Chaos in Fractional-order Autonomous Nonlinear Systems
Wajdi M. Ahmad
Department of Electrical and Electronics Engineering,
University
of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates
J. C. Sprott
Department of Physics, University
of
Wisconsin, Madison, WI 53706, USA
(Accepted 19 September 2002)
ABSTRACT
We numerically investigate chaotic behavior in autonomous nonlinear
models
of fractional order. Linear transfer function approximations
of
the
fractional integrator block are calculated for a set of fractional
orders
in (0, 1], based on frequency domain arguments, and the resulting
equivalent
models are studied. Two chaotic models are considered in this
study;
an electronic chaotic oscillator, and a mechanical chaotic "jerk"
model.
In both models, numerical simulations are used to demonstrate that
for
different types of model nonlinearities, and using the proper
control
parameters,
chaotic attractors are obtained with system orders as low as
2.1.
Consequently, we present a conjecture that third-order chaotic
nonlinear
systems can still produce chaotic behavior with a total system of
order
2 + eps, 1 > eps > 0, using the appropriate control
parameters. The
effect of fractional order on the chaotic range of the control
parameters
is studied. It is demonstrated that as the order is decreased,
the
chaotic range of the control parameter is affected by contraction
and
translation.
Robustness against model order reduction is demonstrated.
Ref: W. M. Ahmad and J. C. Sprott,
Chaos,
Solitons and Fractals
16, 339-351 (2003)
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