Figure 18. Amplitude of the rocking angle *η*_{s} (blue) of steady oscillations and *η*_{t} (orange dashed) at the perturbation threshold as a function of the inclination angle *α* (*a*) or *N*_{h} (*b*). *η*_{0}=0 (not shown) is a stable branch. The (static) suitcase overturns when *η*>1/*N*_{H′} (light blue). The insets show the normalized frequency *f*_{s*} (blue) of the steady oscillations and *f*_{p*} (orange dashed) at the perturbation threshold as a function of *α* (*a*) or *N*_{h} (*b*). (*a*) Other parameters are kept constant with (*a*) *N*_{H′}=1, *N*_{h}=4, *Fr*=4, *N*_{L}=7 and *ρ*=1. At the bifurcation point: *α*≃1.282 *rad*., *η*_{0}≃0.218 *rad*., *ν*′_{0}≃0.0747 *rad*. and *f*_{*}≃0.082. At large values of *η*_{s}, before the suitcase overturns, one observes period-doubling (see [10] for a great overview on dynamical systems). There, *η*_{s} slightly exceeds the static equilibrium limit 1/*N*_{H′}. However, when *α* is smaller than 0.26, *ν*′_{0} is larger than *π*/2 and rapidly increases when *α* decreases. The initial conditions to determine the perturbation threshold are *η*=*η*_{i}, ν′=η∗˙=ν˙∗′=0. When *α* is smaller than 0.03 *rad*., *η* diverges in a couple of rocking rebounds and one cannot really define a period for oscillations. The thin black lines show the results of equations (4.17) and (4.16) for the branches of stable (full line) and unstable (dashed line) limit cycles. (*b*) Other parameters are kept constant: *α*=60^{°}, *Fr*=4, *N*_{L}=7, *N*_{H′}=0.75. The restitution coefficient *e*_{η} tends to *ρ* when *N*_{h} increases. Opaque lines show the results of simulations when *ρ*=1. See-through branches show the results of simulations when *ρ*=0.98. At the bifurcation point: *N*_{h}≃1.59, *η*_{0}≃0.677 *rad*., *ν*′_{0}≃0.452 *rad*. and *f*_{*}≃0.089 when *ρ*=1 and *N*_{h}≃6.53, *η*_{0}≃0.0835 *rad*., *ν*′_{0}≃0.0557 *rad*. and *f*_{*}≃ 0.0904 when *ρ*=0.98. The thin black lines show the results of equations (4.17) and (4.16) for the branches of stable (full line) and unstable (dashed line) limit cycles when *ρ*=1. As in the simulations, the branches of stable and unstable limit cycles join back in a second saddle-node bifurcation for the balance equations (4.17) and (4.16) when *ρ* is smaller than one. When *ρ* decreases the range of *N*_{h} values to observe the instability decreases, and for the considered set of parameter, the range has already vanished when *ρ*=0.98. (Online version in colour.)