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ISSN 1749-3889 (print), 1749-3897 (online)International Journal of Nonlinear Science

Vol.17(2014) No.1,pp.71-79

A Stability Analysis of Logistic Model

Bhagwati Prasad ∗, Kuldip KatiyarDepartment of Mathematics, Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307 India

(Received 18 April 2012, accepted 25 March 2013)

Abstract: The purpose of this paper is to investigate the logistic model using Ishikawa iterative procedurethrough time series and Lyapunov exponent analysis. It is observed that for certain choices there is an appre-ciable growth in the stability of the model even for the higher values of the control parameter.

Keywords: Logistic map; Mann iterate; Ishikawa iterate; Lyapunov exponent; Chaos

1 IntroductionMost of the natural phenomena can be suitably described by the celebrated logistic model, essentially proposed by theBelgian mathematician Verhulst [22, 23]. It is remarkable that this pivoting work of Verhulst was forgotten after hisdeath for a long time and it took more than hundred years to recognize the founding contributions of him towards thepopulation dynamics and nonlinear sciences. For the historical development and the diverse implicational aspect of thismodel, one can refer to [2, 9]. This simple looking map can start from stable points, walk through the spills of stablecycles to a domain where the behavior is in many aspects totally unpredictable and thus possesses various dynamicalcharacteristics. The importance of this map lies in the fact that a minute change in the initial condition may cause adrastic change in the behavior of the function. This extreme sensitivity to the initial condition is the most fascinatingaspect of chaotic maps which make chaotic systems ideal for various applications. Dettmer [5] pointed out that the mostobvious reason for knowing about chaos is to organize and possibly avoid it because the regularity and stability disappearsonce the system becomes chaotic. This chaotic behavior might cause a serious problem to the underlying system, forexample, in communication network the chaotic modes of vibration not only threaten the stability of the system but itmight also break down the entire system. A number of techniques are used for controlling the chaos in the literaturesuch as feedback linearization, variable structure controller, fuzzy method and neural networks [21]. The vagaries ofthe logistic maps have attracted a number of authors since Verhulst used it as a demographic model for his studies. Theinterest was further spurred by the advancement of computational tools and proliferation of digital computers in the latterhalf of the twentieth century. A number of papers have signified the importance of the logistic maps in chaos, fractals,cryptography, optimization, discrete dynamics, population dynamics etc. (see for instance, [6, 7, 12, 17, 18] and severalreferences thereof). Recently Rani and Agarwal [20] studied the comparative behavior of the logistic maps with Picardorbit, Norland orbit and Mann orbit and enhanced the stable behavior of the logistic map for higher values of the controlparameter using Mann iterative procedures. Bresten and Jung [4] obtained the interesting geometric patterns and studiedthe speed of convergence of such maps to expose the underlying complexity in some specific region for the parameterr. In this paper, we investigate the stability of the logistic map using Ishikawa iterative procedure. On the basis of ourexplorations in terms of time series analysis and the Lyapunov exponents of the map we observe that the unstable andchaotic behavior of the orbits transforms into periodic and stable behavior even for higher values of the control parameterr. We use Matlab programs for the computational and graphical requirements.

2 PreliminariesThis section is primarily devoted to the notations and definitions used in the sequel.

∗Corresponding author. E-mail address: b [email protected], [email protected]

Copyright c⃝World Academic Press, World Academic UnionIJNS.2014.02.15/787

72 International Journal of Nonlinear Science, Vol.17(2014), No.1, pp. 71-79

The following one dimensional difference equation represents the discrete version of the Verhulst model also calledlogistic map:

xn+1 = rxn(1− xn), (1)

here 0 ≤ xn ≤ 1 denotes population size at time n = 1, 2, 3, . . . and non-negative real number r is a control parameterthat represents a combined rate for reproduction and starvation.

The equation (1) is found to be the most suitable model for the study of the surplus production of the populationbiomass of species in the presence of limiting factors such as food supply or disease. The above logistic model can pos-sess stable, unstable, periodic and chaotic behaviours and thus receives wide attention due to the great implications of itin chaos theory (see May [14], May and Oster [15]).

