basin of attraction

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  • The basin of attraction is the set of states that results in movement towards a certain attractor.
  • If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.
  • For a stable linear system, every point in the phase space is in the basin of attraction.
  • As can be seen, the combined basin of attraction for a particular root can have many disconnected regions.
  • Each root has a basin of attraction in the complex plane; these basins can be mapped as in the image shown.
  • Often, humans influence stable states by reducing the resilience of basins of attraction.
  • Usually, the wada property can be seen in the basin of attraction of dissipative dynamical systems.
  • Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems.
  • The basins of attraction can be infinite in number and arbitrarily small.
  • Each zero has a basin of attraction in the complex plane, the set of all starting values that cause the method to converge to that particular zero.
  • By their very nature, basins of attraction display resilience.
  • Therefore, the only way to infer the presence of members and to measure the properties of the invariant set is through the basins of attraction.
  • For many complex functions, the boundaries of the basins of attraction are fractals.
  • In some cases there are regions in the complex plane which are not in any of these basins of attraction, meaning the iterates do not converge.
  • The search space is therefore subdivided into basins of attraction, each consisting of all initial points which have a given local optimum as the final point of the local search trajectory.
  • Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set.
  • Such oscillation (or a set of oscillations) is called an attractor, and its attracting set is called the basin of attraction.
  • The basins of attraction -- the regions of the real number line such that within each region iteration from any point leads to one particular root -- can be infinite in number and arbitrarily small.
  • Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction (i.e. is less risky).
  • In nonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in one or another basin of attraction and map asymptotically into one or another attractor.
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