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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:

与线性系统相比，[[非线性系统|非线性]]的方程或系统可以产生更丰富的行为。一个例子是迭代到非线性表达式根的[[牛顿方法]]。如果表达式有多个[[实数|实]]根，则迭代算法的某些起始点将渐近地导致其中一个根，而其他起点将导致另一个根。表达式根的吸引域通常并不简单，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。吸引的区域可以是无限的，可以任意小。例如，<ref>dance，Thomas，“Cubics，chaos and Newton's method”，“[[mathematic Gazette]]”811997年11月，403–408。</ref>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域中：

与线性系统相比，[[非线性系统|非线性]]的方程或系统可以产生更丰富的行为。一个例子是迭代到非线性表达式根的[[牛顿方法]]。如果表达式有多个[[实数|实]]根，则迭代算法的某些起始点将渐近地导致其中一个根，而其他起点将导致另一个根。表达式根的吸引域通常并不简单，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。吸引的区域可以是无限的，可以任意小。例如，<ref>Dence, Thomas, "Cubics, chaos and Newton's method", ''[[Mathematical Gazette]]'' 81, November 1997, 403–408.</ref> 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域中：

2.352836327  converges to −3;

2.352836327  converges to −3;

2.352836327  converges to −3;

2.352836327  收敛到 −3;

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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>

对于具有周期[[边界条件]]s的三维不可压缩Navier–Stokes方程，如果它有一个全局吸引子，那么这个吸引子将是有限维的。<ref>[[Geneviève Raugel]], Global Attractors in Partial Differential Equations, ''Handbook of Dynamical Systems'', Elsevier, 2002, pp. 885–982.</ref>

对于具有周期[[边界条件]]s的三维不可压缩Navier–Stokes方程，如果它有一个全局吸引子，那么这个吸引子将是有限维的。<ref>[[Geneviève Raugel]], Global Attractors in Partial Differential Equations, ''Handbook of Dynamical Systems'', Elsevier, 2002, pp. 885–982.</ref>

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

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

[[蔡氏电路|蔡氏系统]]中的混沌[[隐藏吸引子]]（绿域）。

[[蔡氏电路|蔡氏系统]]中的混沌[[隐吸引子]]（绿域）。

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).