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Chaotic Behavior Of The Logistic Equation

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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