Definition 1 [3] Let X be a non empty space and let f : X → X . A point p ∈ X is called a periodic point of f of periodn ≥ 1, n ∈ N , iff fn(p) = p and fk(p) ̸= p for all k = 1, 2, . . . n−1, where fk(p) := f(f(. . . (f︸︷︷︸

k times

(p)) . . . )). A periodic

point of f of period 1 is simply a fixed point of f .

Definition 2 Let (X, d) be a metric space and f : X → X . The orbit of a point x in X under the transformation f isdefined as a sequence {fn(x) : n = 0, 1, 2, . . . }. This transformation f may also be called a dynamical system denotedby {X, f} .

Definition 3 Let X be a non empty set and f : X → X . For a point x0 in X , construct a sequence {xn} in the followingmanner.

xn = αnf(yn−1) + (1− αn)xn−1,yn−1 = βnf(xn−1) + (1− βn)xn−1,

, (2)

for n = 1, 2, 3, . . . , where 0 < αn ≤ 1 and 0 ≤ βn ≤ 1 and the sequence {αn} is convergent away from 0. Thesequence {xn} constructed above is essentially due to Ishikawa [11]. The corresponding Ishikawa orbit consisting of alliterates of the point x0 is denoted by IO(f, x0, αn, βn).

We shall study the Ishikawa orbit for αn = α and βn = β. It is remarked that (2) becomes Mann or superior orbit [13]when βn = 0 and it gives Picard orbit when βn = 0 and αn = 1 .

Rani and Agarwal [20] and Prasad and Katiyar [19] studied the logistic map for Mann and Picard orbits.

Definition 4 [9] Let X be a non empty set, f : X → X and a point p of X be a periodic point of f with primeperiod k. Then x is called forward asymptotic to p if the sequence {x, fk(x), f2k(x), f3k(x), . . . } converges to p i.e.limn→∞ fnk(x) = p. The stable set of p, denoted by W s(p), consists of all points which are forward asymptotic to p. Ifthe sequence {|x|, |fk(x)|, |f2k(x)|, |f3k(x)|, . . . } grows without bound, then x is forward asymptotic to ∞. The stableset of ∞, denoted by W s(∞), consists of all points which are forward asymptotic to ∞.

Following Alligood et al. [1], we define the Lyapunov exponent in the following manner.

Definition 5 Let f be a continuous map of the real line R. The Lyapunov exponent LE(x1) of the orbit {x1, x2, x3, ...}is defined as

LE(x1) = limn→∞

(1/n)[ln f ′(x1) + ln f ′(x2) + · · ·+ ln f ′(xn)]

if this limit exists. The orbit {xn} is generated by the rule (2).

3 Stability analysis through time seriesIn this section we study the behavior of logistic orbits generated by the iteration scheme (2) using time series. Forcomparing the behavior of the logistic model via new iterative scheme, we tabulate the values of x0 and α as taken byRani and Agarwal [20] and choose a value of β to obtain the maximum value of r up to which the map exhibits the stablebehavior.

IJNS email for contribution: [email protected]

Bhagwati Prasad, Kuldip Katiyar: A Stability Analysis of Logistic Model 73

We observe that the maximum value of r for which logistic map shows stable behavior depends on the values ofthe parameters α and β in the range (0, 1). Considering x0 in [0, 1], we attempt to obtain this value of r correctableup to four decimal places. For some specific choices of parameters α and β, the optimum values of r for initial valuesx0 = 0.15, 0.25, 0.35 and 0.50 are tabulated (see Table 1-5). The corresponding time series for the function with selectedinitial choices for specific r are shown in Figs. 1, 3, 5, 6, 8-10, 12 - 14, 16. For α = 0.9, β = 0.2, we find that thelogistic map is convergent to a fixed point for 0 < r ≤ 4.2991 and the optimum value of r is 5.6363 at initial choicex0 = 0.15 (see Table 1 and Fig. 1). When r > 5.6363, it cannot be described as Ishikawa orbit because xn becomesgreater than 1 for all selected initial choices. At the choice α = 0.64, β = 0.1 the logistic map is convergent to a fixedpoint for 0 < r ≤ 5.2653 and the optimum value of r is 8.1338 for the selected initial choice of x0 = 0.15 (see Fig. 2).This map cannot be described as Ishikawa orbit for r > 8.1338 because xn /∈ [0, 1]. At the specific value x0 = 0.25 withα = 0.17, β = 0.1 the orbit converges to a fixed point at r = 17.5001 while at other selected choices it shows cyclic butstable behavior for 18.4290 < r ≤ 20.3503. Tables 3-4 show the stable and converging behavior under 100 and 10000iterations for the respective optimum choices of r with α = 0.17, β = 0.1 and α = 0.1, β = 0.1. Again, it shows cyclicbut stable behavior for 21.3467 ≤ r ≤ 26.3738 and for r > 26.3738 it leaves the orbit. Table 5 depicts the behaviors ofthe logistic map for smaller values of the parameters α and β.

