1585 words - 6 pages

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

1

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 logistic equations.

To derive a discrete-time version of the logistic equation, considering a

situation where there are no predators and there is unlimited supply of food.

Assuming, on the average, each member of the population gives birth to ¾

new members in one unit of time and that no member dies. Let N(k) be the

number of the members in the population at time t = k; then the change in

the population over a time period between times k and k + 1 is

N(k + 1) ¡ N(k) = ¾N(k) (1)

Thus,

N(k + 1) = (1 + ¾)N(k): (2)

Equation (2) is a linear model of population growth. Given the number N(0)

of members of the population at time k = 0, the number of members at time

k is

N(k) = (1 + ¾)kN(0): (3)

The eqution(3) predicts the exponential growth of the population.

In a more realistic situation, let b be the birth rate, then the number of

births at time k is bN(k). Assuming the number of deaths proportionals to

the population size , let d be the death rate; then the number of deaths at

time k id dN(k). The change in population in a time period between times

k and k + 1 is

N(k + 1) ¡ N(k) = bN(k) ¡ dN(k): (4)

This can rewritten as

N(k + 1) = (1 + b ¡ d)N(k): (5)

Let ¾ = b ¡ d: Then, equation(5)s the same form as...

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