Research Article | Open Access
Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps
The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications. This paper investigates a set of four generalized logistic maps where the conventional map is a special case. The proposed maps have extra degrees of freedom which provide different chaotic characteristics and increase the design flexibility required for many applications such as quantitative financial modeling. Based on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. Mathematical analysis for each generalized map includes bifurcation diagrams relative to all parameters, effective range of parameters, first bifurcation point, and the maximum Lyapunov exponent (MLE). Independent, vertical, and horizontal scales of the bifurcation diagram are discussed for each generalized map as well as a new bifurcation diagram related to one of the added parameters. A systematic procedure to design two-constraint logistic map is discussed and validated through four different examples.
Chaotic iterated maps are essential in both modeling and information processing in many fields. This explains the need for their hardware analog and digital realizations, for example, [1–10]. The conventional logistic map is a famous iterative map based on first order nonlinear difference equation which can model growth rate and is given bywhere is the population growth rate or fertility coefficient and is the relative population size at a discrete time instant .
The map was initially popularized in a paper by the biologist May  as a discrete time demographic model, analogous to the logistic equation first created by Verhulst . A detailed mathematical analysis of the map and its properties has also been presented in . Specifically, the applications of the logistic map have increased during the last few decades, for example, in fields as biology , chemistry , physics , secure data and image transfer [17, 18], random number generation for chaos based communication [19–23], circuit applications , traffic , financial modeling, and business cycle theory [26–28].
The bifurcation diagram shown in Figure 1 is a basic characteristic of the logistic map which represents the sudden appearance of qualitatively different solutions as is varied. For , the system has one stable fixed point. For , the output oscillates between two fixed points; that is, the recurrence converges to a period- orbit. This period’s doubling, quadrupling, and so forth, are called bifurcation which accompanies the onset of chaos. Starting at , the recurrence converges to a period- orbit which implies the presence of chaos according to Sharkovskii’s theorem . Consequently, the output continues bifurcation until reaches the value which shows chaos in the whole range as shown in Figure 1. A chaotic system exhibits infinite number of periodic orbits, some of which could be of arbitrarily long period.
Conventional chaotic systems, especially discrete 1D maps, are highly rich in information and indications that constitute a large portion of the basic study of chaos theory. However, modeling dynamics in natural phenomena and stochastic processes, as well as demand on random number generation for multiple purposes, requires novel chaotic systems. Consequently, generalized maps have been proposed in previous researches [30–33]. Yet, they mostly depend on rather complicated equations that might even be considered 2D maps. Moreover, the parameters of generalized chaotic function proposed in  have been adapted to fit specific encryption algorithm. Thus, the map has not been mathematically analyzed in a generic way. Other generalized logistic, tent, and sine maps have been recently investigated in [34–36]. In this paper, generalizations providing extra degrees of freedom are proposed that could be adapted to fit certain specifications, keeping the advantage of simple mathematical relation using 1D maps only. Requirements on the key points of the resulting bifurcation diagram or the range of the system output could be achieved directly and simply through controlling the values of the parameters which are governed by simple relations. The simplest of which does not require any modifications from the viewpoint of implementation other than using a signed register for both the system parameter and the output . However, for a legible comprehensive analysis, we define the sign of the parameter explicitly confining its value to the set of positive reals . Moreover, further generalizations are proposed that allow scaling of the bifurcation diagrams.
The rest of the paper is organized as follows. Section 2 lists the possible variations on the relation representing the map, according to the signs of different control parameters and the resulting output ranges. The generalized form of these variations is given byBased on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. These maps differ in the ranges of both and , the symmetry, the key points of the bifurcation diagram, and the possible applications that could employ the resulting ranges. Section 3 analyzes the generalized logistic maps and studies how to design any of them to suit certain specifications, proposing three scaling cases. The parameters may take one of three cases: , called the independent scaling case, called the vertical scaling case, and called the zooming case. The choice of these names depends on the effect of the added parameter(s) on the bifurcation diagram. The three proposed parameterized versions are analyzed for each map from the viewpoint of iteration effect, ranges of and , the fixed points, the bifurcation diagrams, and the maximum Lyapunov exponent with respect to all system parameters. A new bifurcation diagram versus the parameter is introduced. Section 4 summarizes the results and provides the general schematic of the bifurcation diagram versus as a function of the other system parameters, as well as the general schematic of the newly proposed bifurcation diagram versus . Four different design examples are presented in Section 5 to verify the provided design procedure according to the general schematic. The designed maps are validated for usage in encryption applications through a simple text encryption scheme. Finally, the last section concludes the contributions of the paper.
