Chaotic Behavior Of The Logistic Equation

Abstract

Chaotic systems are nonlinear dynamical systems that exhibit a

random, unpredictable behavior. Trajectories of chaotic dynamical

systems are sensitive to initial conditions in the sense that starting

from slightly di®erent initial conditions the trajectories diverge expo-

nentially. To study chaos, the behavior of solution to logistic equation

is considered. In this paper, for di®erent parameters, the solutions for

the logistic equation is analyzed. At a certain point, the solution di-

verges to multiple equilibrium points, the periodicities increase as the

parameter increases. To verify the analytical prediction of the math-

ematical model, several computer experiments are run. At a certain

value of the parameter, the solution has theoretical in¯nite periodici-

ties, that is it behaves randomly, the system has turned chaotic.

1 Introduction

The behavior of the solutions of the logistic equation for certain range of

parameters is complex, sometimes of di®erent periodicities or aperiodic. The

aperiodic solutions are called chaotic solutions or chaotic motions. Quoting

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chaos is in the behavior that is not an equilibrium, a cycle or even a quasi-

periodic motion-there is more to be said about chaos. Chaotic motion has

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some aspect that is provably as random as a coin toss. The randomness arises

from sensitive dependence on imperfectly known initial conditions, . . . ".

2 Mathematical Modelling

In the analysis of growth of a population, the behavior of population can

be modeled by di®erential equations known as...

... middle of paper ...

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Bifurcation Diagram

Figure 4: Bifurcation diagram

5 Discussion

The bifurcation diagram, shown in Fig.4 is obtained by getting the solu-

tions of equation for all values of ½. For ½ > 3, there are no attracting ¯xed

points. As row increases, the solutions of the logistic equation exhibit in-

creasing complexity. For certain range of ½, just above 3, the solution settles

down into a steady oscillation of period 2. Then, as ½ is further increased, pe-

riodic solutions of period 4,8,16,. . . appear.For ½ > 3:57, the solution becomes

aperiodic, chaotic.

Works Cited
[1] James Glick.,Chaos

[2] Zak S.,Systems and Control,Oxford University Press.

[3] Chin-Teng Lin & C.S.George Lee, Neural Fuzzy Systems,Prentice Hall

Internation Inc.

[4] http://hypertextbook.com/chaos

[5] http://www.duke.edu/ mjd/chaos/chaosp.html

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