Basin of attraction of triangular maps with applications
We consider planar triangular maps . These maps preserve the fibration of the plane given by . We assume that there exists an invariant attracting fiber for the dynamical system generated by and we study the limit dynamics of those points in the basin of attraction of this invariant fiber, assuming that either it contains a global attractor, or it is filled by fixed or -periodic points. Finally, we apply our results to a variety of examples, from particular cases of triangular systems to some planar quasi-homogeneous maps, and some multiplicative and additive difference equations, as well.
2000 Mathematics Subject Classification: 39A10, 39A11, 39A20
Keywords: Attractors; difference equations; discrete
dynamical systems; triangular maps; periodic solutions;
In this paper we consider triangular systems of the form
where and are real sequences, and , and are continuous functions.
Observe that system (1) preserves the fibration of the plane given by , that is, it sends fibers of to other ones. We will assume that there exists a point which is a local stable attractor of the subsystem . In this case, we say that system (1) has a local attractive fiber . Our objective is to know if the asymptotic dynamics of the orbits corresponding to points in the basin of attraction of the limit fiber is characterized by the dynamics on this fiber, the limit dynamics. In all the cases, we will assume that the limit dynamics is very simple, that is: either (A) the fiber contains a global attractor, see Proposition 2; (B) the fiber is filled by fixed points, Theorem 6(a); or (C) it is filled by -periodic orbits, Theorem 6(b). Observe that there is no need to consider the case in which there is a global repellor on the fiber , since in this situation it is clear that any orbit with initial conditions in the basin of attraction of this fiber is unbounded (otherwise there should be accumulation points in the fiber).
Section 4 is devoted to show some applications of the main results. For instance, in Section 4.1 we study the limit dynamics of some linear quasi-homogeneous maps. The example considered there shows that, for this class of maps, the shape of the basins of attraction of the origin can have a certain level of complexity.
In Section 4.2 we apply our results to study the global dynamics of difference equations of multiplicative type , with being a function. In particular, we reobtain the results in  concerning the difference equations The more general recurrences are studied in Section 4.3. We also apply the results to study the additive difference equations in Section 4.4.
Finally, we also present several different type of recurrences for which the theory developed in this work applies in order to obtain a full characterization of their dynamics.
2 Motivating examples and main results
We started motivated by the following clarifying example, inspired in the ones developed for continuous systems in  (a particular case is also considered in , see also  and ), which shows that there are systems of type
with , such that is a global limit fiber, having a global attractor for the restricted dynamics on , but having unbounded orbits , with , in their the global dynamics.
Example A: Hyperbolic globally attracting fiber with a restricted global attractor but having unbounded solutions. Consider the system
with , and . Of course, this system preserves the fibration , and it has a global attracting limit fiber . On this fiber the dynamics is the following: the origin is attractive if ; the fiber is a continuum of fixed points if ; and the fiber is a continuum of –periodic orbits if . However, as shown in Proposition 1, if then there are initial conditions giving rise to unbounded solutions, and therefore the global dynamics is not characterized by the dynamics on the global limit fiber.
is invariant for system (3). Moreover, for any initial condition the associated orbit is given by , . If , then any initial condition gives rise to unbounded solutions, with .
Imposing that system (3) has solutions of the form , , we easily get
Observe that if (that is when ) we obtain that is an invariant curve. Moreover, if , then , and the orbits on are unbounded solutions.
Example A shows that, in general, for systems of type (2) with terms of degree greater or equal than in , we cannot expect that the global dynamics in the basin of attraction of the limit fiber is characterized by the dynamics on the fiber. However, it remains to explore the case of systems of type (1) with affine terms in . For these systems, the possible limit dynamics that we consider are the cases (A), (B) and (C) mentioned in the introduction.
The first result concerns the case (A), when the the limit fiber contains a global attractor. This case is characterized by the fact that , and the attractor is ;
Consider the system (1) with and continuous and . Suppose that is an attractive point of . Then, for all initial condition with in the basin of attraction of , we have .
We want to point out that the above one is not a local result. In fact, the convergence is guaranteed for all such that is in the basin of attraction of
It is not difficult to see that in the case (B), when the limit fiber is filled by fixed points, there are systems such that the orbits corresponding to initial conditions in the basin of attraction of the limit fiber tend to some of the fixed points on the limit fiber, which depend on the initial conditions. Indeed:
Example B: Hyperbolic attracting fiber and fast enough convergence to the limit dynamics. Consider the system given by
with , and . Then
Observe that the infinite product is convergent since, as , we have
So, for each initial condition the associated orbit converges to the fixed point .
The next example shows, however, that the above situation does not occur when the convergence of to is too slow.
Example C: Hyperbolic attracting fiber but slow convergence to the limit dynamics. Consider the system
with , and
Again, an straightforward computation gives
Using that , which is a divergent series, we get that the infinite product is also divergent. Hence each initial condition gives rise to an unbounded sequence and, therefore, the dynamics on the basin of attraction of the limit fiber is not characterized by the dynamics on the fiber.
The above example leads us to introduce the following definition. As we will see in Example E below, it gives an optimal characterization of the speed of convergence of the terms and to guarantee the convergence of the orbits in the basin of attraction of the limit fiber.
Definition 3 (Fast enough convergence to the limit dynamics property).
If is a stable attractor of , we say that the attracting fiber of system (1) with , has the fast enough convergence to the limit dynamics property if there exists such that if is a solution of satisfying for some , then there exists such that for all , , and also there exist two functions such that for all it is satisfied:
and are non decreasing for ;
Let be a stable attractive fixed point of . We say that is locally contractive at if there exists an open neighborhood of such that any
Prior to state the next results, we state the following result about local contractivity.