2. Generalizations of the Logistic Map and Their Applications
Properties of the conventional discrete 1D logistic map and its applications were extensively discussed in the literature. Recent models of stochastic behaviors represent a great challenge and strong motivation to devise methods for designing generalized maps. Requirements on the key points of the resulting bifurcation diagram or the range of the system output could be achieved in many ways. In this section, we propose a simple and efficient way through controlling the values of multiple parameters governed by simple relations. The proposed maps and their usefulness to real world applications are discussed.
2.1. The Proposed Maps
We define the following parameterized maps controlled by the set of parameters such that . The names of the maps have been chosen carefully to represent the maximum chaotic range of the output.
Definition 1. Variations on the sign of the generalized logistic map represented by (2) can be subdivided into two types according to the maximum chaotic range of the output: (1)Single sign maps:(a)Positive logistic map given by(b)Negative logistic map given bywhere . The bifurcation diagram of this map is the reflected image of about the horizontal axis (), yielding exactly the same range of output magnitude, but with opposite sign.(2)Alternating sign maps:(a)Mostly positive logistic map given by(b)Mostly negative logistic map given bywhere . Thus, the relation between the two maps is similar to that between and .
Figure 2 shows the bifurcation diagrams for three maps in the case of unity scaling , and the graphs of their equations at maximum . The properties of have been previously shown in Figure 1. Table 1 illustrates the main differences between the proposed maps in the case of unity scaling. The output ranges of the proposed maps allow more flexibility in chaotic modeling of finance [26, 27], traffic, weather forecasting [37, 38], and many other fields. The wider output range and the asymmetry recognized in the resulting bifurcation diagrams of these maps could also be added to their advantages as they imply more unpredictability, in addition to the earlier onset of chaos in both of the alternating sign maps compared to the other two maps. The rest of the paper provides a detailed analysis of three different parameterized versions for each of the proposed maps. But first, we devote the rest of this section to present a quick, but not exhaustive, review of related real-world applications, followed by a study of the main properties of the negative and alternating sign maps which have not been covered before in the literature to the best of our knowledge. Various experiments have shown that some phenomena exhibit dynamical behaviors which could only be modeled by generalized maps with modulated parameters as conventional maps are not sufficient to model [39, 40]. On the other hand, proposed generalizations on existing maps enhance their unpredictability and increase their reliability in secure communication and encryption. We believe that there is still a gap which needs to be filled between areas of research in which some biologists, physicists, and economists propose new models and the field of nonlinear dynamics and chaos in which researchers could come up with chaotic systems suitable for predicting the behavior of these models. Examples of models that could employ the proposed maps are discussed below.
2.2. Negative Probabilities and Probabilities above Unity
Axioms of Kolmogorov’s probability theory  have the rule since describes the probability of occurrence of event or the probability of the outcome of an experiment which must be a positive real number. However, these axioms do not provide the complete picture of stochastic reality. Possibilities to extend the probability theory to describe numerous physical models with negative probabilities and even probabilities more than one have been mathematically discussed, as in , and have been used in solving several problems [43–45] and paradoxes, for example, Einstein-Podolsky-Rosen paradox . These out of the expected range probabilities are allowed by quasiprobability distributions that may apply to unobservable events or conditional probabilities. The idea of negative probabilities first arised in physics and particularly in quantum mechanics based on ideas of Wigner , Dirac , and Feynman . Later on, many physicists and scientists [50–53] have argued for the necessity of extended probabilities in quantum theories and that their appearance is one of the main differences between classical and quantum theories. For instance, the probabilities of existence of positive and negative-energy photons or the emissions and absorptions of photons equal and , respectively .
