Exercise 7.2.23

Here is one way to do the numerics for part (a) of exercise 7.2.3.

soln[gamma_,v_]:=NDSolve[{theta'[t]==z[t],z'[t]==-6 Sin[theta[t]]-gamma z[t],theta[0]==0,z[0]==v},{theta,z},{t,-10,10}]

first=soln[1/4,3]

second=soln[1/4,6]

a=ParametricPlot[{theta[t],z[t]}/.first,{t,0,5},PlotStyle->Thick]

b=ParametricPlot[{theta[t],z[t]}/.second,{t,0,10},PlotStyle->Thick]

Show[a,b,PlotRange->All]

c=Plot[theta[t]/.first,{t,0,10}]

d=Plot[theta[t]/.second,{t,0,10}]

Show[c,d,PlotRange->All]

e=VectorPlot[{y,-6 Sin[x]-y/4}/Norm[{y, -6 Sin[x]-y/4}],{x,-6,10},{y,-2,9}]

Show[a,b,e,PlotRange->All]