Parrondian Games in Discrete Dynamic Systems

Steve A. Mendoza and Enrique Peacock-López

#### Abstract

An interesting problem in nonlinear dynamics is the stabilization of chaotic trajectories, assuming that such chaotic behavior is undesirable. The method described in this chapter is based on the Parrondo's paradox, where two losing games can be alternated, yielding a winning game. The idea of alternating parameter values has been used in chemical systems, but for these systems, the undesirable behavior is not chaotic. In contrast, ecological relevant map in one and two dimensions, most of the time, can sustain chaotic trajectories, which we consider as undesirable behaviors. Therefore, we analyze several of such ecological relevant maps by constructing bifurcation diagrams and finding intervals in parameter space that satisfy the conditions to yield a desirable behavior by alternating two undesirable behaviors. The relevance of the work relies on the apparent generality of method that establishes a dynamic pattern of behavior that allows us to state a simple conjecture for two-dimensional maps. Our results are applicable to models of seasonality for 2-D ecological maps, and it can also be used as a stabilization method to control chaotic dynamics.

Keywords: chaos control, Parrondo's paradox, switched dynamic systems, ecological maps, seasonality

#### 1. Introduction

In population dynamics, discrete dynamic systems have been used to model the dynamics of ecological systems. One of the first maps used in ecology that suggested to study the new, ð Þ Xnþ<sup>1</sup> , and the old, ð Þ Xn , non-overlapping populations is the logistic map. [1] Although a simple one-dimensional (1-D) map, the logistic map shows complex dynamics including chaos. Furthermore, the analyses of the logistic map gave us a better understanding of the properties of chaotic dynamics [2–7].

In the case of 1-D discrete dynamics, for the last 18 years, alternate dynamics strategies have been the center of attention due to the so-called Parrondo paradox [8–10], where two losing games can be combined to yield a winning game. Furthermore, the idea that "lose + lose = win" has been extended to "chaos +chaos = periodic" in one-dimensional maps [11]. Just recently and for the first time, we were able to find the Parrondo dynamics in two 2-D maps [12]. In the contest of seasonality, we consider the alternation of undesirable dynamical behaviors yield a desirable behavior [13, 14]. So in the context of population dynamics we have considered cases where "undesirable + undesirable = desirable" dynamical behaviors occur as a result of a simple alternation of parameters [15–21].

#### Fractal Analysis

In our present discussion, we extend our seasonality modeling strategy to several two-dimensional ecologically relevant maps and find that the "undesirable + undesirable = desirable", the "chaos + chaos = periodic", as well as, the "periodic + periodic = chaos" behaviors are not unique to 1-D maps. In Section 2, we consider a delayed logistic map, and in Section 3, we analyze a Lotka-Volterra map. In Section 4, we study a modified 2-D Ricker map, and in Section 5, we analyze the Beddington map. In Section 6, we discuss a modified Lotka-Volterra map, which includes a logistic prey growth. We conclude in Section 7 with a discussion and a summary of our results.

#### 2. Delayed logistic equation

In our analysis of two-dimensional maps, we begin with the extended logistic map that incorporates a delay in population growth, defined by the following relation:

$$X\_{n+1} = Y\_n \tag{1}$$

look for regions that have periodic oscillations that are normally associated with chaos, thus resulting in the case "chaos + chaos = order." Hence, for every switching map that we study, in the figures we also show the unswitched map for the same Ce parameter space. We make a note that the alternation of parameter values as defined by our switching strategy may result in an extension of C parameters yielding oscillations; that is we may see oscillations for C values greater than 2.27 as in the case for lagged logistic map. To compare our maps with Eqs. (1) and (2), we

For our analysis, we pick a Co as in the "chaos + chaos = order" case; however, we may choose a parameter within the periodic windows in the chaotic region, and when using the switched map, we focus on the Ce values that are less than the onset of chaos, which for the case of the lagged logistic pap is C = 2. We use the analysis

