual and calculated cotton yield values, taking into account the cyclic oscillations. years- 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Ya 34,0 32,0 34,9 32,2 33,4 28,6 29,3 29,5 31,3 27,3 27,1 Yc 34,7 31,3 33,9 30,6 33,2 29,9 32,6 29,3 32,1 28,8 31,6 Ya - Yc -0,7 0,7 1,0 1,6 0,2 -1,3 -3,3 0,2 -0,8 -1,5 -4,5 years 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Ya 28,2 29,3 30,6 30,9 29,4 30,3 25,8 28,6 31,0 32,6 Yc 28,5 31,3 28,2 31,1 28 30,9 27,9 30,9 27,9 31 Ya - Yc -0,3 -2,0 2,4 -0,2 1,4 -0,7 -2,1 -2,3 3,1 1,6 American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 76 | P a g e years 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 Ya 31,5 31,4 31,3 32,6 30,2 29,3 28,3 30,1 27,7 33,7 36,6 27,6 Yc 28 31,1 28,2 31,4 28,5 31,7 28,9 32,2 29,5 32,7 30 33,4 Ya - Yc 3,5 0,3 3,1 1,2 1,7 -2,4 -0,6 -2,1 -1,8 1,0 6,6 -5,8 Let's consider the actual and calculated cotton yield values on the graph. Figure 2. Graphical comparison of actual and calculated cotton yield values in the Bukhara region. Figure 3. Graphical forecast of cotton yield considering fluctuations. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Уф 34 32 34 32 33 28 29 29 31 27 27 28 29 30 30 29 30 25 28 31 32 31 31 31 32 30 29 28 30 27 33 37 28 Ур 35 31 34 31 33 30 33 29 32 29 32 28 31 28 31 28 31 28 31 28 31 28 31 28 31 29 32 29 32 29 33 30 33 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 Уф Ур 30.7438 34.1282 31.537 34.9702 32.4278 30.5 31 31.5 32 32.5 33 33.5 34 34.5 35 35.5 2024 2025 2026 2027 2028 American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 38, March - 2025 77 | P a g e It can be seen that the cyclical nature in forecasting ensures reliable results. Conclusion Thus, the following results were obtained in this article: 1. If experimental data characterizing the state of the object at points xi in the form of yi, are available, it is possible to determine whether the process is harmonic or non-harmonic. The necessary condition for harmonicity is: ∆2𝑦 ∆𝑥2 /𝑦 = −𝜔2 = ±𝐴. If А> 0, the process is an oscillatory process. 2. When the value of ⍵ = 2𝜋 𝑇 is known, the period of oscillation T is automatically determined as 𝑇 = 2𝜋 ⍵ , and the frequency of oscillation 𝑦 = 1 𝑇 3. If А<0, then the general solution of the equation and the desired dependency is a harmonic function: 𝑦 = 𝐴𝑠𝑖𝑛√𝜔𝑥 + 𝐵𝑐𝑜𝑠√𝜔𝑥. 4. When А>0 , this process is not harmonic and has the following form: 𝑦 = 𝐴𝑒𝑥 + 𝐵𝑒−𝑥 5. In any case, the coefficients А and of the harmonic function are determined based on the initial conditions y(𝑥0) = 𝑦0 and 𝑦′(𝑥0) = 𝑦0 ′ ; In this case, a system of two equations with two unknowns is solved, and the coefficients А and В are determined. 6. 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