M. A. RAMADAN, S. M. EL-KHOLY 129
Figure 2. Numerical solutions if example 2, x1 component (a)
and x2 component (b).
We solved these two examples numerically using
th order Runge-Kutta method. The results of example 1
are drawn as shown in Figure 1 and the results of exam-
own in Figure 2. The curves are
rawn for different initial values of
d 2 demonstrate time increases
all the compents of the solutions tends to zero.
means, that te aptotically stable. W
verify the rightness of our proved theorems.
ifo ly stable as well
nd T. Furumochi, “A Note on Stability by
 J. Hale, “Ordina,” Wiley Inter-
science, John Woken, 1969.
ple 2 are drawn as sh
that, as the
Figures 1 an
he solutions arsymhich
We introduced two theorems which provide the sufficient
conditions for the boundedness of solution of the nonlin-
ear damped vectorial oscillator and the conditions for the
stability of the zero solution to be unrm
asymptotically stable. We verified our theoretical re-
sults by solving two examples satisfying the assumptions
of the two proved theorems.
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