For our first example of "chaos + chaos = periodic", we consider the parameter value, Co ¼ 2:10 and Eqs. (3) and (4). In our bifurcation diagram for Eqs. (3) and (4), in Figure 2, we consider Ce greater than 2 and look for Ce values that give us periodic oscillations. Figure 2 shows two maps at once, the left hand showing Eqs. (1) and (2), and the right hand graph shows Eqs. (1) and (2) with Co = 2.1. In this case, from Figure 2, we can see one region of periodicity from Ce = 2.26 to 2.27. Another combination of parameters yielding "chaos+chaos = periodic" uses Co = 2.15 and Eqs. (3) and (4), where Figure 3 shows a range of Ce values for which "chaos + chaos = periodic" holds, from Ce = 2.36 to 2.38. The same figure also shows other values for which the "chaos + chaos = periodic" relation holds, but these bands are not as prominent as the one we focus on. Through out the paper, we make a point that different Ce values give widely different behaviors and these differences in dynamic behaviors reveals the differences in the Ce values that give us desirable behaviors. For the rest of the paper, the approximate ranges of Ce will be given for

only study bifurcation maps using Eqs. (3) and (4) from C = 0 to 2.27.

discussed above for all of the cases in this paper.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

Figure 2.

71

Figure 1.

Bifurcation diagram for Eqs. (1) and (2) and Eqs. (3) and (4), using Co 2.1.

Lagged logistic map model, Eqs. (1,2), with C = 0 to C = 2.27 and the region from C = 1.90 to C = 2.27.

$$Y\_{n+1} = C \, Y\_n (1 - X\_n) \tag{2}$$

where C is our bifurcation parameter. For the Lagged Logistic Equation, we consider C values from 0 to 2.27 in the original map, although with alternation, we can obtain a bifurcation diagram showing larger C values. Figure 1, shows the regular bifurcation map of the lagged logistic model. For all of the maps we study, both the X and Y graphs for a given show the same dynamics; for instance, parameters associated with chaotic dynamics in the X map are also associated with chaotic dynamics in the Y map; since we focus on a map's dynamics, we only show the X function map.

From Figure 1, we define our parameter value regions associated with complex or non-complex dynamics. The map on the left of the figure shows the whole bifurcation map; while the right magnifies the complex region. On the right hand figure, which is the magnified map, we can clearly see some periodic windows, but we pick parameter values associated with complex dynamics.

Next, we switch, or alternate, the parameter values between even and odd iterations through the following relation:

$$X\_{n+1} = \begin{cases} f\_n(X\_n, Y\_n) = Y\_n & \text{if } \ n \text{ even} \\\\ f\_n(X\_n, Y\_n) = Y\_n & \text{if } \ n \text{ odd} \end{cases} \tag{3}$$

$$Y\_{n+1} = \begin{cases} g\_n(X\_n, Y\_n) = \mathcal{C}\_\epsilon \, Y\_n (1 - X\_n) & \text{if } \ n \text{ even} \\\\ g\_n(X\_n, Y\_n) = \mathcal{C}\_\sigma \, Y\_n (1 - X\_n) & \text{if } \ n \text{ odd} \end{cases} \tag{4}$$

The equation above describes our switching strategy in which we pick one parameter for every odd iteration, which we name Co, and use the even parameter, Ce, as our bifurcation parameter, for every even iteration. For the first type of behavior, we pick one parameter associated with complex dynamics as our Co value and switch it with our even parameter, Ce, in areas associated with chaotic dynamics. In our case, we see chaotic dynamics for C values greater than 2.0 when we construct the bifurcation diagram for Eqs. (1) and (2). In the resulting alternated, or switched, bifurcation diagram, represented by Eq. (3) for the current section, we

#### Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

In our present discussion, we extend our seasonality modeling strategy to several two-dimensional ecologically relevant maps and find that the "undesirable + undesirable = desirable", the "chaos + chaos = periodic", as well as, the "periodic + periodic = chaos" behaviors are not unique to 1-D maps. In Section 2, we consider a delayed logistic map, and in Section 3, we analyze a Lotka-Volterra map. In Section

4, we study a modified 2-D Ricker map, and in Section 5, we analyze the

summary of our results.

Fractal Analysis

relation:

function map.

