Editing Attractor (section)

==Basins 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|bibcode=2006PhRvL..96d4101S }}</ref>

===Linear equation or system===

An univariate linear homogeneous difference equation <math>x_t=ax_{t-1}</math> diverges to infinity if <math>|a|>1</math> from all initial points except 0; there is no attractor and therefore no basin of attraction. But if <math>|a|<1</math> 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.

Likewise, a linear [[matrix difference equation]] in a dynamic [[coordinate vector|vector]] <math>X</math>, of the homogeneous form <math>X_t=AX_{t-1}</math> in terms of [[square matrix]] <math>A</math> will have all elements of the dynamic vector diverge to infinity if the largest [[eigenvalue]]s of <math>A</math> 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 <math>n</math>-dimensional space of potential initial vectors is the basin of attraction.

Similar features apply to linear [[differential equation]]s. The scalar equation <math> dx/dt =ax</math> causes all initial values of <math>x</math> except zero to diverge to infinity if <math>a>0</math> but to converge to an attractor at the value 0 if <math>a<0</math>, 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 <math>A</math> 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.

===Nonlinear equation or system===

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—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:

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

:2.35287527 converges to 4;
:2.35284172 converges to −3;
:2.35283735 converges to 4;
:2.352836327 converges to −3;
:2.352836323 converges to 1.

Newton's method can also be applied to [[complex analysis|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 [[fractal]]s.