Negative probabilities have more recently been applied to mathematical finance. In quantitative finance, most probabilities are not real probabilities but theoretical ones called pseudoprobabilities. The concept of risk-neutral or pseudoprobabilities is a popular concept in finance which has been numerously applied as in [55, 56]. It has been shown how negative probabilities can be applied to financial option pricing. For example, calculating the probability of a price going up or down could be simplified by allowing such pseudoprobabilities to be negative in certain cases as pointed out by Haug . A generalized version of the well-known Cox, Ross, and Rubinstein binomial tree (CRR tree)  has been considered. The tree is often used to price a variety of derivatives instruments where different probabilities could be yielded which might lie outside the interval . For example, one can get a risk-neutral up probability of and a down probability of . A rigorous mathematical definition of negative probabilities in finance and their properties has been recently derived by Burgin and Meissner . The authors provide several situations in which negative probabilities occurred in finance as well as negative interest rates: overgenerated electricity in Norway that resulted in its price going negative a few hours during the night, the lender paying the bank interest rate in addition to the money in the 1970s in Switzerland, “repos,” that is, repurchase agreements traded at negative interest rates which took place in Japan and USA in 2003, and negative nominal interest rates that occurred in the worldwide 2008/2009 financial crisis.
2.3. Alternating Sign Logistic Maps
A study of the properties of the logistic map with negative control parameter that is quite similar to our mostly positive logistic map has been conducted in . Consider the recurrence , graphically represented by the parabola ; the critical point of the parabola is such thatFor stable steady state response, all the outputs of the recurrence relation should be limited within a closed set. This closure property could be guaranteed by setting a maximum value for . From the symmetry of the curve, this value can be obtained by forcing
Figure 3 shows the graph of the equation for different values of , and , where the map exhibits bounded output if and only if as expected before. For , the values of at and are and , respectively; that is, closed set responses are yielded as . Figure 3 also shows the values of carrying out 10,000 iterations of the recurrence. The time waveforms indicate the stability of the recurrence for as it exhibits a bounded solution. On the other hand, the range shows complete instability of the solution as it diverges to starting the fifteenth iteration. This illustrative discussion can be rather expressed more formally as follows.
Definition 2. Consider the map given by , the point is the one whose coordinates equal the map minimum value , and is the nontrivial fixed point or the intersection of the curves and . The map is stable if and only if the image of () is less than or equal to , such that . This is achieved as .
The previous method for calculation and verification of the range of values and that guarantee bounded responses has not been conducted in . Instead, the value and the interval were used directly in addition to a computer aided plot of the bifurcation diagram that ensures those values. However, our previous method requires iterating several values of and/or an initial assumption. That is why we propose a more generic procedure of specifying the parameters’ ranges that could be detailed as follows with the aid of Figure 4 where a map is defined as a function whose domain (input) space and range (output) space are the same .(1)The lower bound on the range of the map is which is the same as the lower bound on the domain . From (7),(2)The upper bound on the range is , that is, . So, the upper bound is equal towhich is a function of .(3)Then, we equate this value to the map equation to get the corresponding solutions for the values of : and , respectively:Rename the left hand side of the previous equation to ; it reduces toThus, (4)The domain of the map is , while its range is . We equate the lower and upper bounds of both intervals, respectively, to get :This equation could be solved to get three values among which one value is positive which is .(5)Hence, we get and by substitution for as follows:The mathematical analysis of the fixed and periodic points of mostly positive logistic map in the case of unity scaling is quite similar to that of the conventional logistic map. For , the recurrence converges to a period- orbit, followed by a sequence of periodic doubling. Starting at , the recurrence converges to a period- orbit. The output continues bifurcation until reaches the value which exhibits the maximum chaotic behavior. In addition to the bifurcation diagram and the graph of the equation shown in Figure 2(a), the time waveforms starting at different initial points and are shown in Figure 5. These responses demonstrate the maximum chaotic behavior of the map at . Figure 5(c) shows the difference between these two responses; the large number of nonzero points indicates the sensitive dependence on initial conditions which is a main property of chaotic systems.