The numerical convergence of logistic map for various values of r upto 15th decimal places is depicted by the plots ofnumber of iteration verses the initial choice x0 by fixing the parameters r, α and β (see Figs. 2, 4, 7, 11, 15)

x0 r Point of convergence

0.15 5.6363 0.82260.25 4.6291 0.78400.35 4.2991 0.76740.50 4.5048 0.7780

Table 1: α = 0.9 and β = 0.2

x0 r Point of convergence

0.15 8.1338 0.87710.25 6.3073 0.84150.35 5.5395 0.81950.50 5.2653 0.8101

Table 2: α = 0.64 and β = 0.1

4 Lyapunov exponents and stability of orbitsIn this section we attempt to investigate the behaviour of logistic map by estimating its Lyapunov exponent (LE). Lya-punov exponent of a function measures its sensitive dependence upon the initial condition and gives the stretching rate

IJNS homepage: http://www.nonlinearscience.org.uk/


Figure 1: (r, x0, α, β) = (5.6363, 0.15, 0.9, 0.2).

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

x

Itera

tions

unt

ill c

onve

rgen

ce

Figure 2: (r, x0, α, β) = (4.2991, 0.9, 0.2).

Figure 3: (r, x0, α, β) = (8.1338, 0.15, 0.64, 0.1).

0 0.2 0.4 0.6 0.8 10

20

40

60

80

x

Itera

tions

unt

ill c

onve

rgen

ce

Figure 4: (r, α, β) = (5.2653, 0.64, 0.1).

x0 Cyclic Optimum r under 100 iterates Optimum r under 10000 iterates

r r Point of convergence r Point of convergence

0.15 20.3503 18.2490 0.5809 18.4290 0.57490.25 17.5001 17.5001 0.6074 17.5001 0.60740.35 20.8331 18.2490 0.5809 18.4290 0.57490.50 20.8331 18.2490 0.5809 18.4290 0.5749

Table 3: α = 0.17 and β = 0.1



0.15 26.3738 21.1021 0.4981 21.3467 0.49210.25 26.3738 21.1027 0.4981 21.3467 0.49210.35 26.3738 21.1022 0.4981 21.3467 0.49210.50 26.3738 21.1985 0.4957 21.3467 0.4921

Table 4: α = 0.1 and β = 0.1



Figure 5: (r, x0, α, β)= (18.2490, 0.15, 0.17, 0.1). Figure 6: (r, x0, α, β) = (18.4290, 0.15, 0.17, 0.1).

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

x

Itera

tions

unt

ill c

onve

rgen

ce

Figure 7: (r, α, β) = (17.5001, 0.17, 0.1). Figure 8: (r, x0, α, β) = (20.8331, 0.35, 0.17, 0.1)

Figure 9: (r, x0, α, β) = (21.1985, 0.5, 0.1, 0.1) . Figure 10: (r, x0, α, β) = (21.3467, 0.15, 0.1, 0.1).

0 0.2 0.4 0.6 0.8 10

5000

10000

15000

x

Itera

tions

unt

ill c

onve

rgen

ce

Figure 11: (r, α, β) = (21.3467, 0.1, 0.1) . Figure 12: (r, x0, α, β) = (26.3738, 0.15, 0.1, 0.1).





0.15 257.0801 93.0232 0.9892 201.3177 0.49920.25 257.0801 93.0232 0.9892 201.3177 0.49920.35 254.0681 93.0232 0.9892 201.3177 0.49920.50 254.1403 93.0232 0.9892 201.3074 0.4992

Table 5: α = 0.01 and β = 0.01

Figure 13: (r, x0, α, β) = (93.0232, 0.15, 0.01, 0.01). Figure 14: (r, x0, α, β) = (201.3074, 0.5, 0.01, 0.01).