70

2. Delayed logistic equation

Beddington map. In Section 6, we discuss a modified Lotka-Volterra map, which includes a logistic prey growth. We conclude in Section 7 with a discussion and a

In our analysis of two-dimensional maps, we begin with the extended logistic map that incorporates a delay in population growth, defined by the following

where C is our bifurcation parameter. For the Lagged Logistic Equation, we consider C values from 0 to 2.27 in the original map, although with alternation, we can obtain a bifurcation diagram showing larger C values. Figure 1, shows the regular bifurcation map of the lagged logistic model. For all of the maps we study, both the X and Y graphs for a given show the same dynamics; for instance, parameters associated with chaotic dynamics in the X map are also associated with chaotic dynamics in the Y map; since we focus on a map's dynamics, we only show the X

From Figure 1, we define our parameter value regions associated with complex

or non-complex dynamics. The map on the left of the figure shows the whole bifurcation map; while the right magnifies the complex region. On the right hand figure, which is the magnified map, we can clearly see some periodic windows, but

Next, we switch, or alternate, the parameter values between even and odd

The equation above describes our switching strategy in which we pick one parameter for every odd iteration, which we name Co, and use the even parameter, Ce, as our bifurcation parameter, for every even iteration. For the first type of behavior, we pick one parameter associated with complex dynamics as our Co value and switch it with our even parameter, Ce, in areas associated with chaotic dynamics. In our case, we see chaotic dynamics for C values greater than 2.0 when we construct the bifurcation diagram for Eqs. (1) and (2). In the resulting alternated, or switched, bifurcation diagram, represented by Eq. (3) for the current section, we

f <sup>n</sup>ð Þ¼ Xn; Yn Yn if n even

f <sup>n</sup>ð Þ¼ Xn; Yn Yn if n odd

gnð Þ¼ Xn; Yn Ce Ynð Þ 1 � Xn if n even

gnð Þ¼ Xn; Yn Co Ynð Þ 1 � Xn if n odd

we pick parameter values associated with complex dynamics.

iterations through the following relation:

Xnþ<sup>1</sup> ¼

Ynþ<sup>1</sup> ¼

8 ><

>:

8 ><

>:

Xnþ<sup>1</sup> ¼ Yn (1)

(3)

(4)

Ynþ<sup>1</sup> ¼ C Ynð Þ 1 � Xn (2)

look for regions that have periodic oscillations that are normally associated with chaos, thus resulting in the case "chaos + chaos = order." Hence, for every switching map that we study, in the figures we also show the unswitched map for the same Ce parameter space. We make a note that the alternation of parameter values as defined by our switching strategy may result in an extension of C parameters yielding oscillations; that is we may see oscillations for C values greater than 2.27 as in the case for lagged logistic map. To compare our maps with Eqs. (1) and (2), we only study bifurcation maps using Eqs. (3) and (4) from C = 0 to 2.27.

For our analysis, we pick a Co as in the "chaos + chaos = order" case; however, we may choose a parameter within the periodic windows in the chaotic region, and when using the switched map, we focus on the Ce values that are less than the onset of chaos, which for the case of the lagged logistic pap is C = 2. We use the analysis discussed above for all of the cases in this paper.

For our first example of "chaos + chaos = periodic", we consider the parameter value, Co ¼ 2:10 and Eqs. (3) and (4). In our bifurcation diagram for Eqs. (3) and (4), in Figure 2, we consider Ce greater than 2 and look for Ce values that give us periodic oscillations. Figure 2 shows two maps at once, the left hand showing Eqs. (1) and (2), and the right hand graph shows Eqs. (1) and (2) with Co = 2.1. In this case, from Figure 2, we can see one region of periodicity from Ce = 2.26 to 2.27.

Another combination of parameters yielding "chaos+chaos = periodic" uses Co = 2.15 and Eqs. (3) and (4), where Figure 3 shows a range of Ce values for which "chaos + chaos = periodic" holds, from Ce = 2.36 to 2.38. The same figure also shows other values for which the "chaos + chaos = periodic" relation holds, but these bands are not as prominent as the one we focus on. Through out the paper, we make a point that different Ce values give widely different behaviors and these differences in dynamic behaviors reveals the differences in the Ce values that give us desirable behaviors. For the rest of the paper, the approximate ranges of Ce will be given for

Figure 1. Lagged logistic map model, Eqs. (1,2), with C = 0 to C = 2.27 and the region from C = 1.90 to C = 2.27.

