The radius of convergence (Julia set) is an important characteristic of the corresponding basin of attraction
. Julia set is the boundary of the area of complex variable values of [X.sub.n], belonging to which ensures the convergence of the iteration expression (2) to the attractor.
If this run converges, algorithm records the start point and the end point for an initial estimate on the radius of a basin of attraction
As for the definition of basin of attraction
and the stable manifold and so on, see [35-39].
Basin of attraction
for coexistence of two types of 1-periodic cycles is shown in Figure 10(c); the red region and the dark blue region represent the initial values which make supercavitating vehicle fall into different periods of oscillation eventually.
In nine of these regions the global dynamics is in competitive exclusion case, which means that all solutions converge to one of the equilibrium points on the axes and in only two situations we have competitive coexistence case, which means that the interior equilibrium points have substantial basin of attraction
. In all 11 cases, the zero equilibrium has some basin of attraction
which is a part of y-axis so we can say that in these cases system (1) exhibits weak Allee's effect.
The more adaptive walks terminating at an optimum, the larger the basin of attraction
. In this way the accessibility of a local optimum is a measure of its basin of attraction
It is known that the basin of attraction
of an iterative method on a particular function is an interesting tool to visually get to know how an iteration function behaves as a function of different initial points .
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Basin of attraction
of a fixed point ([bar.x], [bar.y]) of a map T, denoted as B(([bar.x],[bar.y])), is defined as the set of all initial points ([x.sub.0], [y.sub.0]) for which the sequence of iterates [T.sup.n](([x.sub.0], [y.sub.0])) converges to ([bar.x], [bar.y]).
In case 3, [E.sub.3] is stable equilibrium, which biologically implies that the two populations will eventually co-exist at [E.sub.3] if the initial size of the two populations is on the basin of attraction
The basin of attraction
(referred to hereafter as simply the basin) is the collection of points such that, when the system is initiated at any of them, it winds up at the associated attractor.
In 1992, Alexander and his colleagues found a theoretical example in which the set of initial conditions, or basin of attraction
, leading to one attractor is riddled with points corresponding to initial conditions leading to another outcome.