0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000

x

Itera

tions

unt

ill c

onve

rgen

ce

Figure 15: (r, α, β) = (201.3074, 0.01, 0.01). Figure 16: (r, x0, α, β)=(257.0801, 0.15, 0.01, 0.01).



per iteration averaged over the trajectory and gives an indication of divergence or convergence of the orbits starting closetogether. Thus it has a crucial role in the theory of dynamical systems for measuring the average rate of the divergence (orconvergence) spread over the trajectory for a chaotic behaviour (or stable periodic behaviour). If the LE for a given r isless than zero the orbit attracts to a stable fixed point or stable periodic orbit. The negative LE characterize an asymptoticstability that is, the more negative the exponent, the greater the stability of the orbit. If the LE for a given r is zero theorbit is a neutral fixed point or an eventually fixed point and it indicates that the system is in some sort of steady statemode. The unstable and chaotic behavior for a given r is characterized by the positive value of the LE which indicates thatthe nearby points, no matter how close, will diverge to any arbitrary separation as we increase the number of iterations. Anumber of authors have explored the LE and studied the behaviour of the dynamical systems see for instance, Giesel et al[8], Huberman and Rudnick [10] and McCartney [16] and references thereof. We obtain the LE of the logistic map for thesame values of the parameters α, β as taken in section 3 by varrying the parameter r and fixing x0 at 0.1. The Lyapunovexponents and the expected behavior of the orbits at the chosen values of the parameters is shown in Table 6 and Fig. 17.

α β r LE Nature of orbit

0.9 0.2 r = 1 0 Natural0 < r < 6.6045 except r = 1 Negative Stabler = 5.8539 least negative More stable

0.64 0.1 r = 1 0 Natural0 < r < 10.5706 except r = 1 Negative Stabler = 5.8539 least negative More stable

0.17 0.1 r = 1, 12 0 Natural0 < r < 19.7808 except r = 1, 12 Negative Stabler = 15.8118 least negative More stable19.7808 < r < 20.8332 Positive Chaotic

0.1 0.1 r = 1, 12 0 Natural0 < r < 24.7838 except r = 1, 12 Negative stabler = 17.7057 least negative More stable24.7838 < r < 26 Positive Chaotic

0.01 0.01 241 < r < 242 Positive Chaoticfor all other 0 < r < 249.9894 Negative stabler = 163.2883 least negative More stable

Table 6:

5 ConclusionIn view of the Lyapunov exponents and the time series analysis of the logistic map studied under the Ishikawa iterativescheme, we conclude that the model exhibits stable behavior for higher values of r as compared to the similar resultsavailable in the literature (see [19], [20]). In this way we could achieve stability of the map for higher values of theparameter r. It is also noticed that as the values of the parameters α and β move closer to zero, the logistic model showsstable behavior for even higher values of r. For the choice of α = 0.01, β = 0.01 under 100 iterations it shows convergentbehavior up to r = 93.0232 for all initial choices of x0. It is remarked that the map has convergent behaviors even atr = 201.3074 although the convergence is slower in this case.

AcknowledgmentsThe authors thank the anonymous referees.



0 2 4 6 8 100

0.2

0.4

0.6

0.8

1Bifurcation Diagram for Logistic Map: [α, β] = [0.9 0.2]

0 2 4 6 8 10−15

−10

−5

0

5Lyapunov exponent for Logistic Map: [α, β] = [0.9 0.2]

(i).(α, β) = (0.9, 0.2) (ii).(α, β) = (0.64, 0.1)

0 5 10 15 20 250

0.2

0.4

0.6

0.8


0 5 10 15 20 25−10

−5

0


0 5 10 15 20 25 300

0.2

0.4

0.6

0.8


0 5 10 15 20 25 30−10

−5

0


(iii).(α, β) = (0.17, 0.1) (iv).(α, β) = (0.1, 0.1)

0 50 100 150 200 2500

0.2

0.4

0.6

0.8


0 50 100 150 200 250−6

−4

−2

0


(v).(α, β) = (0.01, 0.01)

Figure 17: Bifurcation diagrams and lyapunov exponents



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