The other two proposed maps: negative logistic map and mostly negative logistic map, are related to positive and mostly positive logistic maps, respectively, as previously mentioned. Thus, for parameterized maps, we analyze both positive and mostly positive logistic maps because the results for the other two maps could be deduced similarly.
3. Analysis and Results of the Parameterized Maps
For each map of those presented in the previous section, the resulting bifurcation diagram can be designed with the desired key points using as scaling parameters. The parameterized cases studied are , such that , , and . The following subsections provide the analysis and results of these parameterized cases.
3.1. Independent Scaling
In this map, two parameters and are added to be capable of scaling both the horizontal and the vertical axes of the bifurcation diagram independently. The analysis presented in this subsection is used later on in Section 5 to design any logistic map with specific characteristics.
3.1.1. Positive Logistic Map
(1) Range of . To ensure bounded output for all iterations, the value of should be limited to . The roots of the map, its critical point , and its maximum value are given by This inequality provides us with information on not only the maximum value of parameter but also that of parameter . Therefore, the maximum values for the parameters of the map are
(2) Fixed Points and Stability Condition. The fixed points are given by ; then The study of the stability of the fixed points is based on the calculation of the absolute value of the first derivative with respect to at these points . They are either stable or unstable depending on whether this value is less or greater than “one,” respectively. Otherwise, if the absolute value equals “one,” then it is called a bifurcation point. The derivative of this map at the fixed points is given byTherefore, the values of at which the system bifurcates and their function values are
3.1.2. Mostly Positive Logistic Map
(1) Range of . Similarly, for mostly positive logistic map, the solution should be limited to ,
(2) Fixed Points and Stability Condition. Similarly, for mostly positive logistic map, the value of at which the system bifurcates and the corresponding function value are
3.2. Vertical Scaling
In this map, an extra parameter is added which affects the vertical scaling of the bifurcation diagram; that is, it is a special case of the independent scaling map in which . This map could be used to control the vertical axis, that is, the values of the output, not the bifurcation points. In this subsection, the range of , fixed points and their stability analysis, and behavior dependence on different system parameters for the proposed maps are discussed.
3.2.1. Positive Logistic Map
(1) Range of . Substituting in (17a) yieldsThe function iterations at are shown in Figure 6, where the maximum value of the output and the parameter affects the vertical axis only. The number of peaks increases as increases and as approaches its maximum value which exhibits maximum chaotic behavior. On the other hand, for , no peaks could be noticed even in higher iterations.
(2) Fixed Points and Stability Condition. Substituting in (20c) yields
(3) Steady State Solutions versus System Parameters. Figure 7 shows the steady state solutions versus both system parameters and for vertical scaling positive logistic map. The bifurcation diagram versus exhibits similar behavior to the conventional case but with vertical scaling by the factor as shown in Figure 7(a). The nontrivial solution appears once exceeds the first bifurcation point , while the second bifurcation point is with corresponding steady state value . Figure 7(a) also shows the Cobweb plot of the map at and , which differs as the parameters change. The Cobweb is a rough plot for an orbit of , where the graph of the map function is sketched together with the diagonal line . The plot shows that the map exhibits chaotic behavior, since the orbit has a nonperiodic sequence and generates multiple outputs that cover the whole range of .
Figure 7(b) shows the behavior dependence on for a fixed value of . Theoretically, the only condition on is to be positive. It should be noted that for larger values of , extremely small initial points are needed such that they belong to the allowable range. Disregarding this note would yield results that might be falsely interpreted as a case of instability. Plotting starts at to avoid larger values of that decrease the clarity of the figure. The range of output results depends on ; that is, , as proposed by our previous analysis. The type of solution is either fixed, period-, multi-periods, or chaos depending on the value of . As increases, the response covers more values belonging to the allowable range until it covers the whole range at .
