On some generalized discrete logistic maps

ORIGINAL ARTICLEOn some generalized discrete logistic mapsAhmed G.Radwan*Engineering Mathematics Department,Faculty of Engineering,Cairo University,12613,Egypt Nanoelectronics Integrated Systems Center (NISC),Nile University,Cairo,Egypt Received 11February 2012;revised 7May 2012;accepted 15May 2012Available online 28June 2012KEYWORDS Logistic map;Bifurcation diagram;Stability;Generalized 1D map;Arbitrary power;ChaosAbstract Recently,conventional logistic maps have been used in different vital applications like modeling and security.However,unfortunately the conventional logistic maps can tolerate only one changeable parameter.In this paper,three different generalized logistic maps are introduced with arbitrary powers which can be reduced to the conventional logistic map.The added parameter (arbitrary power)increases the degree of freedom of each map and gives us a versatile response that can fit many applications.Therefore,the conventional logistic map is considered only a special case from each proposed map.This new parameter increases the flexibility of the system,and illustrates the performance of the conventional system within any required neighborhood.Many cases will be illustrated showing the effect of the arbitrary power and the equation parameter on the number of equilibrium points,their locations,stability conditions,and bifurcation diagrams up to the chaotic behavior.ª2012Cairo University.Production and hosting by Elsevier B.V.All rights reserved.IntroductionSince 1930until now iterated maps are still considered very important in the modeling and processing of many fields such as in population biology,encryption,communication and business cycle theory [1–11].One of the most famous maps comes from the so called continuous logistic equation which was introduced by Pierre Verhulst in the middle of the 19th century.The dynamical behavior of this continuous equation is trivial compared with that one presented by thediscrete logistic map introduced in the 1960s,although it was popularized in the 1970s by Robert May in his well known paper published in the journal Nature.Another complex map based on the iterated empirical reproduction curves of fish was introduced by William Ricker in 1954.Furthermore,the analysis of many iterated maps was stud-ied such as generating random numbers from the logistic map by John von Neumann in 1940[2].One of the basic classifications of logistic maps can be done with the help of bifurcation diagrams which display some char-acteristic properties of the asymptotic solution of a dynamical system as a function of a control parameter.According to the Sarkovskii theorem [2,3],if the function has a periodic point of period three then it has all periods as well which means chaos can be achieved at a certain range of the control parameter.The major property of any chaotic system is that it exhibits a great sensitivity to initial conditions.The most common lo-gistic map is that showing a non-linear recurrence relation with*Tel.:+201224647440.E-mail address:agradwan@Peer review under responsibility of CarioUniversity2090-1232ª2012Cairo University.Production and hosting by Elsevier B.V.All rights reserved./10.1016/j.jare.2012.05.003a single control parameter l and describes the population size x relative to the time t as followsx nþ1¼k x nð1Àx nÞð1Þwhere k is the growth rate of the population,as discussed before. As k<3the system has afixed stable point(nontrivial solution). However as k increases,the system outputfluctuates between different periodic points.For3<k<3.45the output oscillates between twofixed points(first bifurcation happens at k=3) Moreover,as3.45<k<3.545the system oscillates between four stable points,and so on as k increases until it reaches4 which shows infinitefixed points or chaos.This diagram which describes this process is called the bifurcation diagram.Many recent applications used the logistic map as a model or a data source such as the following examples:In biology [5,6],if the biologist can predict very accurately the population rate of living organisms such that when the population sizefluc-tuates betweenfixed