Figure 2. Bifurcation diagram for Eqs. (1) and (2) and Eqs. (3) and (4), using Co 2.1.

Xnþ<sup>1</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup>ð Þ¼ Xn; Yn ð Þ <sup>r</sup> <sup>þ</sup> <sup>1</sup> ð Þ� Xn r Xð Þ<sup>n</sup> <sup>2</sup> � CeXnYn if n even <sup>f</sup> <sup>n</sup>ð Þ¼ Xn; Yn ð Þ <sup>r</sup> <sup>þ</sup> <sup>1</sup> ð Þ� Xn r Xð Þ<sup>n</sup> <sup>2</sup> � CoXnYn if n odd (

diagram for roughly Ce = 2.58–2.65.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

Figure 5.

Figure 6.

Figure 7.

73

region.

Ynþ<sup>1</sup> <sup>¼</sup> gnð Þ¼ Xn; Yn CoXnYn if n odd gnð Þ¼ Xn; Yn CeXnYn if n even �

As before, we pick a Co value associated with a chaotic trajectory and alternate with Ce, using Eq. (7), which we use as the bifurcation parameter, illustrated in Figure 6. For this figure, we use Co = 2.1, and we can easily find conditions in which "chaos + chaos = order." In particular, we see this phenomena for parameter values of Ce = 2.33–2.40. Figure 7, shows another example of "chaos + chaos = order" using Eqs. (7) and (8) with a Co value of 2.22, and in the corresponding bifurcation

We conclude the present section with an example of "periodicity + periodicity = chaos", using Eqs. (7) and (8) and Ce = 2.44. In this case, in Figure 8, we see

Bifurcation diagram for Eqs. (5) and (6), showing the interval studied, as well as a close up of the chaotic

Bifurcation diagram for Eqs. (5) and (6) and Eqs. (7) and (8) with the Co value 2.1.

Bifurcation diagram for Eqs. (5) and (6) and Eqs. (7) and (8) with the Co value 2.22.

(7)

(8)

Figure 3. Bifurcation diagram for Eqs. (1) and (2), and Eqs. (3) and (4), using Co ¼ 2:15.

Figure 4. Bifurcation diagram for Eqs. (1) and (2), and Eqs. (3) and (4), using Co 2.19.

one window that satisfies the "chaos + chaos = order" or "periodicity + periodicity = chaos", since there are sometimes a variety of parameters meeting the relevant criteria for switching.

We complete our analysis of delayed logistic map with one case in which "periodic + periodic = chaos". As mentioned beforehand, we pick our value associated with periodic trajectories from the area associated with chaotic trajectories, and focus on Ce values less than the chaotic region for comparison. In particular, we choose Co ¼ 2:19 as our periodic parameter for Eqs. (3) and (4); Figure 4 shows the corresponding bifurcation map, and, from the figure, we see one prominent example of "periodic+periodic = chaos" for Ce = 1.85 to 2.00.

#### 3. Lotka-Volterra model

We begin our next section by discussing a discretized form of the Lotka-Volterra model. The Lotka-Volterra map describes predator prey interactions, assuming that the prey has a relatively high initial population, and that the predator's growth rate is directly proportional to the prey's growth rate.

The model follows a relation defined by the map below

$$X\_{n+1} = (1+r)X\_n - rX\_n^2 - CX\_n \ Y\_n \tag{5}$$

$$Y\_{n+1} = \mathbb{C} \, X\_n \, \, Y\_n \, \, \tag{6}$$

In Figure 5, showing Eqs. (5) and (6), we look at the unswitched map, defined by showing the ranges of periodic and aperiodic behavior. As in the previous section, we use the unswitched bifurcation map as a comparison to the switched map when using certain parameters. For this section, we focus on the interval C = 0 to 2.8, and set r = 2 for this map and the rest of the maps that have an r parameter.