Let be a continuous function and let be a stable attractive fixed point of .
If is an orientation preserving function, then is locally contractive at .
There exist orientation reversing functions which are not locally contractive at .
If and is an hyperbolic fixed point of then is locally contractive at .
Now we are ready to present our result about the cases (B) and (C). Observe that if and , then the limit fiber is filled by fixed points (if then there are unbounded solutions). When , the fiber is filled by -periodic orbits and there is a fixed point given by .
Consider system (1) where and are continuous functions, and . Suppose that is an attracting fiber satisfying the fast enough convergence to the limit dynamics property, and consider initial conditions with in the basin of attraction of for the recurrence Then
If and , then there exists such that
If and additionally is locally contractive at , then and
Notice that the above result must not be interpreted in the sense that the limit is different for each initial condition .
Example D: Hyperbolic attracting fiber and fast enough convergence to the limit dynamics via Theorem 6. Consider the systems given by
with , , and .
Fixing and taking ; setting , , , with , we have that hypothesis and are trivially fulfilled. With respect hypothesis we have
On the other hand and
so is also fulfilled. From Theorem 6, for each initial condition there exists such that , and also there exists such that: if then , and if , then and .
The next example shows that Theorem 6 is optimal when is not a hyperbolic attractor of .
Example E: Non-hyperbolic attracting fiber. Optimal characterization of fast enough and slow convergence to the limit dynamics. Consider the systems given by
with , , and .
Take small enough and such that is in the basin of attraction of . Now, we claim that if then there exists such that ; and if then is a divergent sequence if , and if . To prove this claim, we will use the following result that we learned from S. Stević . Its proof is attributed to E. Jacobsthal, see also [17, Problem 174, page 217]:
Theorem 7 (E. Jacobsthal).
Let with , be a continuous function such that for every and , where , and are positive. Let and . Then
Hence if , then is a convergent sequence, and if then , and the claim is proved.
At this point we will see that the criterium given by Theorem 6 is optimal. By fixing small enough, taking , and using Theorem 7, we have that for the sequence defined by (5) and for all , there exists and such that if then
, , , and we have that for any , the hypothesis – are trivially fulfilled, and with respect we have that and
Observe that is convergent if and only if for all or, in other words, if and only if
Hence, Theorem 6 guarantees convergence of the sequence if for all , which is the optimal value.
The next result shows that if system (1) is a differentiable one, and is a hyperbolic attractor of , then the hypotheses of Theorem 6 are fulfilled. Furthermore, each point in the attracting fiber is the limit of an orbit of the basin of attraction of the fiber.
Consider system (1) where where is a neighborhood of , and . Suppose that is a hyperbolic attractor of . Then, for any in the basin of attraction of there exists such that:
If and , then .
If , then and .
Furthermore, for any point there exists an initial condition in the basin of attraction of the limit fiber such that .
3 Proofs of the main results
Proof of Proposition 2.
First observe that, from the continuity of , there exists such that for all such that , we have with . Consider such that is in the basin of attraction of . From the hypotheses, we can assume that there exists such that for all , we have hence, since is continuous, . Thus we have . Applying this last inequality we obtain
and therefore the sequence is bounded. As a consequence and . Hence, there are subsequences and . For these subsequences, from equation (1), and using the continuity of and , we have
hence we have that .
Proof of Proposition 5.
To prove , since is a stable attractor, we can take such that for all with In particular, is the only fixed point in the neighborhood Observe that since preserves orientation, if then
Assume that statement is not true. Given a sequence , then for each with we can find a point such that
Taking, if necessary, a subsequence, we can assume that for all (the opposite assumption can be treated in a similar manner), and also that the subsequence is monotone decreasing. Notice that, since and preserves orientation, also and the second inequality in (6) reads as
On the other hand, we can consider the sequence of iterates by of one point such that , obtaining for all and that That is, we can choose a subsequence with the property that and, as before, being monotone decreasing. Considering the continuous function we have that for all
Hence, for all it exists such that
It implies that and is not isolated as a fixed point of . A contradiction with our assumptions.
In order to see consider the function This map has two fixed points and Since the point is a repellor of the function The point is not an hyperbolic fixed point because but it is easy to see that the interval is invariant under the action of and that for all and Then, is an attracting fixed point of
On the other hand, if then satisfies hence condition (4) is not satisfied.
Statement (c) is a simple consequence of the mean value theorem.
Proof of Theorem 6.
In order to prove statement consider such that is in the basin of attraction of . First we start proving that the sequence is bounded. From the hypothesis, if is small enough, there exists such that for all , . Hence, to simplify the notation, in the following we will assume that the point is such that . Furthermore,
From the fact that is non decreasing for , if is large enough
In summary, we have . Applying this last inequality we obtain
Observe that from hypothesis we have , and by hypothesis , for all we have that . So
Observe that , because
Hence, regarding that if is large enough then , we have
so the sequence is bounded, and therefore also the sequence is bounded.
Now we are going to see that is a Cauchy sequence and therefore it has a limit . First observe that if is a bound of then
Therefore, since the series and are convergent, then for all there exists such that for all
and therefore is a Cauchy sequence, which completes the proof of statement (a).
In order to see , observe that
where and . After renaming , and we get the system
We are going to see that system (7) satisfies the hypothesis –, and since and , the result will follow from the convergence of the sequence guaranteed by statement (a).
Adding and subtracting both and to we get,