3.2.2. Mostly Positive Logistic Map
(1) Range of . Substituting in (21b) yields
(2) Fixed Points and Stability Condition. Substituting in (22) yields
(3) Steady State Solutions versus System Parameters. Figure 8 shows the steady state solutions versus both system parameters and and the Cobweb plot at and for mostly positive logistic map. The results could be interpreted and proved to be conforming to our analysis. Ten different snapshots of the bifurcation diagram versus the system parameter for both maps are shown in Figure 9, where the idea of vertical scaling is quite clear.
3.2.3. Maximum Lyapunov Exponent
From the nonlinear analysis of chaos, the maximum Lyapunov exponent should be positive to prove chaotic behavior. is one of the basic properties of chaotic systems and it is given by for a discrete map and the orbit starting at , where is the natural logarithm.
For both vertical scaling maps, the time evolution of for different values of roughly reaches the same steady state value. This constant value is the same as that of the conventional map . Besides, all map variations exhibit the same value at representing maximum chaotic behavior. This value has been proved for almost all orbits of chaotic maps, starting from the tent map and extending to the logistic map and others by conjugacy of maps . Figure 10(a) shows 3D plot of as a function of both and , in addition to the corresponding contour plot, for vertical scaling positive logistic map, while Figure 10(b) shows them for vertical scaling mostly positive logistic map.
In this map, an extra parameter is added which affects the zooming of the bifurcation diagram; that is, it is a special case of the independent scaling map in which . Moreover, there is a new bifurcation diagram presented with respect to the system parameter that will be generalized later on. This map is richer in analysis and could be used for controlling both axes, that is, the bifurcation points in addition to the corresponding output values. This control can be achieved only in a dependent way, such that the area of the bifurcation diagram versus remains constant. In this subsection, the range of , fixed points and their stability analysis, and two different bifurcation diagrams are discussed.
3.3.1. Positive Logistic Map
(1) Range of . Substituting in (17a) yields
(2) Fixed Points and Stability Condition. Substituting in (20c) yields
(3) Steady State Solutions versus System Parameters. Figure 11(a) shows the bifurcation diagram versus for different values of for zooming positive logistic map. The horizontal and vertical axes corresponding to the values of the parameter and the solution are scaled by and , respectively, so that the total area is still the same and equals a constant value of square units. Figure 11(a) also shows the Cobweb plot at and , which exhibits chaotic behavior where the whole range of is covered.
Figure 11(b) shows the bifurcation diagram versus for different values of , where the diagram has equal axes lengths which we will call a square-axis, and its area is a function of . The unity area square results in the case of which corresponds to the maximum value of in the unity scaling case (). Recalling that as given by (28b), then the values of are scaled by . Accordingly, the values of undergo the same scaling and . Thus, for positive logistic map, the horizontal and vertical axes are scaled together by the same ratio which equals .
3.3.2. Mostly Positive Logistic Map
(1) Range of . Substituting in (21b) yields
(2) Fixed Points and Stability Condition. Substituting in (22) yields
(3) Steady State Solutions versus System Parameters. Figure 12(a) shows the bifurcation diagram versus for different values of for zooming mostly positive logistic map. The diagrams are quite similar to those in the case of zooming positive logistic map in having constant area axes. Figure 12(a) also shows the Cobweb plot at and , which shows chaotic behavior where the whole range of is covered.
Figure 12(b) shows the bifurcation diagram versus for different values of for mostly positive map. Similar to positive map, the horizontal and vertical axes are scaled together by the same ratio which equals for mostly positive map. Yet, they do not form a square-axis due to its alternating sign nature, where . Instead, the length of the vertical axis is double that of the horizontal axis but extends to the negative part asymmetrically. Therefore, the bifurcation diagrams with respect to are of equal area for both zooming maps. However, the bifurcation diagrams with respect to are of equal axes for the positive map and a vertical axis double the horizontal axis for the mostly positive map. This enriches the map characteristics and raises its degrees of freedom to be used in control applications according to the required constraints.
