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The dynamical phase diagram of the fractional Langevin equation isinvestigated for harmonically bound particle. It is shown that criticalexponents mark dynamical transitions in the behavior of the system. Fourdifferent critical exponents are found. (i) $\alpha_c=0.402\pm 0.002$ marks atransition to a non-monotonic under-damped phase, (ii) $\alpha_R=0.441...$marks a transition to a resonance phase when an external oscillating fielddrives the system, (iii) $\alpha_{\chi_1}=0.527...$ and (iv)$\alpha_{\chi_2}=0.707...$ marks transition to a double peak phase of the"loss" when such an oscillating field present. As a physical explanation wepresent a cage effect, where the medium induces an elastic type of friction.Phase diagrams describing over-damped, under-damped regimes, motion andresonances, show behaviors different from normal.

Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors