The trajectory of the network consists of the set of states along the evolution path as the network converges toward the attractor state.The basin of attraction is the set of states that results in movement towards a certain attractor.Various types of attractors may be used to model different types of network dynamics.…

If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.In the case of a system of linear equations, the two main classes of iterative methods are the stationary iterative methods, and the more general Krylov subspace methods.…

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 eventually be iterated into the attractor.For a stable linear system, every point in the phase space is in the basin of attraction.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.…

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.…

Resilience can change in stable states when environmental parameters are shifted.Often, humans influence stable states by reducing the resilience of basins of attraction.…

In Chaos theory, wada basins arise very frequently.Usually, the wada property can be seen in the basin of attraction of dissipative dynamical systems.But the exit basins of Hamiltonian system can also show the wada property.…

Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems.A basin having the property that every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin, or said to have the Wada property.…

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.…

When dealing with complex functions, Newton's method can be directly applied to find their zeroes.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.These sets can be mapped as in the image shown.…

Therefore, reforestation is often unsuccessful because conditions are too dry to allow the trees to grow (Wilson and Agnew 1992).By their very nature, basins of attraction display resilience.Ecosystems are resistant to state shifts - they will only undergo shifts under substantial perturbations - but some states are more resilient than others.…

In a chaotic scattering system, the invariant set of the system is usually not directly accessible because it is non-attracting and typically of measure zero.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.Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set.…

These sets can be mapped as in the image shown.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.…

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.For example, if one uses a real initial condition to seek a root of x^2+1, all subsequent iterates will be real numbers and so the iterations cannot converge to either root, since both roots are non-real.…

A trajectory in search space is generated, which maps an initial point to a local optimum, where local search is stuck (no improving neighbors are available).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.A local optimum can be isolated (surrounded by non-locally-optimal points) or part of a plateau, a locally optimal region with more than one point of equal value.…

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.Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set.…

An oscillation in a dynamical system can be easily localized numerically if initial conditions from its open neighborhood lead to long-time behavior that approaches the oscillation.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.

When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria.Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction (i.e. is less risky).…

For a stable linear system, every point in the phase space is in the basin of attraction.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.…