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| 此词条暂由彩云小译翻译,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,未经人工整理和审校,带来阅读不便,请见谅。 |
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− | #REDIRECT [[Periodic point]]
| + | 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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− | REDIRECT Periodic point
| + | In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a 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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− | 重定向周期点
| + | 在数学中,在迭代函数和动力系统的研究中,函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。 |
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| + | == Iterated functions == |
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| + | Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself, |
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| + | Given a mapping f from a set X into itself, |
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| + | 给定一个从集合 x 到自身的映射 f, |
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| + | :<math>f: X \to X,</math> |
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| + | <math>f: X \to X,</math> |
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| + | x 到 x,数学 |
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| + | a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that |
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| + | a point x in X is called periodic point if there exists an n so that |
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| + | X 中的点 x 称为周期点,如果存在一个 n |
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| + | :<math>\ f_n(x) = x</math> |
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| + | <math>\ f_n(x) = x</math> |
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| + | [ math ] f _ n (x) = x |
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| + | where <math>f_n</math> is the ''n''th [[iterated function|iterate]] of ''f''. The smallest positive [[integer]] ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n'' (this is not to be confused with the notion of a [[periodic function]]). |
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| + | where <math>f_n</math> is the nth iterate of f. The smallest positive integer n satisfying the above is called the prime period or least period of the point x. If every point in X is a periodic point with the same period n, then f is called periodic with period n (this is not to be confused with the notion of a periodic function). |
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| + | 其中 f 的第 n 次迭代。满足上述条件的最小正整数 n 称为点 x 的素周期或最小周期。如果 x 中的每一个点都是周期点,周期 n 相同,那么 f 被称为周期点,周期 n (这不能和周期函数的概念混淆)。 |
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| + | If there exist distinct ''n'' and ''m'' such that |
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| + | If there exist distinct n and m such that |
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| + | 如果存在不同的 n 和 m |
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| + | :<math>f_n(x) = f_m(x)</math> |
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| + | <math>f_n(x) = f_m(x)</math> |
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| + | [ math > f _ n (x) = f _ m (x)] |
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| + | then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic. |
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| + | 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。 |
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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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| + | 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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| + | 如果 f 是微分流形的微分同胚,因此定义了导数 f _ n ^ prime </math > ,那么周期点是双曲的,如果 |
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| + | :<math>|f_n^\prime|\ne 1,</math> |
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| + | <math>|f_n^\prime|\ne 1,</math> |
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| + | 数学 | f _ n ^ prime | ne 1,</math > |
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| + | that it is ''[[Attractor|attractive]]'' if |
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| + | that it is attractive if |
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| + | 这是有吸引力的,如果 |
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| + | :<math>|f_n^\prime|< 1,</math> |
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| + | <math>|f_n^\prime|< 1,</math> |
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| + | [ math > | f _ n ^ prime | < 1,</math > |
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| + | and it is ''repelling'' if |
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| + | and it is repelling if |
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| + | 而且它会排斥如果 |
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| + | :<math>|f_n^\prime|> 1.</math> |
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| + | <math>|f_n^\prime|> 1.</math> |
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| + | [ math > | f _ n ^ prime | > 1 |
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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]]. |
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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. |
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| + | 如果周期点或不动点的稳定流形维数为零,则称其为源; 如果不稳定流形维数为零,则称其为汇; 如果稳定流形和不稳定流形都有非零维数,则称其为鞍点或鞍点。 |
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| + | === Examples === |
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| + | A period-one point is called a [[fixed point (mathematics)|fixed point]]. |
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| + | A period-one point is called a fixed point. |
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| + | 一个周期——一个点叫做不动点。 |
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| + | The [[logistic map]] |
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| + | The logistic map |
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| + | 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> |
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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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| + | < 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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| + | 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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| + | 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 attracts all orbits). For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value is an attracting periodic point of period 1. With r greater than 3 but less than 1 + , 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 . 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0到1之间的 r,0是唯一的周期点,周期为1(给出序列0,0,0,... ,吸引所有轨道)。对于1到3之间的 r,值0仍然是周期性的,但不是吸引点,而值是周期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,周期内出现一组周期点,其中任意一个正整数,对于 r 的某些值,这些重复序列中的一个是吸引的,而对于其他的序列,它们都不是吸引的(几乎所有的轨道都是混沌的)。 |
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| + | == Dynamical system == |
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| + | Given a [[real global dynamical system]] ('''R''', ''X'', Φ) with ''X'' the [[Phase space (dynamical system)|phase space]] and Φ the [[evolution function]], |
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| + | Given a real global dynamical system (R, X, Φ) with X the phase space and Φ the evolution function, |
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| + | 给定一个具有 x 相空间和 φ 演化函数的实整体动力系统(r,x,φ) , |
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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> |
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| + | 数学: 数学{ r }乘 x 到 x |
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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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| + | 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 t |
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| + | :<math>\Phi(t, x) = x\,</math> |
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| + | <math>\Phi(t, x) = x\,</math> |
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| + | Phi (t,x) = x,</math > |
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| + | 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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| + | 这个性质的最小正 t 称为点 x 的素周期。 |
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| + | === Properties === |
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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''' |
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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. |
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| + | ==See also== |
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| + | * [[Limit cycle]] |
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| + | * [[Limit set]] |
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| + | * [[Stable manifold|Stable set]] |
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| + | * [[Sharkovsky's theorem]] |
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| + | * [[Stationary point]] |
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| + | * [[Periodic points of complex quadratic mappings]] |
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| + | {{PlanetMath attribution|id=4516|title=hyperbolic fixed point}} |
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| + | [[Category:Limit sets]] |
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| + | Category:Limit sets |
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| + | 类别: 极限集 |
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| <noinclude> | | <noinclude> |
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− | <small>This page was moved from [[wikipedia:en:Periodic orbit]]. Its edit history can be viewed at [[周期点/edithistory]]</small></noinclude> | + | <small>This page was moved from [[wikipedia:en:Periodic point]]. Its edit history can be viewed at [[周期点/edithistory]]</small></noinclude> |
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| [[Category:待整理页面]] | | [[Category:待整理页面]] |