values,a period of2,4or8years,many novel achievements can be presented[2].Also,the logistic map can be used to model some processes in chemistry such ware to create logistic maps[20]and Chaotic circuits[21,22] which has random-like appearance values limited between two bounds.The generation of the logistic map with its bifur-cation properties can also help in the noise analysis for many applications such as modeling of the respiratory system[23].In this paper we will investigate three different cases of the lo-gistic map of arbitrary power.The three cases can be summa-rized byx nþ1¼k x anð1Àx bnÞð2Þwhere(a,b)will take one of the three following cases(a,a),(1, a)and(a,1)for all a e R+.For each case,we will discuss the fixed points,its range,the effect of iteration,arbitrary power a and the bifurcation diagrams with respect to the two parame-ters a and k.The next three sections in this paper will discuss the behavior and properties of the three proposed logistic maps.The summary and comparisons of the three logistic sys-tems will be introduced in the conclusion.First generalized logistic map x nþ1¼k x anð1Àx anÞFig.1(a)The effect of the function iteration f m where fðx;a;kÞ¼4x aðlÀx aÞfor m={1,2,4}and(b)the projection of thefifth iteration for different values of k={3.0,3.5,4.0}.164 A.G.Radwancurves rotate as a changes.It is clear that the surface rotation of the iterated function f m increases as m increases and a de-creases.In addition,as m increases the number of peaks in-creases exponentially in a nonlinear way(not as the conventional case)so that some of them rotate left and others right as shown in Fig.1a when m=4.Effect of k withfixed m=5As known from the conventional case,the parameter k affects the map response.Fig.1b shows the projection of thefifth iter-ated function f5in the a–x plane for different values of k.The range of this function increases as k increases from less than 0.8when k=3.0up to the full range[0,1]when k=4.0. Moreover,the number of peaks increases as shown from Fig.1b from the merging of the red color(high values)with the blue(low values),and the contours become more nonlinear as k increases from3.0up to4.0.The nontrivialfixed points versus kThefixed points can be calculated from xüfðxÃ;k;aÞthen the equation that controls the value of x\is given by kðxÃÞ2aÀkðxÃÞaþxü0.Let us assume a=0.1k,where k e N+and y=x0.1which transforms the previous equation into a polynomial as k y2kÀk y kþy10¼0As long as the parameter k is known the roots of the previous equation can be easily obtained.Fig.2shows the nontrivial solution (x\…0)where x\increases as k increases when a<1.In addi-tion,the nonlinearity of the curve x\also increases.As a be-comes very small,the value of x\becomes closer to zero (trivial solution).The stability criteria of these points is classi-fied based on the derivative at these points for example if j f0ðxÃ;k;aÞj<1then this point is a sink point(stable point), however if j f0ðxÃ;k;aÞj>1this point will be a source(unstable point).The derivative f0¼@f@xis given byf0ðxÃ;k;aÞ¼kaðxÃÞaÀ1ð1À2ðxÃÞaÞð3ÞTherefore,the critical point k s is the value of k when the abso-lute derivative becomes one.This value depends on the rela-tionship between thefixed point x\and x p¼ffiffiffiffiffiffiffi0:5apas follows k s¼1aðxÃÞaÀ1ð1À2ðxÃÞaÞxÃ<x p4x p xüx p1aðxÃÞaÀ1ð2ðxÃÞaÀ1ÞxÃ>x p8>><>>:ð4ÞFig.2Thefixed points and their derivatives versus for different values of(a)a<1and(b)a>1.Table1The values of x k and k c for different a.a 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.92x k0.1040.1980.2760.340.3970.4440.4840.51950.550.5774 k c 1.3674 1.6136 1.81 1.976 2.117 2.238 2.34479 2.43896 2.5228 2.5981Fig.3The bifurcation diagram of thefirst proposed logistic map versus A for different values of a={0.1,0.3,1.1,2.5}. Fig.4The bifurcation diagram of thefirst proposed logistic map versus a for different values of X=1,2,3,and4.When a=0.1,the system has a singlefixed point in the full range of k and it is always stable.However,as a increases, the absolute derivative j f0ðxÃ;k;aÞj begins to exceed one as k increases as shown from Fig.2a.For example when a=0.5, then fðx;k;aÞ¼kffiffiffix pð1Àffiffiffix