Abstract
In this paper, a fractional 3-dimensional
(3-D) 4-wing quadratic autonomous system (Qi system)
is analyzed. Time domain approximation method
(Grunwald–Letnikov method) and frequency domain
approximationmethod are used together to analyze the
behavior of this fractional order chaotic system. It is
found that the decreasing of the system order has great
effect on the dynamics of this nonlinear system. The
fractional Qi system can exhibit chaos when the total order less than 3, although the regular one always
shows periodic orbits in the same range of parameters.
In some fractional order, the 4 wings are decayed
to a scroll using the frequency domain approximation
method which is different from the result using time
domain approximation method. A surprising finding
is that the phase diagrams display a character of local
self-similar in the 4-wing attractors of this fractional
Qi system using the frequency approximation
method even though the number and the characteristics
of equilibria are not changed. The frequency spectrums
show that there is some shrinking tendency of
the bandwidth with the falling of the system states order.
However, the change of fractional order has little
effect on the bandwidth of frequency spectrum using
the time domain approximation method. According to
the bifurcation analysis, the fractional order Qi system
attractors start from sink, then period bifurcation
to some simple periodic orbits, and chaotic attractors,
finally escape from chaotic attractor to periodic orbits
with the increasing of fractional order α in the interval
[0.8, 1]. The simulation results revealed that the
time domain approximation method is more accurate
and reliable than the frequency domain approximation
method.