Article Contents
Article Contents

# Properties of basins of attraction for planar discrete cooperative maps

• * Corresponding author: M. R. S. Kulenović
• It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction $\mathcal{B}$ with relatively simple geometry: the boundary of each component of $\mathcal{B}$ consists of the union of two unordered curves, and the components of $\mathcal{B}$ are not comparable as sets. The boundary curves are Lipschitz if the map is of class $C^1$. Further, if a periodic point is in $\partial \mathcal{B}$, then $\partial\mathcal{B}$ is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

Mathematics Subject Classification: Primary: 37C70, 37D05, 37D10; Secondary: 39A23, 39A30.

 Citation:

• Figure 1.  (a) The basin of attraction $\mathcal{B}$ of the zero fixed point $\mathrm o$ of the map $T(x, y) = (y, x^3+y^3)$. Note that $\mathcal{B}$ is unbounded, and $\partial \mathcal{B}$ contains two fixed points $\mathrm{p}_1$ and $\mathrm{p}_2$ which are saddle points. The union of the stable manifolds of $\mathrm{p}_1$ and $\mathrm{p}_2$ gives $\partial \mathcal{B}$. (b) The basin of attraction $\mathcal{B}$ of the zero fixed point of the map $T(x, y) = (y^3, x^3+y^3)$. The set $\mathcal{B}$ is bounded, and $\partial \mathcal{B}$ contains two fixed points $\mathrm{p}_1$ and $\mathrm{p}_2$ (saddles) and a repelling minimal period-two point ${\mathrm q}_1$ and ${\mathrm q}_2$. The union of the stable manifolds of $\mathrm{p}_1$ and $\mathrm{p}_2$ gives $\partial \mathcal{B}$

Figure 2.  (a) the basin $\mathcal{B}$ of the fixed point $\bar{\mathrm{x}}$ has three components $\mathcal{B}^\prime$, $\mathcal{B}_*$, $\mathcal{B}^{\prime\prime}$ whose closure is in $\mbox{int}(\mathcal{R})$ and such that $\mathcal{B}^\prime <\!\!<_{se}\mathcal{B}_* <\!\!<_{se}\mathcal{B}^{\prime\prime}$. Each component has boundary curves $\mathcal{C}_+$ and $\mathcal{C}_-$. (b) The set $\mathcal{B}$ has only one component, which has part of its boundary in $\partial \mathcal{R}$. Also, $\bar{\mathrm{x}} \in \partial \mathcal{R}$. The point $\mathrm{p}$ is an endpoint of both boundary curves $\mathcal{C}_-$ and $\mathcal{C}_+$. The point $\mathrm{p}$ is a fixed point of $T$.

Figure 3.  Basin of attraction of the origin $\mathrm{o}$ for the map $U$ in (8)]Basin of attraction of the origin $\mathrm{o}$ for the map $U$ in (8). The points $\mathrm{p}$ and $\mathrm{q}$ are saddle fixed points.

Figure 4.  Graphs of $\phi$ from (12) and the identity function on the nonnegative semi axis.]Graphs of $\phi$ from (12) and the identity function on the nonnegative semi axis. $\phi$ has locally asymptotically stable fixed points $0$, $b = 2.06$, and a repelling fixed point $a = 0.95$. The real numbers $c = 6.03$ and $d = 12.80$ are pre-images of $a$. The basin of attraction of $0$ on the semi-axis consists of the intervals $0\leq t < a$ and $c< t < d$. All decimal numbers have been rounded to two decimals

Figure 5.  The partial derivatives of $V(x, y)$

Figure 6.  (a) Three components of the basin of attraction of the zero fixed point $\mathrm o$ of the map $V(x, y)$ in Example 1. Here $\mathrm r$, $\mathrm s$ are saddle fixed points, ${\mathrm p}_1$ and ${\mathrm q}_1$ are a saddle period-two point, ${\mathrm p}_2$ and ${\mathrm q}_2$ are repelling fixed points, and ${\mathrm p}_3$, ${\mathrm p}_4$, ${\mathrm q}_3$, ${\mathrm q}_4$ are eventual period-two points. The boundary of the invariant part of the basin of attraction consist of stable manifolds of saddle fixed points with a period-two endpoints. In addition, there are two eventually period-two points which are end points of another piece of the basin of attraction which is mapped into the invariant part. (b) The invariant component of the basin of attraction of the origin $\mathrm o$.

Figure 7.  Global dynamics for map (13) as given in Proposition 1. Here $\alpha = 0.4$ and $\delta = 0.7$

Figure 8.  The four cases in the definition of $\mathcal{C}_\pm$

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