A Stability Analysis of Logistic Model

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.17(214) No.1,pp.71-79 A Stability Analysis of Logistic Model Bhagwati Prasad, Kuldip Katiyar Department of Mathematics, Jaypee Institute of Information Technology, A-1, Sector-62, Noida, UP-2137 India (Received 18 April 212, accepted 2 March 213) Abstract: The purpose of this paper is to investigate the logistic model using Ishikawa iterative procedure through time series and Lyapunov eponent analysis. It is observed that for certain choices there is an appreciable growth in the stability of the model even for the higher values of the control parameter. Keywords: Logistic map; Mann iterate; Ishikawa iterate; Lyapunov eponent; Chaos 1 Introduction Most of the natural phenomena can be suitably described by the celebrated logistic model, essentially proposed by the Belgian mathematician Verhulst [22, 23]. It is remarkable that this pivoting work of Verhulst was forgotten after his death for a long time and it took more than hundred years to recognize the founding contributions of him towards the population dynamics and nonlinear sciences. For the historical development and the diverse implicational aspect of this model, one can refer to [2, 9]. This simple looking map can start from stable points, walk through the spills of stable cycles to a domain where the behavior is in many aspects totally unpredictable and thus possesses various dynamical characteristics. The importance of this map lies in the fact that a minute change in the initial condition may cause a drastic change in the behavior of the function. This etreme sensitivity to the initial condition is the most fascinating aspect of chaotic maps which make chaotic systems ideal for various applications. Dettmer [] pointed out that the most obvious reason for knowing about chaos is to organize and possibly avoid it because the regularity and stability disappears once the system becomes chaotic. This chaotic behavior might cause a serious problem to the underlying system, for eample, in communication network the chaotic modes of vibration not only threaten the stability of the system but it might also break down the entire system. A number of techniques are used for controlling the chaos in the literature such as feedback linearization, variable structure controller, fuzzy method and neural networks [21]. The vagaries of the logistic maps have attracted a number of authors since Verhulst used it as a demographic model for his studies. The interest was further spurred by the advancement of computational tools and proliferation of digital computers in the latter half 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 several references thereof). Recently Rani and Agarwal [2] studied the comparative behavior of the logistic maps with Picard orbit, Norland orbit and Mann orbit and enhanced the stable behavior of the logistic map for higher values of the control parameter using Mann iterative procedures. Bresten and Jung [4] obtained the interesting geometric patterns and studied the speed of convergence of such maps to epose the underlying compleity in some specific region for the parameter r. In this paper, we investigate the stability of the logistic map using Ishikawa iterative procedure. On the basis of our eplorations in terms of time series analysis and the Lyapunov eponents of the map we observe that the unstable and chaotic behavior of the orbits transforms into periodic and stable behavior even for higher values of the control parameter r. We use Matlab programs for the computational and graphical requirements. 2 Preliminaries This section is primarily devoted to the notations and definitions used in the sequel. Corresponding author. E-mail address: b prasad1@yahoo.com, bhagwati.prasad@jiit.ac.in Copyright c World Academic Press, World Academic Union IJNS.214.2.1/787

