Symmetry breaking in a driven and strongly damped pendulum

We examine the conditions for appearance of a symmetry breaking bifurcation in damped and periodically driven pendulums in the case of strong damping. We show that symmetry breaking, unlike other nonlinear phenomena, can exist at high dissipation. We prove that symmetry breaking phases exist between phases of symmetric normal and symmetric inverted oscillations. We find that symmetry broken solutions occupy a smaller region of the pendulum’s parameter space in comparison to the statements made in earlier considerations [McDonald and Plischke, Phys. Rev. B 27, 201 (1983)]. Our research on symmetry breaking in a strongly damped pendulum is relevant to an understanding of the phenomena of dynamic symmetry breaking and rectification in pure ac driven semiconductor superlattices.

Figure 1

(Color online) First order pendulum, with $\beta =1∕2$: regions of inverted and normal mode oscillations. White areas correspond to normal oscillations with $⟨\theta ⟩=0$ and gray areas to the inverted mode $⟨\theta ⟩=\pi$. No symmetry broken solutions are found. Solid (red) lines indicate the analytic prediction for mode boundaries, Eq. (8), which are plotted for $n=1,\dots ,6$.

Figure 2

(Color online) Phase-space portrait of a typical symmetry broken solution. To highlight the asymmetry both solutions ${\theta }_1$ [solid (red)] and ${\theta }_2$ [dashed (blue)], which are related via the symmetry ${\theta }_1\left(t\right)=-{\theta }_2(t+T∕2)$, are plotted. Taking $f$ as the control parameter, the insets on the right show the orbit just before and after the symmetry breaking. This is the $n=2$ transition; therefore it is from inverted to normal mode. The fixed parameters are $\gamma =2$ and $\omega =0.4$.

Figure 3

(Color online) Second order pendulum equation with $\gamma =2$: regions of inverted and normal state oscillations. Sandwiched between them are the regions of symmetry breaking. Solid (red) lines indicate analytic prediction of Eqs. (12, 10).

Figure 4

(Color online) The amplitudes of the two first harmonic components $A_0$ and $A_1$ plotted against $f$. $A_0$ [lower curve (blue)] alternates between $0$ and $\pi$, while $A_1$ [upper curve (red)] shows nearly linear dependence on the parameter $f$. Small deviations from this are seen near the transitions between normal and inverted states, along with an $A_0$ component that is not equal to 0, $\pi$. To guide the eye, dashed lines connecting the points $A_1\left(f\right)=A^{\left(n\right)}$ on the vertical axis to the corresponding $f$ on the horizontal axis are drawn. The fixed pendulum parameters are $\gamma =3$ and $\omega =0.2$.

Figure 5

(Color online) The amplitudes of the two first harmonic components as functions of the drive amplitude $f$. (a) Low frequency and weak drive, $\gamma =3$ and $\omega =0.2$; (b) high frequency and strong drive, $\gamma =3$ and $\omega =4$. Notice the relatively large amplitude of the second harmonic component in the low-frequency case.