7. Discussion

Aside from the r parameter, this map also has the h parameter, which we set equal to unity. Unlike the previous two maps we study that have relevant behaviors past C = 10, the max value of the unswitched map is C = 3.85, but chaos is only

Our first chaotic point is Co = 3.3, and the corresponding bifurcation diagram is shown in Figure 18. There is a somewhat small region of periodicity from Ce = 3.704

The second to last figure, Figure 19 shows our final odd switching parameter, Co = 3.1 and the corresponding bifurcation diagram, which shows a similar range of

Our last figure, Figure 20, shows an example of "periodic + periodic = chaos", where we switch with Co = 3.57 and see chaos for most values between Ce = 2.5

Bifurcation diagram for Eqs. (17) and (18) showing the interval studied, as well as a close up of the chaotic

Bifurcation diagram for Eqs. (17) and (18) and Eqs. (19) and (20) with the Co value 3.3.

Bifurcation diagram for Eqs. (17) and (18) and Eqs. (19) and (20) with the Co value 3.1.

present above C = 3.0, as shown in Figure 17.

periodic parameter values, specifically, Ce = 3.70–3.72.

to 3.724.

Fractal Analysis

and 3.0.

Figure 17.

Figure 18.

Figure 19.

78

region..

In previous sections, we have analyzed five relevant ecological 2-D maps, setting a pattern of dynamic behavior similar to the well studied "chaos + chaos = periodic" in switched 1-D maps. Therefore, with the results discussed in this chapter, we can extend the 1-D maps conjecture to 2-D maps. The conjecture asserts that given a map with chaotic dynamics, we can find two parameters associated to chaotic trajectories that, when alternated yield a periodic trajectory. In general, we can consider these kinds of maps as nonautonomous maps because one of the parameters is a function of the iterations. In most case, we pick a parameter value for the even iterations and a different parameter for the odd iterations. But the connection with the Parrondo's paradox is associated with the kind of alternating parameters, which in the conjecture are parameter associated with chaotic, or, in general, complex trajectories.

The case of "chaos + chaos = periodic" was presented for the first time by Almeida et al. [16] for simple 1-D maps, and just recently for 2-D maps by Mendoza et al. [12]. The implication of the so-called Parrondo's dynamics has been used to model seasonality, but with the observation that, under the Parrondo dynamics, the case of "periodic + periodic = chaos" is also possible [15]. As generalization we have consider cases of "undesirable + undesirable = desirable" dynamics behaviors to analyze simple models of seasonality [23–25], which include migration or immigration [13, 14].

In the present analysis, we emphasize the use of bifurcation diagrams to find intervals of values in parameter space that could satisfy the "undesirable + undesirable = desirable" or "periodic + periodic = chaos" dynamics. Although we are interested in modeling ecological systems and in particular the effect of seasonality, one could use our results to look at the switched maps as a way to control chaotic dynamics. In particular an extension to continuous dynamic systems may be relevant or applicable to chemical and mechanical systems [26].

In summary, our approach of building bifurcation diagrams readily yield intervals of parameter values that can show the so-called Parrondian dynamics for 1-D and 2-D maps. We have concentrated on ecological relevant maps, but the approach applies to any kind of maps. In particular, we can easily find parameters that show desirable dynamics in switched maps, controlling complex or undesirable dynamics, with the by product that we can also avoid the alternation of desirable dynamics that could yield undesirable dynamics in switched maps. Finally, we believed that we have stablished a pattern of dynamic behavior that supports the conjecture described in previous paragraphs.