72 International Journal of Nonlinear Science, Vol.17(214), No.1, pp. 71-79 The following one dimensional difference equation represents the discrete version of the Verhulst model also called logistic map: n+1 = r n (1 n ), (1) here n 1 denotes population size at time n = 1, 2, 3,... and non-negative real number r is a control parameter that 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 population biomass of species in the presence of limiting factors such as food supply or disease. The above logistic model can possess stable, unstable, periodic and chaotic behaviours and thus receives wide attention due to the great implications of it in chaos theory (see May [14], May and Oster [1]). 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 period n 1, n N, iff f n (p) = p and f k (p) p for all k = 1, 2,... n 1, where f k (p) := f(f(... (f(p))... )). A periodic }{{} k times point of f of period 1 is simply a fied point of f. Definition 2 Let (X, d) be a metric space and f : X X. The orbit of a point in X under the transformation f is defined as a sequence {f n () : n =, 1, 2,... }. This transformation f may also be called a dynamical system denoted by {X, f}. Definition 3 Let X be a non empty set and f : X X. For a point in X, construct a sequence { n } in the following manner. n = α n f(y n 1 ) + (1 α n ) n 1, y n 1 = β n f( n 1 ) + (1 β n ) n 1,, (2) for n = 1, 2, 3,..., where < α n 1 and β n 1 and the sequence {α n } is convergent away from. The sequence { n } constructed above is essentially due to Ishikawa [11]. The corresponding Ishikawa orbit consisting of all iterates of the point is denoted by IO(f,, α n, β n ). We shall study the Ishikawa orbit for α n = α and β n = β. It is remarked that (2) becomes Mann or superior orbit [13] when β n = and it gives Picard orbit when β n = and α n = 1. Rani and Agarwal [2] 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 prime period k. Then is called forward asymptotic to p if the sequence {, f k (), f 2k (), f 3k (),... } converges to p i.e. lim n f nk () = p. The stable set of p, denoted by W s (p), consists of all points which are forward asymptotic to p. If the sequence {, f k (), f 2k (), f 3k (),... } grows without bound, then is forward asymptotic to. The stable set of, denoted by W s ( ), consists of all points which are forward asymptotic to. Following Alligood et al. [1], we define the Lyapunov eponent in the following manner. Definition Let f be a continuous map of the real line R. The Lyapunov eponent LE( 1 ) of the orbit { 1, 2, 3,...} is defined as LE( 1 ) = lim n (1/n)[ln f ( 1 ) + ln f ( 2 ) + + ln f ( n )] if this limit eists. The orbit { n } is generated by the rule (2). 3 Stability analysis through time series In this section we study the behavior of logistic orbits generated by the iteration scheme (2) using time series. For comparing the behavior of the logistic model via new iterative scheme, we tabulate the values of and α as taken by Rani and Agarwal [2] and choose a value of β to obtain the maimum value of r up to which the map ehibits the stable behavior. IJNS email for contribution: editor@nonlinearscience.org.uk

Bhagwati Prasad, Kuldip Katiyar: A Stability Analysis of Logistic Model 73 We observe that the maimum value of r for which logistic map shows stable behavior depends on the values of the parameters α and β in the range (, 1). Considering in [, 1], we attempt to obtain this value of r correctable up to four decimal places. For some specific choices of parameters α and β, the optimum values of r for initial values =.1,.2,.3 and. are tabulated (see Table 1-). The corresponding time series for the function with selected initial choices for specific r are shown in Figs. 1, 3,, 6, 8-1, 12-14, 16. For α =.9, β =.2, we find that the logistic map is convergent to a fied point for < r 4.2991 and the optimum value of r is.6363 at initial choice =.1 (see Table 1 and Fig. 1). When r >.6363, it cannot be described as Ishikawa orbit because n becomes greater than 1 for all selected initial choices. At the choice α =.64, β =.1 the logistic map is convergent to a fied point for < r.263 and the optimum value of r is 8.1338 for the selected initial choice of =.1 (see Fig. 2). This map cannot be described as Ishikawa orbit for r > 8.1338 because n / [, 1]. At the specific value =.2 with α =.17, β =.1 the orbit converges to a fied point at r = 17.1 while at other selected choices it shows cyclic but stable behavior for 18.429 < r 2.33. Tables 3-4 show the stable and converging behavior under 1 and 1 iterations for the respective optimum choices of r with α =.17, β =.1 and α =.1, β =.1. Again, it shows cyclic but stable behavior for 21.3467 r 26.3738 and for r > 26.3738 it leaves the orbit. Table depicts the behaviors of the logistic map for smaller values of the parameters α and β. The numerical convergence of logistic map for various values of r upto 1th decimal places is depicted by the plots of number of iteration verses the initial choice by fiing the parameters r, α and β (see Figs. 2, 4, 7, 11, 1) r Point of convergence.1.6363.8226.2 4.6291.784.3 4.2991.7674. 4.48.778 Table 1: α =.9 and β =.2 r Point of convergence.1 8.1338.8771.2 6.373.841.3.39.819..263.811 Table 2: α =.64 and β =.1 4 Lyapunov eponents and stability of orbits In this section we attempt to investigate the behaviour of logistic map by estimating its Lyapunov eponent (LE). Lyapunov eponent of a function measures its sensitive dependence upon the initial condition and gives the stretching rate IJNS homepage: http://www.nonlinearscience.org.uk/

