Pendulum Motion:  θ vs (OverTilde[t])

Manipulate[
Module[{θ,t,sol},
sol = NDSolve[{((((ω/(2 π))))^2) θ''[t] + ((γ ω)/(2 π)) θ'[t] + Sin[θ[t]] == f Cos[2 π t],
θ[0] == θ0, θ'[0] == θp0},θ,{t,0,T}];
Plot[Evaluate[θ[t] /. sol],{t,0,T},
AxesLabel -> {"t","θ"},
RotateLabel -> False]
],
{{f,(9/10),"(OverTilde[f])"},0,2,Appearance -> "Labeled"},
{{γ,(1/2),"(OverTilde[γ])"},0,1,Appearance -> "Labeled"},
{{ω,(2/3),"(OverTilde[ω])"},0,2,Appearance -> "Labeled"},
{{T,6,"(Subscript[(OverTilde[t]), max])"},0,30,Appearance -> "Labeled"},
{{θ0,0,"θ(0)"},-(π/2),(π/2),Appearance -> "Labeled"},
{{θp0,0,"(OverDot[θ])(0)"},0,2,Appearance -> "Labeled"}
]

Sensitivity to Initial Conditions

Manipulate[
Module[{t,θ1,θ2,sol1,sol2,γ,ω},
γ = (1/2);
ω = (2/3);
sol1 = NDSolve[{((((ω/(2 π))))^2) θ1''[t] + ((γ ω)/(2 π)) θ1'[t] + Sin[θ1[t]] == f Cos[2 π t],
θ1[0] == θ10, θ1'[0] == 0},θ1,{t,0,T}];
sol2 = NDSolve[{((((ω/(2 π))))^2) θ2''[t] + ((γ ω)/(2 π)) θ2'[t] + Sin[θ2[t]] == f Cos[2 π t],
θ2[0] == θ20, θ2'[0] == 0},θ2,{t,0,T}];
Plot[
{
Evaluate[θ1[t] /. sol1],Evaluate[θ2[t] /. sol2]
},{t,0,T},
AxesLabel -> {"t","θ"},
PlotStyle -> {Blue,Red},
RotateLabel -> False]
],
{{f,(9/10),"(OverTilde[f])"},0,2,Appearance -> "Labeled"},
{{T,6,"(Subscript[(OverTilde[t]), max])"},0,30,Appearance -> "Labeled"},
{{θ10,0,"(Subscript[θ, 1])(0)"},-(π/2),(π/2),Appearance -> "Labeled"},
{{θ20,0,"(Subscript[θ, 2])(0)"},-(π/2),(π/2),Appearance -> "Labeled"}
]

Logarithmic Plot of |Δθ((OverTilde[t]))| vs (OverTilde[t]).

Manipulate[
Module[{t,θ1,θ2,sol1,sol2,γ,ω},
γ = (1/2);
ω = (2/3);
sol1 = NDSolve[{((((ω/(2 π))))^2) θ1''[t] + ((γ ω)/(2 π)) θ1'[t] + Sin[θ1[t]] == f Cos[2 π t],
θ1[0] == θ10, θ1'[0] == 0},θ1,{t,0,T},
MaxSteps->∞];
sol2 = NDSolve[{((((ω/(2 π))))^2) θ2''[t] + ((γ ω)/(2 π)) θ2'[t] + Sin[θ2[t]] == f Cos[2 π t],
θ2[0] == θ20, θ2'[0] == 0},θ2,{t,0,T},
MaxSteps->∞];
Plot[
Log10[Abs[Evaluate[θ1[t] /. sol1]-Evaluate[θ2[t] /. sol2]]],{t,0,T},
AxesLabel -> {"t","log \[VerticalSeparator]Δθ((OverTilde[t]))\[VerticalSeparator]"},
PlotStyle -> Red,
RotateLabel -> False]
],
{{f,(107/100),"(OverTilde[f])"},0,2,Appearance -> "Labeled"},
{{T,40,"(Subscript[(OverTilde[t]), max])"},0,60,Appearance -> "Labeled"},
{{θ10,-(π/2),"(Subscript[θ, 1])(0)"},-(π/2),(π/2),Appearance -> "Labeled"},
{{θ20,-(π/2)+ (1/10),"(Subscript[θ, 2])(0)"},-(π/2),(π/2),Appearance -> "Labeled"}
]

