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第9行： − <font color="red">'''算子 Operators'''是一个函数空间到函数空间上的映射O：X→X，广义的讲，对任何函数进行某一项操作都可以认为是一个算子，如求幂次、求微分等。</font> <font color="blue">这句话的英文原文在哪里？</font> + − − --[[用户:嘉树|嘉树]]（[[用户讨论:嘉树|讨论]]） 这个是补充内容，摘自百度百科 https://baike.baidu.com/item/%E7%AE%97%E5%AD%90 − − 某些情况下，也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。 第262行：
第258行： − − − − === In human development 在人类发展中的应用=== − − − − In [[Developmental psychology|human development]], dynamical systems theory has been used to enhance and simplify Erik Erikson's [[eight stages of psychosocial development]] and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and [[fractal]] properties of the [[Fibonacci sequence]].<ref name=Sacco>{{cite journal|last=Sacco|first=R.G.|title=Re-envisaging the eight developmental stages of Erik Erikson: The Fibonacci Life-Chart Method (FLCM)|journal=Journal of Educational and Developmental Psychology|date=2013|volume=3|issue=1|pages=140–146|doi=10.5539/jedp.v3n1p140|doi-access=free}}</ref> Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years).<ref name=Sacco/> − − In human development, dynamical systems theory has been used to enhance and simplify Erik Erikson's eight stages of psychosocial development and offers a standard method of examining the universal pattern of human development. This method is based on the self-organizing and fractal properties of the Fibonacci sequence. Using mathematical modeling, a natural progression of human development with eight life stages has been identified: early infancy (0–2 years), toddler (2–4 years), early childhood (4–7 years), middle childhood (7–11 years), adolescence (11–18 years), young adulthood (18–29 years), middle adulthood (29–48 years), and older adulthood (48–78+ years). − − 在人类发展方面，动力系统理论已经被用来增强和简化 Erik Erikson 的'''社会心理发展8阶段理论 Eight Stages of Psychosocial Development'''，并提供了一个检验人类发展普遍模式的标准方法。该方法基于斐波那契数列的'''自组织性 self-organizing'''和'''分形 Fractal''' 特性。利用数学模型，人类发展的自然进程被分为8个生命阶段: 早期婴儿期(0-2岁)、幼儿期(2-4岁)、童年早期(4-7岁)、童年中期(7-11岁)、青春期(11-18岁)、成年早期(18-29岁)、成年中期(29-48岁)和老年成年期(48-78岁及以上)。 − − --[[用户:木子二月鸟|木子二月鸟]]（[[用户讨论:木子二月鸟|原wiki里没有这个应用呀？是被删除了吗？]] − − According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from [[adolescence]] to [[young adulthood]], and after reaching the critical point of 18 years of age (young adulthood), a peak in [[testosterone]] is observed in males<ref>{{cite journal |last1=Kelsey |first1=T. W. |title=A validated age-related normative model for male total testosterone shows increasing variance but no decline after age 40 years. |journal=PLOS One |volume=9 |issue=10 |pages=e109346 |doi=10.1371/journal.pone.0109346|pmid=25295520 |year=2014 |pmc=4190174 |bibcode=2014PLoSO...9j9346K }}</ref> and the period of optimal [[fertility]] begins in females.<ref>{{cite book |last1=Tulandi |first1=T. |title=Preservation of fertility |date=2004 |publisher=Taylor & Francis |pages=1–20}}</ref> Similarly, at age 30 optimal fertility begins to decline in females,<ref name="Social Science 2008">{{cite journal |last1=Blanchflower |first1=D. G. |title=Is well-being U-shaped over the life cycle? |journal=Social Science & Medicine |volume=66 |issue=8 |pages=1733–1749 |doi=10.1016/j.socscimed.2008.01.030|pmid=18316146 |year=2008 |citeseerx=10.1.1.63.5221 }}</ref> and at the stage transition from [[middle adulthood]] to older adulthood at 48 years, the average age of onset of [[menopause]] occurs.<ref name="Social Science 2008"/> − − According to this model, stage transitions between age intervals represent self-organization processes at multiple levels (e.g., molecules, genes, cell, organ, organ system, organism, behavior, and environment). For example, at the stage transition from adolescence to young adulthood, and after reaching the critical point of 18 years of age (young adulthood), a peak in testosterone is observed in males and the period of optimal fertility begins in females. Similarly, at age 30 optimal fertility begins to decline in females, and at the stage transition from middle adulthood to older adulthood at 48 years, the average age of onset of menopause occurs. − − 根据这个模型，年龄的阶段转换代表了多层次的自组织过程(例如，分子、基因、细胞、器官、器官系统、生物体、行为和环境)。例如，在从青春期向成年早期过渡的阶段中，在达到18岁这一关键年龄之后，男性的睾丸激素达到高峰，女性的最佳生育期开始。同样，在30岁时，女性的最佳生育能力开始下降；在从成年中期过渡到老年成年期时，48岁是绝经的平均年龄。 − − − − These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for [[weather forecasting]]. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological [[DNA]]<ref>{{cite journal |last1=Perez |first1=J. C. (2010). |title=Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618. |journal=Interdisciplinary Sciences: Computational Life Sciences |volume=2 |issue=3 |pages=228–240 |doi=10.1007/s12539-010-0022-0|pmid=20658335 |year=2010 }}</ref> and self-organizing properties of the Fibonacci numbers that converge on the [[golden ratio]]. − − These events are physical bioattractors of aging from the perspective of Fibonacci mathematical modeling and dynamically systems theory. In practical terms, prediction in human development becomes possible in the same statistical sense in which the average temperature or precipitation at different times of the year can be used for weather forecasting. Each of the predetermined stages of human development follows an optimal epigenetic biological pattern. This phenomenon can be explained by the occurrence of Fibonacci numbers in biological DNA and self-organizing properties of the Fibonacci numbers that converge on the golden ratio. − − 从斐波那契数学模型和动力系统理论的角度来看，上述事件是衰老的'''物理生物吸引子Physical Bioattractors'''。实际上，正如一年中不同时间的平均气温和降水量可以用来预测天气，预测人类的发展在统计意义上同样是可能的。人类发展的每个'''预定阶段Predetermined Stages'''都遵循最佳的'''表观遗传生物模式 Epigenetic Biological Pattern'''。这种现象可以用 DNA 中的斐波那契数和收敛于黄金分割比的斐波那契数的自组织特性来解释。 − −
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'''动力系统理论 Dynamical Systems Theory'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量运行在一个某些区间离散、其他区间连续的集合、或者像cantor集一样任意的时间集合上时，人们就能得到时间尺度上的动力方程。
'''动力系统理论 Dynamical Systems Theory'''是一个用来描述复杂动力系统行为的数学领域，通常使用微分方程或差分方程。当采用微分方程时，该理论被称为连续动力系统。从物理学的角度来看，连续动力系统是经典力学的推广，也是运动方程的推广，不受极小作用原理Euler–Lagrange方程的约束。当采用差分方程时，该理论被称为离散动力系统。当时间变量运行在一个某些区间离散、其他区间连续的集合、或者像cantor集一样任意的时间集合上时，人们就能得到时间尺度上的动力方程。
'''算子 Operators'''是一个函数空间到函数空间上的映射O：X→X，广义的讲，对任何函数进行某一项操作都可以认为是一个算子，如求幂次、求微分等。某些情况下，也可以用'''混合算子 Mixed Operators'''来建模，如微分-差分方程。
== Applications 应用==
== Applications 应用==
=== In biomechanics 在运动生物力学中的应用===
=== In biomechanics 在运动生物力学中的应用===