74 International Journal of Nonlinear Science, Vol.17(214), No.1, pp. 71-79 3 2 2 1 1.2.4.6.8 1 Figure 1: (r,, α, β) = (.6363,.1,.9,.2). Figure 2: (r,, α, β) = (4.2991,.9,.2). 8 6 4 2.2.4.6.8 1 Figure 3: (r,, α, β) = (8.1338,.1,.64,.1). Figure 4: (r, α, β) = (.263,.64,.1). Cyclic Optimum r under 1 iterates Optimum r under 1 iterates r r Point of convergence r Point of convergence.1 2.33 18.249.89 18.429.749.2 17.1 17.1.674 17.1.674.3 2.8331 18.249.89 18.429.749. 2.8331 18.249.89 18.429.749 Table 3: α =.17 and β =.1 Cyclic Optimum r under 1 iterates Optimum r under 1 iterates r r Point of convergence r Point of convergence.1 26.3738 21.121.4981 21.3467.4921.2 26.3738 21.127.4981 21.3467.4921.3 26.3738 21.122.4981 21.3467.4921. 26.3738 21.198.497 21.3467.4921 Table 4: α =.1 and β =.1 IJNS email for contribution: editor@nonlinearscience.org.uk

Bhagwati Prasad, Kuldip Katiyar: A Stability Analysis of Logistic Model 7 Figure : (r,, α, β)= (18.249,.1,.17,.1). Figure 6: (r,, α, β) = (18.429,.1,.17,.1). 4 3 2 1.2.4.6.8 1 Figure 7: (r, α, β) = (17.1,.17,.1). Figure 8: (r,, α, β) = (2.8331,.3,.17,.1) Figure 9: (r,, α, β) = (21.198,.,.1,.1). Figure 1: (r,, α, β) = (21.3467,.1,.1,.1). 1 1.2.4.6.8 1 Figure 11: (r, α, β) = (21.3467,.1,.1). Figure 12: (r,, α, β) = (26.3738,.1,.1,.1). IJNS homepage: http://www.nonlinearscience.org.uk/

76 International Journal of Nonlinear Science, Vol.17(214), No.1, pp. 71-79 Cyclic Optimum r under 1 iterates Optimum r under 1 iterates r r Point of convergence r Point of convergence.1 27.81 93.232.9892 21.3177.4992.2 27.81 93.232.9892 21.3177.4992.3 24.681 93.232.9892 21.3177.4992. 24.143 93.232.9892 21.374.4992 Table : α =.1 and β =.1 Figure 13: (r,, α, β) = (93.232,.1,.1,.1). Figure 14: (r,, α, β) = (21.374,.,.1,.1). 1 8 6 4 2.2.4.6.8 1 Figure 1: (r, α, β) = (21.374,.1,.1). Figure 16: (r,, α, β)=(27.81,.1,.1,.1). IJNS email for contribution: editor@nonlinearscience.org.uk

