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齐次形式的单变量（单变量）线性差分方程<math>x_t=ax{t-1}</math>从除0以外的所有初始点| A>1发散到无穷大；没有吸引子，因此没有吸引池。但是如果| a |<1，则数线图上的所有点渐进地（或在0的情况下直接）到0；0是吸引子，整个数线是吸引域。

齐次形式的单变量（单变量）线性差分方程<math>x_t=ax{t-1}</math>从除0以外的所有初始点| A>1发散到无穷大；没有吸引子，因此没有吸引池。但是如果| a |<1，则数线图上的所有点渐进地（或在0的情况下直接）到0；0是吸引子，整个数线是吸引域。

==Basins of attraction吸引区==

==Basins of attraction吸引池==

Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form <math>X_t=AX_{t-1}</math> in terms of square matrix A will have all elements of the dynamic vector diverge to infinity if the largest eigenvalue of A is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n-dimensional space of potential initial vectors is the basin of attraction.

Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form <math>X_t=AX_{t-1}</math> in terms of square matrix A will have all elements of the dynamic vector diverge to infinity if the largest eigenvalue of A is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n-dimensional space of potential initial vectors is the basin of attraction.

An attractor's '''basin of attraction''' is the region of the [[phase space]], over which iterations are defined, such that any point (any [[initial condition]]) in that region will [[asymptotic behavior|asymptotically]] be iterated into the attractor. For a [[stability (mathematics)|stable]] [[linear system]], every point in the phase space is in the basin of attraction. However, in [[nonlinear system]]s, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.<ref>{{cite journal|last1=Strelioff|first1=C.|last2=Hübler|first2=A.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|page=044101}}</ref>

An attractor's '''basin of attraction''' is the region of the [[phase space]], over which iterations are defined, such that any point (any [[initial condition]]) in that region will [[asymptotic behavior|asymptotically]] be iterated into the attractor. For a [[stability (mathematics)|stable]] [[linear system]], every point in the phase space is in the basin of attraction. However, in [[nonlinear system]]s, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.<ref>{{cite journal|last1=Strelioff|first1=C.|last2=Hübler|first2=A.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|page=044101}}</ref>

吸引子的“吸引域”是[[相空间]]的区域，在该区域上定义迭代，因此该区域中的任何点（任何[[初始条件]]）将[[渐近行为|渐进]]迭代到吸引子中。对于一个[[稳定性（数学）|稳定]][[线性系统]]，相空间中的每一点都在吸引域中。然而，在[[非线性系统]]s中，有些点可能直接或渐近地映射到无穷大，而另一些点可能位于不同的吸引域中并渐进地映射到不同的吸引子；其他初始条件可能在或直接映射到一个非吸引点或循环中。<ref>{{cite journal|last1=Strelioff|first1=C.|last2=Hübler|first2=A.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|page=044101}}</ref>

吸引子的“吸引池”是[[相空间]]的区域，在该区域上定义迭代，因此该区域中的任何点（任何[[初始条件]]）将[[渐近行为|渐进]]迭代到<font color="#ff8000"> 吸引子</font>中。对于一个[[稳定性（数学）|稳定]][[线性系统]]，相空间中的每一点都在吸引池中。然而，在[[非线性系统]]s中，有些点可能直接或渐近地映射到无穷大，而另一些点可能位于不同的吸引池中并渐进地映射到不同的吸引子；其他初始条件可能位于或直接映射到一个非吸引点或循环中。<ref>{{cite journal|last1=Strelioff|first1=C.|last2=Hübler|first2=A.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|page=044101}}</ref>

Similar features apply to linear differential equations. The scalar equation <math> dx/dt =ax</math> causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system <math>dX/dt=AX</math> gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

Similar features apply to linear differential equations. The scalar equation <math> dx/dt =ax</math> causes all initial values of x except zero to diverge to infinity if a > 0 but to converge to an attractor at the value 0 if a < 0, making the entire number line the basin of attraction for 0. And the matrix system <math>dX/dt=AX</math> gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.