Parrondian Games in Discrete Dynamic Systems DOI: http://dx.doi.org/10.5772/intechopen.81499

$$X\_{n+1} = \begin{cases} f\_n(X\_n, Y\_n) = (r+1)(X\_n) - r(X\_n)^2 - \mathcal{C}X\_n Y\_n & \text{if } n \text{ even} \\ f\_n(X\_n, Y\_n) = (r+1)(X\_n) - r(X\_n)^2 - \mathcal{C}X\_n Y\_n & \text{if } n \text{ odd} \end{cases} \tag{7}$$

$$Y\_{n+1} = \begin{cases} g\_n(X\_n, Y\_n) = \mathcal{C}\_o X\_n Y\_n & \text{if } n \text{ odd} \\ g\_n(X\_n, Y\_n) = \mathcal{C}\_t X\_n Y\_n & \text{if } n \text{ even} \end{cases} \tag{8}$$

As before, we pick a  $C\_{o}$  value associated with a chaotic trajectory and alternate with  $C\_{o}$ , using Eq. (7), which we use as the bifurcation parameter, illustrated in \*\*f\*\*igure 6\*\*.\*\* For this figure, we use  $C\_{o} = 2.1$ , and we can easily find conditions in which "chaos + chaos = order." In particular, we see this phenomenon for parameter values of  $C\_{e} = 2.33 - 2.40$ . \*\*Figure 7\$, shows another example of "chaos + chaos = order "using Eqs. (7) and (8) with a  $C\_{o}$  value of 2.22, and in the corresponding bifurcation diagram for roughly  $C\_{e} = 2.58 - 2.65$ .

We conclude the present section with an example of "periodicity + periodicity = chaos", using Eqs. (7) and (8) and Ce = 2.44. In this case, in Figure 8, we see

Figure 5.

one window that satisfies the "chaos + chaos = order" or "periodicity + periodicity = chaos", since there are sometimes a variety of parameters meeting the relevant

ple of "periodic+periodic = chaos" for Ce = 1.85 to 2.00.

Bifurcation diagram for Eqs. (1) and (2), and Eqs. (3) and (4), using Co ¼ 2:15.

Bifurcation diagram for Eqs. (1) and (2), and Eqs. (3) and (4), using Co 2.19.

is directly proportional to the prey's growth rate.

The model follows a relation defined by the map below

Xnþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>r</sup> Xn � rX<sup>2</sup>

by showing the ranges of periodic and aperiodic behavior. As in the previous section, we use the unswitched bifurcation map as a comparison to the switched map when using certain parameters. For this section, we focus on the interval C = 0 to 2.8, and set r = 2 for this map and the rest of the maps that have an r parameter.

We complete our analysis of delayed logistic map with one case in which "periodic + periodic = chaos". As mentioned beforehand, we pick our value associated with periodic trajectories from the area associated with chaotic trajectories, and focus on Ce values less than the chaotic region for comparison. In particular, we choose Co ¼ 2:19 as our periodic parameter for Eqs. (3) and (4); Figure 4 shows the corresponding bifurcation map, and, from the figure, we see one prominent exam-

We begin our next section by discussing a discretized form of the Lotka-Volterra model. The Lotka-Volterra map describes predator prey interactions, assuming that the prey has a relatively high initial population, and that the predator's growth rate

In Figure 5, showing Eqs. (5) and (6), we look at the unswitched map, defined

<sup>n</sup> � C Xn Yn (5)

Ynþ<sup>1</sup> ¼ C Xn Yn (6)

criteria for switching.

Figure 3.

Fractal Analysis

Figure 4.

72

3. Lotka-Volterra model

Bifurcation diagram for Eqs. (5) and (6), showing the interval studied, as well as a close up of the chaotic region.

Figure 6. Bifurcation diagram for Eqs. (5) and (6) and Eqs. (7) and (8) with the Co value 2.1.

Figure 7. Bifurcation diagram for Eqs. (5) and (6) and Eqs. (7) and (8) with the Co value 2.22.

Figure 8. Bifurcation diagram for Eqs. (5) and (6) and Eqs. (7) and (8) with the Co value 2.44.

some chaotic behavior for values of Ce in the interval 1½ � :75; 2:00 , a region that is periodic when using Eqs. (5) and (6).