Bhagwati Prasad, Kuldip Katiyar: A Stability Analysis of Logistic Model 77 per iteration averaged over the trajectory and gives an indication of divergence or convergence of the orbits starting close together. Thus it has a crucial role in the theory of dynamical systems for measuring the average rate of the divergence (or convergence) spread over the trajectory for a chaotic behaviour (or stable periodic behaviour). If the LE for a given r is less than zero the orbit attracts to a stable fied point or stable periodic orbit. The negative LE characterize an asymptotic stability that is, the more negative the eponent, the greater the stability of the orbit. If the LE for a given r is zero the orbit is a neutral fied point or an eventually fied point and it indicates that the system is in some sort of steady state mode. The unstable and chaotic behavior for a given r is characterized by the positive value of the LE which indicates that the nearby points, no matter how close, will diverge to any arbitrary separation as we increase the number of iterations. A number of authors have eplored the LE and studied the behaviour of the dynamical systems see for instance, Giesel et al [8], Huberman and Rudnick [1] and McCartney [16] and references thereof. We obtain the LE of the logistic map for the same values of the parameters α, β as taken in section 3 by varrying the parameter r and fiing at.1. The Lyapunov eponents and the epected 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.9.2 r = 1 Natural < r < 6.64 ecept r = 1 Negative Stable r =.839 least negative More stable.64.1 r = 1 Natural < r < 1.76 ecept r = 1 Negative Stable r =.839 least negative More stable.17.1 r = 1, 12 Natural < r < 19.788 ecept r = 1, 12 Negative Stable r = 1.8118 least negative More stable 19.788 < r < 2.8332 Positive Chaotic.1.1 r = 1, 12 Natural < r < 24.7838 ecept r = 1, 12 Negative stable r = 17.77 least negative More stable 24.7838 < r < 26 Positive Chaotic.1.1 241 < r < 242 Positive Chaotic for all other < r < 249.9894 Negative stable r = 163.2883 least negative More stable Table 6: Conclusion In view of the Lyapunov eponents and the time series analysis of the logistic map studied under the Ishikawa iterative scheme, we conclude that the model ehibits stable behavior for higher values of r as compared to the similar results available in the literature (see [19], [2]). In this way we could achieve stability of the map for higher values of the parameter r. It is also noticed that as the values of the parameters α and β move closer to zero, the logistic model shows stable behavior for even higher values of r. For the choice of α =.1, β =.1 under 1 iterations it shows convergent behavior up to r = 93.232 for all initial choices of. It is remarked that the map has convergent behaviors even at r = 21.374 although the convergence is slower in this case. Acknowledgments The authors thank the anonymous referees. IJNS homepage: http://www.nonlinearscience.org.uk/

78 International Journal of Nonlinear Science, Vol.17(214), No.1, pp. 71-79 Bifurcation Diagram for Logistic Map: [α, β] = [.9.2] 1.8.6.4.2 2 4 6 8 1 Lyapunov eponent for Logistic Map: [α, β] = [.9.2] 1 1 2 4 6 8 1 (i).(α, β) = (.9,.2) (ii).(α, β) = (.64,.1) Bifurcation Diagram for Logistic Map: [α, β] = [.17.1] Bifurcation Diagram for Logistic Map: [α, β] = [.1.1] 1 1.8.8.6.6.4.4.2.2 1 1 2 2 Lyapunov eponent for Logistic Map: [α, β] = [.17.1] 1 1 1 1 2 1 1 2 (iii).(α, β) = (.17,.1) 1 1 1.8.6.4.2 1 1 2 2 Lyapunov eponent for Logistic Map: [α, β] = [.1.1] 2 2 4 6 2 (iv).(α, β) = (.1,.1) Bifurcation Diagram for Logistic Map: [α, β] = [.1.1] 2 2 3 Lyapunov eponent for Logistic Map: [α, β] = [.1.1] 1 1 2 2 (v).(α, β) = (.1,.1) Figure 17: Bifurcation diagrams and lyapunov eponents IJNS email for contribution: editor@nonlinearscience.org.uk 2 3

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