类似的特征也适用于线性微分方程。标量方程 < math > dx/dt = ax </math > 导致除0以外的所有 x 的初始值在 a > 0时发散到无穷大，但在 a < 0时收敛到吸引子，使得整个数列沿着吸引盆的方向为0。矩阵系统 < math > dX/dt = AX </math > 如果矩阵 a 的任何特征值是正的，则该矩阵系统从除零向量以外的所有初始点发散; 但如果所有特征值都是负的，则零向量是吸引域为整个相空间的吸引子。

类似的特征也适用于线性微分方程。标量方程<math> dx/dt =ax</math> 导致除0以外的所有 x 的初始值在 a > 0时发散到无穷大，但在 a < 0时收敛到<font color="#ff8000"> 吸引子</font>，使整条数线成为0的吸引池。矩阵系统  <math>dX/dt=AX</math>如果矩阵 A 的任何特征值是正的，则该矩阵系统从除零向量以外的所有初始点发散; 但如果所有特征值都是负的，则零向量是吸引池为整个相空间的<font color="#ff8000"> 吸引子</font>。

===Linear equation or system线性方程或系统===

===Linear equation or system线性方程或系统===

A single-variable (univariate) linear [[difference equation]] of the [[homogeneous equation|homogeneous form]] <math>x_t=ax_{t-1}</math> diverges to infinity if |''a''| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |''a''| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.

A single-variable (univariate) linear [[difference equation]] of the [[homogeneous equation|homogeneous form]] <math>x_t=ax_{t-1}</math> diverges to infinity if |''a''| > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if |''a''| < 1 all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.

[[齐次方程|齐次形式]]<math>x|t=ax{t-1}</math>的单变量（单变量）线性[[差分方程]]发散到无穷大，如果除了0以外的所有初始点|“A”>>1；没有吸引子，因此没有吸引池。但如果|“a”小于1，则数线图上的所有点渐进地（或在0的情况下直接映射）到0；0是吸引子，整个数线是吸引域。

[[齐次方程|齐次形式]]<math>x|t=ax{t-1}</math>的单变量（单变量）线性[[差分方程]]发散到无穷大，如果除了0以外的所有初始点|“A”>>1；没有<font color="#ff8000"> 吸引子</font>，因此没有吸引池。但如果 |''a''| < 1，则数线图上的所有点渐进地（或在0的情况下直接映射）到0；0是吸引子，整个数线是吸引池。

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple&mdash;it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function <math>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins of attraction:

Equations or systems that are nonlinear can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one real root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple&mdash;it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example, for the function <math>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins of attraction:

与线性系统相比，非线性方程或系统可以产生更多种类的行为。一个例子是牛顿迭代非线性表达式根的方法。如果表达式有多个实根，则迭代算法的某些起始点会渐近地导致其中一个根，而其他起始点会导致另一个根。表达式根部的吸引盆地通常不是简单的---- 它不是简单地将最靠近一个根部的点全部映射到那里，形成一个由附近点组成的吸引盆地。吸引力的盆地可以是无限的，也可以是任意的小。例如，对于函数 <math>f(x)=x^3-2x^2-11x+12</math> ，下面的初始条件是连续的吸引池:

与线性系统相比，非线性方程或系统可以产生更多种类的行为。一个例子是非线性表达式根的牛顿迭代法。如果表达式有多个实根，则迭代算法的某些起始点会渐近地得出其中一个根，而其他起始点会得出另一个根。表达式根的吸引池通常并不简单，最接近一个根的点都映射到那里，从而形成由附近点组成的吸引区。吸引的区域在数值上可以是无限的，可以任意小。例如，对于函数<math>f(x)=x^3-2x^2-11x+12</math>，以下初始条件在连续的吸引池中：

Likewise, a linear [[matrix difference equation]] in a dynamic [[coordinate vector|vector]] ''X'', of the homogeneous form <math>X_t=AX_{t-1}</math> in terms of [[square matrix]] ''A'' will have all elements of the dynamic vector diverge to infinity if the largest [[eigenvalue]] of ''A'' is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire ''n''-dimensional space of potential initial vectors is the basin of attraction.

Likewise, a linear [[matrix difference equation]] in a dynamic [[coordinate vector|vector]] ''X'', of the homogeneous form <math>X_t=AX_{t-1}</math> in terms of [[square matrix]] ''A'' will have all elements of the dynamic vector diverge to infinity if the largest [[eigenvalue]] of ''A'' is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire ''n''-dimensional space of potential initial vectors is the basin of attraction.

