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Dynamic Stabilization in Forced Nonlinear Oscillators with Humped Potentials [PPT]

  The best-known 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 vertically-up 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 multi-well 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:

[1] S.-Y. Kim and K. Lee, "Multiple transitions to chaos in a parametrically forced pendulum," Phys. Rev. E 53, 1579-1586 (1996).
[2] S.-Y. Kim, S.-H. Shin, J. Yi, and C. Jang, "Bifurcations in a parametrically forced magnetic pendulum," Phys. Rev. E 56, 6613-6619 (1997).
[3] S.-Y. Kim, "Symmetry and dynamics of a magnetic oscillator," J. Korean Phys. Soc. 32, 735-738 (1998).
[4] S.-Y. Kim, "Nonlinear dynamics of a damped magnetic oscillator," J. Phys. A 32, 6727-6739 (1999).
[5] S.-Y. Kim and Y. Kim, "Dynamic stabilization in the double-well Duffing oscillator," Phys. Rev. E 61, 6517-6520 (2000).
[6] Y. Kim, S.-Y. Lee, and S.-Y. Kim, "Experimental observation of dynamic stabilization in a double-well Duffing oscillator," Phys. Lett. A 275, 254-259 (2000).
[7] S.-Y. Kim, "Nonlinear dynamics in a parametrically forced damped pendulum," Sae Mulli (New Physics) 38, S33-S37 (1998) (in Korean).
[8] W. Lim and S.-Y. Kim, "Nonlinear dynamics of a magnetic oscillator," Sae Mulli (New Physics) 39, 353-356 (1999). (in Korean).
[9] S.-Y. Lee, Y. Kim and S.-Y. Kim, "Stabilization of an unstable orbit in a double-well Duffing oscillator," Sae Mulli (New Physics) 39, 365-368 (1999) (in Korean).

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 saddle-node 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 self-similar resonance horns is a "Qualitatively Universal" Feature in The Bifurcation Structure of many driven nonlinear oscillators such as the Toda, Morse and double-well Duffing oscillators (asymmetric soft oscillators), the directly-driven pendulum (symmetric soft oscillator), and the single-well Duffing oscillator (symmetric hard oscillator).  For more details, see the following publications:

[1] J. Jeong and S.-Y. Kim, "Bifurcations in a horizontally driven pendulum," J. Korean Phys. Soc. 35, 393-398 (1999).
[2] S.-Y. Kim, "Bifurcation structure of the double-well Duffing oscillator," Int. J. Mod. Phys. B 14, 1801-1813 (2000).

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