(* regular newmark *)
A = Inverse[(({{1, γ (ω^2) Δt }, {0, 1 + β (ω^2)((Δt)^2)}}))].(({{1, (γ-1)(ω^2)Δt}, {Δt, 1+(β-(1/2))(ω^2) ((Δt)^2)}}))/.Δt->(τ/ω);
λ = FullSimplify[Eigenvalues[A], Assumptions->{τ>0, ω>0, β>0, γ>0}];
requirements = And @@ (Norm[#] <= 1& /@ λ);
Block[{γ=(1/2)}, 
RegionPlot[requirements, {β, 0, (1/2)},{τ,0, 20}, 
Axes->True,Frame->False,  AxesLabel->{Style["β", Medium], Style["τ", Medium]}]
]

(* displacement-controlled newmark *)
 λ = FullSimplify[Eigenvalues[(({{1-(1/(2 β)), -(1/(β Δt))}, {Δt-((γ Δt)/(2 β)), 1-(γ/β)}}))], Assumptions->{Δt > 0, β > 0}];
requirements = And @@ (Norm[#] <= 1& /@ λ);
RegionPlot[requirements, {β, 0, (1/2)},{γ,0, 1}, 
Axes->True,Frame->False,  AxesLabel->{Style["β", Medium], Style["γ", Medium]}]

(* velocity-controlled newmark *)
A = Inverse[(({{γ Δt, 0 }, {- β ((Δt)^2), 1}}))].(({{(γ-1) Δt, 0}, {((1/2)-β)((Δt)^2), 1}}));
λ = FullSimplify[Eigenvalues[A]]