同样地，在动态[[坐标向量|向量]''X''中的线性[[矩阵差分方程]]在[[平方矩阵]]''a'中的齐次形式<math>X|t=AX{t-1}</math>中，如果“a”的最大[[特征值]]在绝对值上大于1，则动态向量的所有元素将发散到无穷大；没有吸引子，也没有吸引池。但如果最大特征值小于1，则所有初始向量将渐近收敛于零向量，即吸引子；潜在初始向量的整个n维空间就是吸引池。

同样地，在动态[[坐标向量|向量]''X''中的线性[[矩阵差分方程]]在[[平方矩阵]]''a'中的齐次形式<math>X|t=AX{t-1}</math>中，如果“a”的最大[[特征值]]在绝对值上大于1，则动态向量的所有元素将发散到无穷大；没有吸引子，也没有吸引池。但如果最大特征值小于1，则所有初始向量将渐近收敛于零向量，即零为吸引子；潜在初始向量的整个n维空间就是吸引池。

Basins of attraction in the complex plane for using Newton's method to solve x⁵&nbsp;−&nbsp;1&nbsp;=&nbsp;0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

Basins of attraction in the complex plane for using Newton's method to solve x⁵&nbsp;−&nbsp;1&nbsp;=&nbsp;0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

Newton's method can also be applied to complex functions to find their roots. Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction are fractals.

牛顿法也可以应用于求复变函数的根。在复杂的平面上，每个根部都有一个吸引盆; 这些盆地可以如图所示绘制出来。可以看出，组合盆地的吸引力为一个特定的根可以有许多不相连的地区。对于许多复杂的函数，吸引盆地的边界是分形。

牛顿法也可以应用于求复变函数的根。在复杂的平面上，每个根部都有一个吸引池; 这些区域可以如图所示绘制出来。可以看出，组合区域的吸引力为一个特定的根可以有许多不相连的地区。对于许多复杂的函数，吸引池的边界是分形。

:2.35287527  converges to 4;

:2.35287527  converges to 4;

Parabolic partial differential equations may have finite-dimensional attractors.  The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.

Parabolic partial differential equations may have finite-dimensional attractors.  The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.

抛物型偏微分方程可能具有有限维吸引子。方程的扩散部分阻尼更高的频率，在某些情况下导致一个全局吸引子。Ginzburg-Landau 方程、 Kuramoto-Sivashinsky 方程和二维强迫 Navier-Stokes 方程都具有有限维的全局吸引子。

<font color="#ff8000"> 抛物型偏微分方程</font>可能具有有限维<font color="#ff8000"> 吸引子</font>。方程的扩散部分阻尼更高的频率，在某些情况下导致一个全局吸引子。<font color="#ff8000"> Ginzburg-Landau 方程、 Kuramoto-Sivashinsky 方程和二维强迫 Navier-Stokes 方程</font>都具有有限维的全局吸引子。

:2.352836327  converges to −3;

:2.352836327  converges to −3;

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< ! -- 一旦隐藏吸引子中的 < nowiki > </nowiki > 得到解决，这应该被取消评论。更多信息请参见演讲页面。

< ! -- 一旦隐藏吸引子中的 < nowiki > </nowiki > 得到解决，应该取消评论。更多信息请参见演讲页面。

[[Parabolic partial differential equation]]s may have finite-dimensional attractors.  The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The ''Ginzburg–Landau'', the ''Kuramoto–Sivashinsky'', and the two-dimensional, forced [[Navier–Stokes equation]]s are all known to have global attractors of finite dimension.

[[Parabolic partial differential equation]]s may have finite-dimensional attractors.  The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The ''Ginzburg–Landau'', the ''Kuramoto–Sivashinsky'', and the two-dimensional, forced [[Navier–Stokes equation]]s are all known to have global attractors of finite dimension.