Phase Space Plot

Manipulate[
Module[{θ,t,sol,NumT,dt},
sol = NDSolve[{((((ω/(2 π))))^2) θ''[t] + ((γ ω)/(2 π)) θ'[t] + Sin[θ[t]] == f Cos[2 π t],
θ[0] == θ0, θ'[0] == θp0},θ,{t,T0,T},MaxSteps->∞][[1]];
NumT = (T-T0)*400;
dt = (1/400);
ListPlot[
Table[
Evaluate[{Mod[θ[T0 + dt*i],2 π,-π],θ'[T0 + dt*i]} /. sol],
{i,0,NumT}
],
AspectRatio -> 1,
PlotRange -> {{-π,π},Automatic},
AxesLabel -> {"θ","(OverDot[θ])"},
RotateLabel -> False,
Ticks->{{-π,-(π/2),0,(π/2),π},Automatic},
PlotStyle -> PointSize[Tiny]]
],
{{f,(539/500),"(OverTilde[f])"},0,2,Appearance -> "Labeled"},
{{γ,(1/2),"(OverTilde[γ])"},0,1,Appearance -> "Labeled"},
{{ω,(2/3),"(OverTilde[ω])"},0,2,Appearance -> "Labeled"},
{{T0,0,"(Subscript[(OverTilde[t]), min])"},0,50,1,Appearance -> "Labeled"},
{{T,60,"(Subscript[(OverTilde[t]), max])"},60,400,10,Appearance -> "Labeled"},
{{θ0,-(π/2),"θ(0)"},-(π/2),(π/2),Appearance -> "Labeled"},
{{θp0,0,"(OverDot[θ])(0)"},0,2,Appearance -> "Labeled"}
]

Poincaré Section Plot

Manipulate[
Module[{θ,t,sol,NumT},
sol = NDSolve[{((((ω/(2 π))))^2) θ''[t] + ((γ ω)/(2 π)) θ'[t] + Sin[θ[t]] == f Cos[2 π t],
θ[0] == θ0, θ'[0] == θp0},θ,{t,T0,T},MaxSteps->∞][[1]];
NumT = T-T0;
ListPlot[
Table[
Evaluate[{Mod[θ[T0 + i],2 π,-π],θ'[T0 + i]} /. sol],
{i,0,NumT}
],
AspectRatio -> 1,
PlotRange -> {{-π,π},{-10,30}},
AxesLabel -> {"θ","(OverDot[θ])"},
RotateLabel -> False,
Ticks->{{-π,-(π/2),0,(π/2),π},Automatic},
PlotStyle -> PointSize[Small]]
],
{{f,(1078/1000),"(OverTilde[f])"},0,2,Appearance -> "Labeled"},
{{γ,(1/2),"(OverTilde[γ])"},0,1,Appearance -> "Labeled"},
{{ω,(2/3),"(OverTilde[ω])"},0,2,Appearance -> "Labeled"},
{{T0,10,"(Subscript[(OverTilde[t]), min])"},10,40,1,Appearance -> "Labeled"},
{{T,60,"(Subscript[(OverTilde[t]), max])"},50,6000,50,Appearance -> "Labeled"},
{{θ0,-(π/2),"θ(0)"},-(π/2),(π/2),Appearance -> "Labeled"},
{{θp0,0,"(OverDot[θ])(0)"},0,2,Appearance -> "Labeled"}
]

Poincaré Section Plot - One Magnification

Module[{θ,t,sol,NumT,f,γ,ω,T0,T,θ0,θp0},
f = (3/2);
γ = (1/4);
ω = (2/3);
T0 = 10;
T = 60000;
θ0 = -(π/2);
θp0 = 0;
sol = NDSolve[{((((ω/(2 π))))^2) θ''[t] + ((γ ω)/(2 π)) θ'[t] + Sin[θ[t]] == f Cos[2 π t],
θ[0] == θ0, θ'[0] == θp0},θ,{t,T0,T},MaxSteps->∞][[1]];
NumT = T-T0;
ListPlot[
Table[
Evaluate[{Mod[θ[T0 + i],2 π,-π],θ'[T0 + i]} /. sol],
{i,0,NumT}
],
AspectRatio -> 1,
PlotRange -> {{-(1/2),(3/2)},{-7,-(7/2)}},
AxesLabel -> {"θ","(OverDot[θ])"},
RotateLabel -> False,
PlotStyle -> PointSize[Tiny]]
]

Poincaré Section Plot - One More Magnification

Module[{θ,t,sol,NumT,f,γ,ω,T0,T,θ0,θp0},
f = (3/2);
γ = (1/4);
ω = (2/3);
T0 = 10;
T = 60000;
θ0 = -(π/2);
θp0 = 0;
sol = NDSolve[{((((ω/(2 π))))^2) θ''[t] + ((γ ω)/(2 π)) θ'[t] + Sin[θ[t]] == f Cos[2 π t],
θ[0] == θ0, θ'[0] == θp0},θ,{t,T0,T},MaxSteps->∞][[1]];
NumT = T-T0;
ListPlot[
Table[
Evaluate[{Mod[θ[T0 + i],2 π,-π],θ'[T0 + i]} /. sol],
{i,0,NumT}
],
AspectRatio -> 1,
PlotRange -> {{-(1/5),(1/5)},{-5,-(7/2)}},
AxesLabel -> {"θ","(OverDot[θ])"},
RotateLabel -> False,
PlotStyle -> PointSize[Tiny]]
]