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| 此词条暂由彩云小译翻译,翻译字数共488,未经人工整理和审校,带来阅读不便,请见谅。 | | 此词条暂由彩云小译翻译,翻译字数共488,未经人工整理和审校,带来阅读不便,请见谅。 |
| + | 此词条由舒寒初步翻译。 |
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| In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time. | | In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time. |
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− | If there exist distinct ''n'' and ''m'' such that | + | If there exist distinct ''n'' and ''m'' such that <math>f_n(x) = f_m(x)</math> |
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− | 如果存在不同的n和m使 | + | 如果存在不同的n和m使:<math>f_n(x) = f_m(x)</math> |
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− | :<math>f_n(x) = f_m(x)</math> | |
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| then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic. | | then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic. |
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| 那么x称为前周期点。所有周期点都是预周期点。 | | 那么x称为前周期点。所有周期点都是预周期点。 |
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| If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if | | If ''f'' is a [[diffeomorphism]] of a [[differentiable manifold]], so that the [[derivative]] <math>f_n^\prime</math> is defined, then one says that a periodic point is ''hyperbolic'' if |
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− | If f is a diffeomorphism of a differentiable manifold, so that the derivative <math>f_n^\prime</math> is defined, then one says that a periodic point is hyperbolic if
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| :<math>|f_n^\prime|\ne 1,</math> | | :<math>|f_n^\prime|\ne 1,</math> |
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− | 如果f是微分流形的微分同胚,则定义了导数f_n,如果f_n^prime不等于1,那么f是双曲周期点,
| + | 如果f是微分流形的微分同胚,则定义了导数<math>f_n^\prime</math>,如果:<math>|f_n^\prime|\ne 1,</math>,那么f是双曲周期点, |
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− | that it is ''[[Attractor|attractive]]'' if
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− | :<math>|f_n^\prime|< 1,</math> | |
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− | 如果f_n^prime < 1,则称周期点f为吸引子,
| + | that it is ''[[Attractor|attractive]]'' if :<math>|f_n^\prime|< 1,</math> |
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− | and it is ''repelling'' if
| + | 如果:<math>|f_n^\prime|< 1,</math>,则称周期点f为吸引子, |
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− | :<math>|f_n^\prime|> 1.</math> | + | and it is ''repelling'' if:<math>|f_n^\prime|> 1.</math> |
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− | 如果f_n^prime > 1,则称周期点f为排斥子。
| + | 如果:<math>|f_n^\prime|> 1.</math>,则称周期点f为排斥子。 |
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| If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]]. | | If the [[dimension]] of the [[stable manifold]] of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its [[unstable manifold]] is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or [[saddle point]]. |
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| The [[logistic map]] | | The [[logistic map]] |
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− | The logistic map
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| 后勤地图 | | 后勤地图 |
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| :<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math> | | :<math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math> |
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− | <math>x_{t+1}=rx_t(1-x_t), \qquad 0 \leq x_t \leq 1, \qquad 0 \leq r \leq 4</math>
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− | x_{t+1}=rx_t(1-x_t),qquad 0 leq x _ t leq 1,qquad 0 leq r leq 4
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| exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' − 1) / ''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' − 1) / ''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic). | | exhibits periodicity for various values of the parameter ''r''. For ''r'' between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which [[attractor|attracts]] all orbits). For ''r'' between 1 and 3, the value 0 is still periodic but is not attracting, while the value {{nowrap|(''r'' − 1) / ''r''}} is an attracting periodic point of period 1. With ''r'' greater than 3 but less than 1 + {{radic|6}}, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and {{nowrap|(''r'' − 1) / ''r''}}. As the value of parameter ''r'' rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of ''r'' one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic). |
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| 参数r的各种值呈现周期性。对于介于0到1之间的r,0是唯一的周期点,周期为1(给出了吸引所有轨道的序列0,0,0,... )。对于介于1到3之间的r,值0仍然是周期性的,但不是吸引点,而该值是周期1的吸引周期点。当r大于3但小于1 + 时,存在一对周期2的点,它们共同构成一个吸引序列,非吸引周期1点为0。当参数r的值上升到4时,会出现周期为正的一组周期点;对于 r 的某些值,这些重复序列中的一个被吸引,而对于其他值,则没有一个被吸引(几乎所有的轨道都是混乱的)。 | | 参数r的各种值呈现周期性。对于介于0到1之间的r,0是唯一的周期点,周期为1(给出了吸引所有轨道的序列0,0,0,... )。对于介于1到3之间的r,值0仍然是周期性的,但不是吸引点,而该值是周期1的吸引周期点。当r大于3但小于1 + 时,存在一对周期2的点,它们共同构成一个吸引序列,非吸引周期1点为0。当参数r的值上升到4时,会出现周期为正的一组周期点;对于 r 的某些值,这些重复序列中的一个被吸引,而对于其他值,则没有一个被吸引(几乎所有的轨道都是混乱的)。 |
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| 给定一个实整体动力系统(R,X,Φ) ,其中X为相空间,Φ为演化函数, | | 给定一个实整体动力系统(R,X,Φ) ,其中X为相空间,Φ为演化函数, |
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− | :<math>\Phi: \mathbb{R} \times X \to X</math>
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| <math>\Phi: \mathbb{R} \times X \to X</math> | | <math>\Phi: \mathbb{R} \times X \to X</math> |
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− | Phi:实数集{R}乘 X → X
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| a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that | | a point ''x'' in ''X'' is called ''periodic'' with ''period'' ''t'' if there exists a ''t'' > 0 so that |
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− | a point x in X is called periodic with period t if there exists a t > 0 so that
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| 如果存在 t > & thinsp; 0,则X中的点x称为周期为t的周期。因此 | | 如果存在 t > & thinsp; 0,则X中的点x称为周期为t的周期。因此 |
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− | :<math>\Phi(t, x) = x\,</math>
| + | <math>\Phi(t, x) = x\,</math> |
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− | Phi (t,x) = x,
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| The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''. | | The smallest positive ''t'' with this property is called ''prime period'' of the point ''x''. |
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− | The smallest positive t with this property is called prime period of the point x.
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| 具有此性质的最小正t称为点x的素数周期。 | | 具有此性质的最小正t称为点x的素数周期。 |
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| * Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R''' | | * Given a periodic point ''x'' with period ''p'', then <math>\Phi(t,x) = \Phi(t+p,x)</math> for all ''t'' in '''R''' |
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| * 给定一个周期为“p”的周期点“x”,则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x) | | * 给定一个周期为“p”的周期点“x”,则对于“R”中所有“t”的Phi(t,x) = \Phi(t+p,x) |
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| * Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period. | | * Given a periodic point ''x'' then all points on the [[orbit (dynamics)|orbit]] <math>\gamma_x</math> through ''x'' are periodic with the same prime period. |
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− | * 给定周期点“x”,则在轨道gamma_x上的所有点都具有相同的素数周期 | + | * 给定周期点“x”,则在轨道 <math>\gamma_x</math>上的所有点都具有相同的素数周期 |
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