pÞ,thefixed point x\is given by xüðk=ðkþ1ÞÞ2and x p=0.25.The derivative at thisfixed point is given by f0ðxÃ;k;0:5Þ¼0:5ð1ÀkÞ.Therefore,this fixed point is stable only in the range k e(0,3).If k>k s=3,then this point will be unstable and bifurcation starts.Similarly for other cases when a<1,the system has a fixed point in the range of k e(0,k max).When a>1.0,the curves of x\become more nonlinear as shown in Fig.2b.By simple differentiation and analysis,thecritical value x k at which@k@xü0is given by x k¼ð1Àa1À2aÞ1=a andthe critical value k c¼x k2ak ak.These critical values are locatedin the acceptable range if a>1.0as shown from Table1, and also from Fig.2b.By using the definition of k c and x k in the calculation of the derivative at that point where j f0ðx k;k c;aÞj¼1is the stability limit of thefixed points.There-in Fig.3.Therefore,forfixed value of k,the behavior of the system will be changed with different a.For example at k=3.8,the system response changed fromfixed points,period two,multiple periods,and chaos when a equals to0.1,0.3,0.5 and a>0.7respectively.As a=1,the conventional bifurca-tion diagram is obtained.The bifurcation diagrams versus k for a>1As discussed in thefixed points subsection,when a>1this lo-gistic map has afixed point for each k in the interval k e(k c,k s). Therefore,the bifurcation diagram suffers from discontinuities once the parameter a>1as shown from Fig.3where thefixed points appeared suddenly at k=k c as discussed in Table1. For example when a=2.5,the bifurcation diagram starts very close to k=2.9with a single solution(fixed point)until k%3.3where bifurcation begins.It is very clear that the bifur-cation diagram has a similar shape to the conventional case. However as a becomes greater than one the range of k for a5The bifurcation diagram of the second proposed logistic map(a)versus A for different values of a={0.1,0.4,0.7,1.0,100} versus a for different values k={1.5,2.5,4.0}.On some generalized discrete logistic maps167different as shown in Fig.3where the range of acceptable a for nontrivial solution is related to k c .For example when k =2the system has a single nontrivial fixed point up to a =1.416.Avery interesting curve is found when k =3where the system has period two in the range a e (0.5,1).Otherwise,parameter a in this case since p ðx ;k ;a Þis different than the pre-vious case f ðx ;k ;a Þwhere this range was fixed.The fixed points168A.G.RadwanThefifth iteration a e[0,2]for different k<k max=2.5981 The effect of k in the5th iteration p5ðx;k;aÞwhere the function value increases as k increases is studied.The function value at k=0.5<1is very small compared to higher values of k max>k>0.5where this value will approach one as k ap-proaches its maximum value k max=2.5981.In addition,the number of peaks increases as k increases and at higher values of a.The bifurcation diagrams versus k for different aThe bifurcation diagrams with respect to k for different values of a are shown in Fig.5.It is clear that thefixed point appears in all cases at k=1however the range of obtaining chaos is widely different with a.When a=0.1thefirst bifurcation hap-pens at k s=21and chaos happens at k max=28.53whereas these values shrink to k s=6and k max=8.117at a=0.4 and shrink more up to k s=3and k max=4at the conven-tional case(a=0.4)as shown in Fig.5.The range from the first bifurcation to chaos in case a=1.5,2.0,3.0,and5.0is gi-ven by(2.33,3.07),(2,2.598),(1.667,2.116),and(1.4,1.717) respectively.For a=100this range will shrink to (1.02,1.0577)where the bifurcation happens at k=1.02and the system behaves chaotically at k=1.0577with an output covering all the range[0,1].The bifurcation diagrams versus a for different kSimilarly the bifurcation diagrams with respect to a are shown in Fig.5for different values of k=1.5,2.5,and4.0.All the bifurcation diagrams are similar to the bifurcation diagram shown before with respect to k.It is noted that,the dependence of the functions f and p are different with respect to a as shown from the bifurcation diagrams of Figs.4and5respectively. Third generalized logistic map x nþ1¼k x anð1Àx nÞLet us define sðx;k;aÞ¼k x að1ÀxÞwhere the peak of this