
Institute for Computational Neuroscience 
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Dynamic Stabilization in Forced Nonlinear Oscillators with Humped Potentials [PPT]
The bestknown example for the
Dynamic Stabilization
of an unstable periodic orbit is the
inverted pendulum. In the parametrically forced pendulum with a vertically
oscillating suspension point, the inverted state, corresponding to the
verticallyup configuration, becomes stabilized when the amplitude of the
vertical oscillation passes through a threshold value. This dynamic
stabilization phenomenon occurs generically in many other oscillators with
humped potentials such as the directly forced pendulums and forced multiwell
potential oscillators. We investigate the
Bifurcation Mechanism for The Dynamic
Stabilization
in many forced oscillators, and find that dynamic
stabilization of the unstable orbits, arising from the unstable equilibrium
points of the potential, occur through reverse subcritical or supercritical
pitchfork bifurcations. These findings are also examined in the real
experiments. Thus we suppose that that is a universal route to dynamic
stabilization. For
more details, see the following publications: ¡¡ Qualitative Universality in The Bifurcation Structure
We
investigate the global bifurcation structure associated with resonances of a
stable periodic orbit arising from the stable equilibrium point of a potential
in the forced nonlinear oscillator by varying the driving amplitude $A$ and
frequency $\omega$. For the primary and superharmonic resonances, the
corresponding saddlenode bifurcation curves form "horns," leaning to the lower
(higher) frequencies for the soft and hard oscillators, respectively. It is
found that with $\omega$ decreasing, resonance horns with successively
increasing torsion numbers repeat in a similar shape in the parameter plane. It
is thus supposed that recurrence of selfsimilar resonance horns is a
"Qualitatively Universal" Feature in The Bifurcation Structure
of many driven nonlinear oscillators
such as the Toda, Morse and doublewell Duffing oscillators (asymmetric soft
oscillators), the directlydriven pendulum (symmetric soft oscillator), and the
singlewell Duffing oscillator (symmetric hard oscillator). For
more details, see the following publications: ¡¡ 
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