Conewise (piecewise) linear, non-gyroscopic and gyroscopic autonomous systems are studied in detail First, the general properties of such systems are examined. Then the non-linear real and complex modes are obtained for both non-gyroscopic and gyroscopic cases, by means of analytical or numerical tools. Several interesting characteristics of non-linear modes are found and compared to those of linear modes. Accurate analytical estimates of non-linear mode periods are formulated for both non-gyroscopic and gyroscopic systems. The validity of such estimates is discussed in relation to a non-linear mode bifurcation phenomenon, which is located by using both characteristic multipliers and Poincare mapping.

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The nutrition status of children is gaining more attention with a rapid nutrition transition. This study aimed to investigate trends and urban-rural differences in dietary energy and macronutrient composition among Chinese children. A total of 7565 participants aged 6 to 17 years were obtained from three rounds (1991, 2004 and 2015) of the Chinese Health and Nutrition Survey (CHNS). The individual diet was evaluated via three consecutive 24-hour dietary recalls and compared with the Chinese Dietary Reference Intakes (DRIs). From 1991 to 2015, there was a significant increase in children’s fat intake, the proportion of energy intake from fat, and the proportion of children with more than 30% of energy from fat and less than 50% of energy from carbohydrates (p < 0.001). Compared with the DRI, the proportion with higher fat and lower carbohydrate intakes were, respectively, 64.7% and 46.8% in 2015. The urban-rural disparities in fat and carbohydrate intake gradually narrowed, while the gap in protein intake increased notably over time (p < 0.001). Chinese children experienced a rapid transformation to a low-carbohydrate and high-fat diet. Urban-rural disparities persistently existed; further nutritional interventions and education were of great significance, so as to ensure a more balanced diet for Chinese children.