[[抛物型偏微分方程]]可能具有有限维吸引子。方程的扩散部分会阻尼更高的频率，在某些情况下会导致全局吸引子。“金茨堡-兰道”、“库拉莫托-西瓦辛斯基”和二维受迫[[纳维-斯托克斯方程]]都具有有限维的全局吸引子。

[[抛物型偏微分方程]]可能具有有限维吸引子。方程的扩散部分会阻尼更高的频率，在某些情况下会导致全局吸引子。<font color="#ff8000"> “金茨堡-兰道”、“库拉莫托-西瓦辛斯基”和二维受迫[[纳维-斯托克斯方程]]</font>都具有有限维的全局吸引子。

Chaotic hidden attractor (green domain) in Chua's system.

Chaotic hidden attractor (green domain) in Chua's system.

From a computational point of view, attractors can be naturally regarded as self-excited attractors or

From a computational point of view, attractors can be naturally regarded as self-excited attractors or

== Numerical localization (visualization) of attractors: self-excited and hidden attractors 吸引子的数值局部化（可视化）：自激吸引子和隐吸引子==

== Numerical localization (visualization) of attractors: self-excited and hidden attractors 吸引子的数值局部化（可视化）：自激吸引子和隐吸引子==

hidden attractors. Self-excited attractors can be localized numerically by standard computational procedures, in which after a transient sequence, a trajectory starting from a point on an unstable manifold in a small neighborhood of an unstable equilibrium reaches an attractor, such as the classical attractors in the Van der Pol, Belousov–Zhabotinsky, Lorenz, and many other dynamical systems.  In contrast, the basin of attraction of a hidden attractor does not contain neighborhoods of equilibria, so the hidden attractor cannot be localized by standard computational procedures.

hidden attractors. Self-excited attractors can be localized numerically by standard computational procedures, in which after a transient sequence, a trajectory starting from a point on an unstable manifold in a small neighborhood of an unstable equilibrium reaches an attractor, such as the classical attractors in the Van der Pol, Belousov–Zhabotinsky, Lorenz, and many other dynamical systems.  In contrast, the basin of attraction of a hidden attractor does not contain neighborhoods of equilibria, so the hidden attractor cannot be localized by standard computational procedures.

隐藏吸引子。自激吸引子可以用标准的计算程序进行数值局部化，在一个瞬态序列之后，从不稳定平衡点的小邻域的不稳定流形上的一个点出发的轨迹到达一个吸引子，如 Van der Pol、 Belousov-Zhabotinsky、 Lorenz 等许多其他动力系统中的经典吸引子。相反，一个隐藏吸引子的吸引盆不包含平衡邻域，因此隐藏吸引子不能被标准的计算程序局部化。

隐藏吸引子。<font color="#ff8000"> 自激吸引子</font>可以用标准的计算程序进行数值局部化，在一个瞬态序列之后，从不稳定平衡点的小邻域的不稳定流形上的一个点出发的轨迹将到达一个吸引子，如 Van der Pol、 Belousov-Zhabotinsky、 Lorenz 等许多其他动力系统中的经典吸引子。相反，一个隐藏吸引子的吸引池不包含平衡邻域，因此隐藏吸引子不能被标准的计算程序局部化。

[[File:Chua-chaotic-hidden-attractor.jpg|thumb|

[[File:Chua-chaotic-hidden-attractor.jpg|thumb|

Chaotic [[hidden attractor]] (green domain) in [[Chua's circuit|Chua's system]].

Chaotic [[hidden attractor]] (green domain) in [[Chua's circuit|Chua's system]].

Trajectories with initial data in a neighborhood of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange).

Trajectories with initial data in a neighborhood of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange).

初始数据位于两个鞍点附近（蓝色）的轨迹趋向于无穷大（红色箭头）或趋向于（黑色箭头）稳定的零平衡点（橙色）。

初始数据位于两个鞍点附近（蓝色）的轨迹趋向于无穷大（红色箭头）或趋向于（黑色箭头）稳定的零平衡点（橙色）。

''[[hidden attractor]]s''.<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |

''[[hidden attractor]]s''.<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |

从计算的角度来看，吸引子可以自然地看作是“自激吸引子”或

从计算的角度来看，吸引子可以自然地看作是“自激吸引子”或''[[隐藏吸引子]]s''<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |

 

'[[隐藏吸引子]]s''<ref name="2011-PLA-Hidden-Chua-attractor">{{cite journal |author1=Leonov G.A. |author2=Vagaitsev V.I. |author3=Kuznetsov N.V. |

year = 2011 |

year = 2011 |