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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==

==Basins of attraction吸引区==

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>

 

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。

===Linear equation or system===

===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.

 

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 f (x) = x ^ 3-2x ^ 2-11x + 12 </math > ，下面的初始条件是连续的吸引盆地:

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:

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.

 

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.

用牛顿法求解 x < sup > 5 </sup >-1 = 0。相似颜色区域中的点映射到同一个根; 颜色较深意味着需要更多的迭代来收敛。

用牛顿法求解 x⁵&nbsp;−&nbsp;1&nbsp;=&nbsp;0。相似颜色区域中的点映射到同一个根; 颜色较深意味着需要更多的迭代来收敛。

Similar features apply to linear [[differential equation]]s. 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 equation]]s. 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.

 

2.35287527  converges to 4;

2.35287527  converges to 4;

2.35287527汇聚为4;

2.35287527汇聚为4;

===Nonlinear equation or system===

===Nonlinear equation or system非线性方程或系统===

2.35284172  converges to −3;

2.35284172  converges to −3;

2.35284172  converges to −3;

2.35284172  收敛到 −3;

2.35283735  converges to 4;

2.35283735  converges to 4;

2.35283735汇聚至4;

2.35283735收敛到4;

Equations or systems that are [[nonlinear system|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 number|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,<ref>Dence, Thomas, "Cubics, chaos and Newton's method", ''[[Mathematical Gazette]]'' 81, November 1997, 403–408.</ref> 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 system|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 number|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,<ref>Dence, Thomas, "Cubics, chaos and Newton's method", ''[[Mathematical Gazette]]'' 81, November 1997, 403–408.</ref> for the function <math>f(x)=x^3-2x^2-11x+12</math>, the following initial conditions are in successive basins of attraction:

2.352836327  converges to −3;

2.352836327  converges to −3;

[[File:newtroot 1 0 0 0 0 m1.png|thumb|Basins of attraction in the complex plane for using Newton's method to solve ''x''⁵&nbsp;−&nbsp;1&nbsp;=&nbsp;0. Points in like-colored regions map to the same root; darker means more iterations are needed to converge.]]

[[File:newtroot 1 0 0 0 0 m1.png|thumb|Basins of attraction in the complex plane for using Newton's method to solve ''x''⁵&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.

== Partial differential equations ==

== Partial differential equations偏微分方程 ==

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

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

[[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.

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

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

For the three-dimensional, incompressible Navier–Stokes equation with periodic [[boundary condition]]s, if it has a global attractor, then this attractor will be of finite dimensions.<ref>[[Geneviève Raugel]], Global Attractors in Partial Differential Equations, ''Handbook of Dynamical Systems'', Elsevier, 2002, pp. 885–982.</ref>

For the three-dimensional, incompressible Navier–Stokes equation with periodic [[boundary condition]]s, if it has a global attractor, then this attractor will be of finite dimensions.<ref>[[Geneviève Raugel]], Global Attractors in Partial Differential Equations, ''Handbook of Dynamical Systems'', Elsevier, 2002, pp. 885–982.</ref>

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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

从计算的角度来看，吸引子可以自然地看作自激吸引子或自激吸引子

从计算的角度来看，吸引子可以自然地看作自激吸引子或自激吸引子

== Numerical localization (visualization) of attractors: self-excited and hidden attractors ==

== 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.

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).

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''[[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. |

year = 2011 |

year = 2011 |

</ref> 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 oscillator|Van der Pol]], [[Belousov–Zhabotinsky reaction|Belousov–Zhabotinsky]], [[Lorenz attractor|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.

</ref> 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 oscillator|Van der Pol]], [[Belousov–Zhabotinsky reaction|Belousov–Zhabotinsky]], [[Lorenz attractor|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.

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