Solonetz and Solonetzic rangelands are widespread in arid regions of Central Kazakhstan, where pasture degradation is often difficult to assess because surface vegetation patterns do not always reflect subsurface soil constraints. This study aimed to evaluate degradation mechanisms in Solonetz pasture ecosystems of the Ulytau region by integrating field soil-profile descriptions, laboratory analyses, vegetation observations, forage productivity data and Sentinel-2A-derived MSAVI. Ten monitoring soil profiles were examined for particle-size distribution, soluble salts, ionic composition, exchangeable cations, available N, P and K, vegetation cover and forage yield. USDA textural classification, salt-distribution analysis, Pearson correlation, PCA, RDA and MSAVI-based mapping were used to link soil-profile properties with vegetation and spectral response. The results showed strong profile heterogeneity, with clay enrichment, subsurface salt accumulation, alkalinity and Na- or Mg-related exchange-complex imbalance forming several degradation pathways. Surface horizons were often weakly saline, whereas deeper layers contained stronger chemical and physical limitations. MSAVI values were low across the monitoring sites and reflected vegetation–soil surface conditions rather than salinity or sodicity directly. Combining soil-profile diagnostics with Sentinel-2A MSAVI improved the interpretation of degradation patterns and provides a practical framework for monitoring spatially heterogeneous Solonetz rangelands in arid pasture systems.

Solonetz and Solonetzic rangelands are widespread in arid regions of Central Kazakhstan, where pasture degradation is often difficult to assess because surface vegetation patterns do not always reflect subsurface soil constraints. This study aimed to evaluate degradation patterns in Solonetz pasture ecosystems of the Ulytau region by integrating field soil-profile descriptions, laboratory analyses, vegetation observations, forage productivity data and Sentinel-2A-derived MSAVI. Ten monitoring soil profiles were examined for particle-size distribution, soluble salts, ionic composition, exchangeable cations, available N, P and K, vegetation cover and forage yield. USDA textural classification, salt-distribution analysis, Pearson correlation, PCA, RDA and MSAVI-based mapping were used to link soil-profile properties with vegetation and spectral response. The first two PCA axes explained 65.5% of the total variance, while selected soil profile constrains accounted for 58% of the variation in vegetation cover, forage yield and MSAVI in the RDA analyses. The results showed strong profile heterogeneity, with clay enrichment, subsurface salt accumulation, alkalinity and Na- or Mg-related exchange–complex imbalance associated with several degradation pathways. Surface horizons were often weakly saline, whereas deeper layers contained stronger chemical and physical limitations. MSAVI values were low across the monitoring sites and reflected vegetation–soil surface conditions rather than salinity or sodicity directly. MSAVI ranged from 0.0888 to 0.2148, with a mean value of 0.1248. Combining soil-profile diagnostics with Sentinel-2A MSAVI improved the reliability of interpreting spatial degradation patterns and provides a practical framework for monitoring spatially heterogeneous Solonetz rangelands, supporting sustainable rangeland management under arid conditions.

In this paper, we study the Henon-Heiles potential and the problem of two fixed centers. In studies of nonlinear systems for which exact solutions are unknown, the Poincare section method is used. For the Henon-Heiles potential, Poincare sections were obtained. At low energies, the Henon-Heiles system looks integrable, since independently of the initial conditions, the trajectories obtained with the help of numerical integration lie on two-dimensional surfaces, i.e. as if there existed a second independent integral. Next, the potential of two fixed centers was investigated. It was shown on the basis of the Poincare section that, in the case μ1 = μ2 = 1 the internal cross-sectional structure decomposes from the values H = –1.7, but the internal cross-sectional structure is preserved in the interval , in the case μ1 = 0.9 and μ1 = 0.1 the internal cross-sectional structure decomposes from the values but the internal cross-sectional structure is preserved in the interval , in the case of μ1 = 0.7 and μ1 = 0.3 the internal cross-sectional structure decomposes from the values , but the internal cross-sectional structure is preserved in the interval . With increasing energy, many of these surfaces decay. It is assumed that the numerical results obtained will serve as the basis for comparison with analytical solutions. Берілген мақалада Хенон-Хейлес потенциалы мен қозғалмайтын екі центр есебі қарастырылады. Сызықты емес жүйелердің нақты шешімдері белгісіз болғанда Пуанкаре қима әдісі қолданылады. Хенон-Хейлес потенциалы үшін Пуанкаре қимасы алынды. Энергияның төмен деңгейлерінде Хенон-Хейлес жүйесі интегралданады, екінші белгісіз интергралы бар секілді бастапқы шарттардан тәуелсіз жүйенің траекториясы сандық интегралдық әдіспен шешіліп, екі өлшемді кеңістікте сипатталады. Сонымен қатар қозғалмайтын екі центр есебі зерттелді. Пуанкаре қимасының негізінде қозғалмайтын екі центр есебіне келесідей тұжырымдамалар алынды: miu1=miu2=1 кезінде H=-1.7 мәнінен бастап ішкі қима ыдырайды, ал H [-0.5, -1.6 ] аралығында ішкі қима сақталады; miu1=-0.9 және miu1=-0.9 и miu2=-0.1 кезінде H=-0.9 мәнінен бастап ішкі қима ыдырайды, ал H [-0.3, -0.8] аралығында ішкі қима сақталады; miu1=-0.7 және miu2=-0.3 кезінде H=-0.8 мәнінен бастап ішкі қима ыдырайды, ал H [-0.2, -0.7] аралығында ішкі қима сақталады. Сонымен қатар, энергияның өсуімен, осы қималардың көпшілігі ыдырайды. Алынған сандық нәтижелер аналитикалық шешімдермен салыстыру үшін негіз болады деп болжануда. В данной работе исследуется потенциал Хенона-Хейлеса и задача двух неподвижных центров. При исследовании нелинейных систем для которых неизвестны точные решения используется метод сечения Пуанкаре. Для потенциала Хенона-Хейлеса были получены сечения Пуанкаре. При малых энергиях система Хенона-Хейлеса выглядит интегрируемой, так как независимо от начальных условий, траектории, полученные с помощью численного интегрирования, лежат на двумерных поверхностях, т.е. так, как если бы существовал второй независимый интеграл. Далее был исследован потенциал задачи двух неподвижных центров. Было показано на основе сечения Пуанкаре что, в случае μ1 = μ2 = 1 внутренняя структура сечений распадается со значений H = –1.7, но внутренняя структура сечений сохраняется в отрезке , в случае μ1 = 0.9 и μ1 = 0.1 внутренняя структура сечений распадается со значений , но внутренняя структура сечений сохраняется в отрезке , в случае μ1 = 0.7 и μ1 = 0.3 внутренняя структура сечений распадается со значений , но внутренняя структура сечений сохраняется в отрезке . С увеличением энергии многие из этих поверхностей распадаются. Предполагается что, полученные численные результаты, послужат основой для сравнения с аналитическими решениями.