17-03-2014, 08:32 PM
are better characterized using a non integer dynamic model
based on fractional calculus or differentiation or integration of
non integer order. Dynamical system at fractional order have
attracted increasingly in recent years. In this paper we
investigate the fractional order of logistic equation on the basis
of fractional calculus. The chaotic behavior in fractional order
is identified by Lyapunov exponent for the order ½ and ¼. The
numerical results show the positive Lyapunov exponent as an
indication that the system is chaotic at fractional order.
Keywords-component; Fractional calculus; Lyapunov
exponent; logistic model; Riemann–Liouville derivative;chaos
I. INTRODUCTION
The nonlinear dynamic system can exhibit the
complicated behavior based on simple rules where the small
different of the initial input can lead to the large differences
in the output. In this paper, we study the chaotic behavior of
fractional order in logistics equation which described the
population growth model at fractional order. The general
logistic model described by ordinary differential equation
that solutions is explained the constant population growth
rate. This exponential growth model is a simple rule that not
include the limitations on resources, disease and others
factors which exhibit the chaotic behavior for certain
parameter values range [5].
The concept of differentiation and integration to non
integer order is by no means new. The fractional calculus or
the theory of integral and derivatives of non integer order
dates back to the same period as classical calculus since
Leibnitz made some remarks on fractional derivative of
order ½ in the 17th century [1]. Many mathematicians have
contributed to the theory and definitions of the fractional
calculus include Leibniz, Riemann, Grunwald and Letnikov
for more than a century [1][7]. There are numerous
applications of fractional calculus growing continuously
during the last few years include the quantum mechanics,
physical kinetics and chaotic dynamics [2]. The applications
to physics involve viscoelastic systems and electromagnetic
waves, but more recently dynamics of fractional order
dynamical systems, involving fractional mechanics and
fractional oscillators has been analyzed [4][7]. These
mathematical phenomena allow describing a real object more
accurate than the classical integer methods. A typical
example of a non-integer or fractional order system is the
voltage-current relation of a semi-infinite loss transmission
line or diffusion of heat into a semi-infinite solid, where the
heat flow is equal to the half-derivative of temperature
according to time [3].There are numerous applications; it
includes the chaotic dynamics, material sciences, mechanics
of fractal and complex media, quantum mechanics, physical
kinetics, long-range interaction, long-range dissipation, non-
Hamiltonian mechanics [3].
In this paper, we have investigated the dynamic of a
fractional generalization of the well-know logistic equation
with Lyapunov exponent for order ½ and ¼.
based on fractional calculus or differentiation or integration of
non integer order. Dynamical system at fractional order have
attracted increasingly in recent years. In this paper we
investigate the fractional order of logistic equation on the basis
of fractional calculus. The chaotic behavior in fractional order
is identified by Lyapunov exponent for the order ½ and ¼. The
numerical results show the positive Lyapunov exponent as an
indication that the system is chaotic at fractional order.
Keywords-component; Fractional calculus; Lyapunov
exponent; logistic model; Riemann–Liouville derivative;chaos
I. INTRODUCTION
The nonlinear dynamic system can exhibit the
complicated behavior based on simple rules where the small
different of the initial input can lead to the large differences
in the output. In this paper, we study the chaotic behavior of
fractional order in logistics equation which described the
population growth model at fractional order. The general
logistic model described by ordinary differential equation
that solutions is explained the constant population growth
rate. This exponential growth model is a simple rule that not
include the limitations on resources, disease and others
factors which exhibit the chaotic behavior for certain
parameter values range [5].
The concept of differentiation and integration to non
integer order is by no means new. The fractional calculus or
the theory of integral and derivatives of non integer order
dates back to the same period as classical calculus since
Leibnitz made some remarks on fractional derivative of
order ½ in the 17th century [1]. Many mathematicians have
contributed to the theory and definitions of the fractional
calculus include Leibniz, Riemann, Grunwald and Letnikov
for more than a century [1][7]. There are numerous
applications of fractional calculus growing continuously
during the last few years include the quantum mechanics,
physical kinetics and chaotic dynamics [2]. The applications
to physics involve viscoelastic systems and electromagnetic
waves, but more recently dynamics of fractional order
dynamical systems, involving fractional mechanics and
fractional oscillators has been analyzed [4][7]. These
mathematical phenomena allow describing a real object more
accurate than the classical integer methods. A typical
example of a non-integer or fractional order system is the
voltage-current relation of a semi-infinite loss transmission
line or diffusion of heat into a semi-infinite solid, where the
heat flow is equal to the half-derivative of temperature
according to time [3].There are numerous applications; it
includes the chaotic dynamics, material sciences, mechanics
of fractal and complex media, quantum mechanics, physical
kinetics, long-range interaction, long-range dissipation, non-
Hamiltonian mechanics [3].
In this paper, we have investigated the dynamic of a
fractional generalization of the well-know logistic equation
with Lyapunov exponent for order ½ and ¼.