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We investigate the dynamical phase diagram of the fractional Langevinequation and show that critical exponents mark dynamical transitions in thebehavior of the system. For a free and harmonically bound particle the criticalexponent $\alpha_c= 0.402\pm 0.002$ marks a transition to a non-monotonicunder-damped phase. The critical exponent $\alpha_{R}=0.441...$ marks atransition to a resonance phase, when an external oscillating field drives thesystem. Physically, we explain these behaviors using a cage effect, where themedium induces an elastic type of friction. Phase diagrams describing theunder-damped, the over-damped and critical frequencies of the fractionaloscillator, recently used to model single protein experiments, show behaviorsvastly different from normal.

The Critical Exponent of the Fractional Langevin Equation is $\alpha_c\approx 0.402$