Figure 13 shows snapshots of the bifurcation diagrams versus at different values of for both zooming maps. Figure 14 shows the other type of bifurcation diagram which is plotted versus at different values of for both maps. The snapshots reflect the inverse proportionality relation between and , where as increases, the value of decreases and vice versa. Similarly, as increases, the value of decreases and vice versa.
3.3.3. Maximum Lyapunov Exponent
Figure 15(a) shows 3D plot of as a function of both and for zooming positive logistic map, while Figure 15(b) shows it for zooming mostly positive logistic map. This continuous surface plot illustrates the dependence of the allowed range of on the value of for both zooming positive and mostly positive logistic maps. This fact is further indicated by discrete snapshots of as a function of for different values of . However, the value of remains the same for maximum chaotic behavior, that is, at . The same steady state value is obtained which equals that of the conventional logistic map as previously explained.
4. General Schematic of the Bifurcation Diagrams
Recalling the analyses and discussions throughout this paper and combining them with the results obtained in the last section, we can sketch a generalized schematic for the bifurcation diagrams. The key points of the two different bifurcation diagrams versus the system parameters and for both independent scaling positive and mostly positive maps can be summarized as follows.
4.1. Positive Logistic Map
4.1.1. Bifurcation Diagram versus
The values of for the first and second bifurcation points are and of , respectively, where . Moreover, the value of the first nontrivial bifurcation point equals of the maximum value , where . Figure 16(a) shows the key points of the bifurcation diagram versus in terms of and which conform to the results expected by our analysis.
4.1.2. Bifurcation Diagram versus
From another point of view, we can design the bifurcation diagram with respect to the parameter where the bifurcation points and maximum values are shown in Figure 16(b) in terms of the other parameters and . The values of at which bifurcation occurs are and of , where . Moreover, the value of the first nontrivial bifurcation point equals the maximum value , where .
4.2. Mostly Positive Logistic Map
4.2.1. Bifurcation Diagram versus
Figure 17(a) shows the key points of the bifurcation diagram versus in terms of and . The value of is of , where . The minimum and maximum values of are and , respectively.
4.2.2. Bifurcation Diagram versus
Figure 17(b) shows the key points of the bifurcation diagram versus in terms of and . The value of is of , where . The minimum and maximum values of are and , respectively.
5. Logistic Map Design Procedure
Table 2 summarizes the results of independent scaling positive and mostly positive logistic maps. As previously detailed, the other two maps exhibit the same properties except for values of that have the same magnitude, but opposite sign. Consequently, any required scaling can be achieved using the two extra parameters and , for bifurcation diagram versus . The horizontal scaling can be controlled by the parameter , while the vertical scaling depends on the value . To achieve horizontal scaling only while keeping the vertical axis with the values corresponding to the unity scaling case, the value of should equal that of . The bifurcation diagram versus could be similarly designed and controlled by the values of and . The remarkable advantage in the proposed variations on the discrete 1D logistic map is not only about scaling or shifting but also about the increased complexity of the output that makes it more unpredictable, that is, more convenient for applications that employ chaos.
5.1. Design Examples
The required specifications on the bifurcation diagram could be realized with respect to one of the parameters and . Both realizations are considered; we denote the value of the parameter at which first, or second, bifurcation occurs as and , respectively, and the maximum value of the parameter as ; where could be or . Let us also define such thatThe design problem involves constraints which are the values of and , in addition to if exist, that is, in the case of positive or negative logistic maps. For single sign maps, and , while for alternating sign maps . In addition, the values of and , in the case of alternating sign maps, are related. For the bifurcation diagram with respect to , , while for the bifurcation diagram with respect to , . For mostly positive map, , while for mostly negative map, . Table 3 summarizes these constraints.
Four different cases of designing a map with certain specifications on the bifurcation diagram versus either parameter are provided in Table 4(a). The design examples have been chosen such that they cover the cases of horizontal and independent scaling, since vertical scaling and zooming maps have been discussed thoroughly. First, the suitable map is chosen according to the required range of output: positive only, negative only, or alternating sign. Then, the design equations are solved simultaneously to get the corresponding values of the other parameters. The bifurcation diagrams, their main key points, and the map equation are shown in Table 4(b).