ex-ists at x¼x p¼a which monotonically increases as a in-creases.As afi0the peak value x p tends to0however as afi1the value of x pfi1.The range of k to guarantee that s mðx;k;aÞis always in the range[0,1]is given by k e(0,k max)where k max¼ðaþ1Þaþ1a a .Then if a increases,k max will increase asshown in ing simple calculations,thefixed point x\ will be stable in the range max(0,aÀ1)<x<aþ1which isequivalent to0<k<ðaþ2Þaðaþ1ÞaÀ1¼k s in case if a<1andk min¼a aðaþ1ÞaÀ1<k<ðaþ2Þaðaþ1ÞaÀ1¼k s when a>1.Therefore,therange of operating k forfixed a is the distance between k=0and the dashed line(horizontal line k max)as clear in the case of a=0.5in Fig.6a.The range of all possible outputs is given from the intersection between the dashed line with the solid line.However,as a>1the curve k c has a minimum (non-invertible)then for constant a thefixed point will not start from k=0and the starting point(k min)increases as a in-creases.Hence,the maximum range of acceptable k for non-trivial solution is(k min,k max).For example when a=1.5,the system has only one trivial solution in the interval(0,2.6), the single nontrivialfixed point exists from k min%2.6and then bifurcation to chaos appears in the interval below k max=5.38 as shown in Fig.6a and its bifurcation diagram in Fig.6b.As a increases both the k min and k max increase where the system has a nontrivial solution as shown in Fig.6a and b in the case a=6.3.The bifurcation diagram of the third proposed logistic map versus the arbitrary power is shown in Fig.6c where the cha-otic response happens at the lower values of a.As a increases the system behavior transforms from chaotic response,to peri-odic until it reachesfixed point.In addition,as k increases both the lower and upper values of a for nontrivial solutions are increased.Comparison and calculation of Lyapunov exponentFrom the previous sections,we found that the proposed logis-tic maps have different characteristics and cover all possible cases.For example,the bifurcation range is:k e(0,4)withfixed limits as in thefirst proposed map when a<1,k e(k c,4)with variable start limit as in thefirst proposed map when a>1,k e(0,k max)with variable end limit as in the third proposed map when a<1,k e(1,k max)with variable end limit as in the second pro-posed map,k e(k min,k max)with variable start and end limits as in the third proposed map when a>1.Moreover,the bifurcation diagram with respect to the new parameter a for the second and third proposed maps has sim-ilar properties as the conventional logistic map which increases the designflexibility.The time domain output and the illustra-tion of Cobweb method for the three proposed maps are shown in Table2when the new parameter a=0.5and in the chaotic range of k.The calculation of Lyapunov exponentTo prove the chaotic behavior of the output response,it is re-quired to calculate the Lyapunov exponent which is the major key for chaotic systems.As known from the nonlinear analysis of chaos,it is necessary to have a positive value of the Lyapu-nov exponent to prove chaotic behavior.Recently[24–28], there are many numerical techniques to calculate the value of the Lyapunov exponent.For the1-D map defined by x k+1=f(x k,k),the Lyapunov exponent for the orbit starting at x o can be calculated byL:E:¼limn!11nX nÀ1i¼0ln j f0ðx iÞj()ð5Þwhere f0ðxÞis the derivative of the function f(x).The Lyapu-nov exponents of the proposed maps are shown in Table2. As known the conventional logistic map at k=3.9has a Lyapunov exponent of order0.496.The effect of the new parameter a in thefirst proposed map(fixed range)on the Lyapunov exponent is shown in Table2where the LE in-creases as the value of a increases for the same value of k.Five170 A.G.Radwandifferent cases of the LE for each of the second and third pro-posed maps are shown in Table2.ConclusionThis paper introduces three independent generalized logistic maps of arbitrary order.The summary of the main factors for each proposed map,the critical points,ranges,and some comments on the bifurcation diagrams with respect to the sys-tem parameter and also to the arbitrary power are introduced. 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