In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling. Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables.
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| - Center manifold (en)
- Varietà centrale (it)
- Центральное многообразие (ru)
- 中心流形 (zh)
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| - In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling. Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables. (en)
- Центра́льное многообра́зие особой точки автономного обыкновенного дифференциального уравнения — инвариантное многообразие в фазовом пространстве, проходящее через особую точку и касающееся инвариантного центрального подпространства линеаризации дифференциального уравнения. Важный объект изучения теории дифференциальных уравнений и динамических систем. В некотором смысле, вся нетривиальная динамика системы в окрестности особой точки сосредоточена на центральном многообразии. (ru)
- 中心流形（center manifold）是動力系統數學理論的一部份，最早是用此概念來判斷退化平衡點的穩定性。之後這個概念成為数学模型的建構基礎。 若將球往上拋。可根據牛顿运动定律預測球的運動，方式是求解有其位置以及速度的微分方程，但在時的行為就無法用牛顿运动定律來描述。在球反彈時，球會有形變，就無法用剛體的牛顿运动定律來預測系統的演進，需要用连续介质力学來描述組成球的所有粒子在形變前後的行為。在反彈後，球的形變會快速消失，球繼續依循牛顿运动定律。若將球視為是由許多互相影響的成份所組成的系統，牛頓運動定律對球的描述，只以位置、速度及旋轉方式呈現，即為變形球的中心流形。若有一系統是由許多互相影響成份所組成，而其影響效應會快速衰減，可以用中心流形，以較簡單的方式來描述系統。 中心流形在分岔理論中有重要的地位，因為系統在中心流形的位置會出現特殊的行為，在中也很重要，微尺度的長時間動態常常會受到相對簡單、變數尺度較大的中心流形吸引。 (zh)
- In matematica, in particolare nello studio dei sistemi dinamici, la varietà centrale di un punto di equilibrio di un sistema dinamico consiste nelle orbite il cui comportamento in prossimità del punto di equilibrio non è soggetto né all'attrazione della varietà stabile né alla repulsione di quella instabile. Sia dato un sistema dinamico: dove è una matrice costante, è di classe con in un intorno del punto di equilibrio isolato e: Se e sono le varietà stabile ed instabile dell'equazione: (it)
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| - In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling. Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables. (en)
- In matematica, in particolare nello studio dei sistemi dinamici, la varietà centrale di un punto di equilibrio di un sistema dinamico consiste nelle orbite il cui comportamento in prossimità del punto di equilibrio non è soggetto né all'attrazione della varietà stabile né alla repulsione di quella instabile. Sia dato un sistema dinamico: dove è una matrice costante, è di classe con in un intorno del punto di equilibrio isolato e: Se e sono le varietà stabile ed instabile dell'equazione: detto lo spazio generato dagli autovettori di associati ad autovalori con parte reale nulla, esiste una varietà invariante , detta varietà centrale, tangente in prossimità del punto di equilibrio. Non è necessariamente unica. (it)
- Центра́льное многообра́зие особой точки автономного обыкновенного дифференциального уравнения — инвариантное многообразие в фазовом пространстве, проходящее через особую точку и касающееся инвариантного центрального подпространства линеаризации дифференциального уравнения. Важный объект изучения теории дифференциальных уравнений и динамических систем. В некотором смысле, вся нетривиальная динамика системы в окрестности особой точки сосредоточена на центральном многообразии. (ru)
- 中心流形（center manifold）是動力系統數學理論的一部份，最早是用此概念來判斷退化平衡點的穩定性。之後這個概念成為数学模型的建構基礎。 若將球往上拋。可根據牛顿运动定律預測球的運動，方式是求解有其位置以及速度的微分方程，但在時的行為就無法用牛顿运动定律來描述。在球反彈時，球會有形變，就無法用剛體的牛顿运动定律來預測系統的演進，需要用连续介质力学來描述組成球的所有粒子在形變前後的行為。在反彈後，球的形變會快速消失，球繼續依循牛顿运动定律。若將球視為是由許多互相影響的成份所組成的系統，牛頓運動定律對球的描述，只以位置、速度及旋轉方式呈現，即為變形球的中心流形。若有一系統是由許多互相影響成份所組成，而其影響效應會快速衰減，可以用中心流形，以較簡單的方式來描述系統。 中心流形在分岔理論中有重要的地位，因為系統在中心流形的位置會出現特殊的行為，在中也很重要，微尺度的長時間動態常常會受到相對簡單、變數尺度較大的中心流形吸引。 (zh)
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