Chapter 3, Section 3.14, Figure 3.32

We are going to construct schematic diagrams illustrating the various types of stability around an equilibrium point in the circular restricted problem.  First we set the coordinates of the eipcentre.

x0 = 3*Cos[Pi/3];
y0 = 2*Sin[Pi/3];

Next we define c and s for use in rotaional transformations.

c = Cos[Pi/6];
s = Sin[Pi/6];

Now we set up graphics objects for each centre and the staring point.

centre1 = Graphics[{PointSize[0.01],Point[{0,0}]}];
centre2 = Graphics[{PointSize[0.01],Point[{x0,y0}]}];
start = Graphics[{PointSize[0.03],
                  Point[{x0,y0}+ {{c ,s}, {-s, c}}.
                  {1.5*Cos[Pi/3], 0.75*Sin[Pi/3]}]}];

Now we set basic to be the basic plot of the libration path and the 2:1epicyclic path.

basic =
ParametricPlot[
 {{3*Cos[t], 2*Sin[t]},
  {x0,y0} + 
  {{c ,s}, {-s, c}}.{1.5*Cos[t], 0.75*Sin[t]}},
 {t,0,2Pi},
 AspectRatio->Automatic]

Now we derive a plot for each type of stability in turn.  First we have asymptotic stability in each of the two modes.

stab1 =
ParametricPlot[
 {{x0,y0} + 
  {{c ,s}, {-s, c}}.
   (Exp[-t/8]*{1.5*Cos[-t+Pi/3], 0.75*Sin[-t+Pi/3]}),
  Exp[-t/8]*{3*Cos[-t+Pi/3], 2*Sin[-t+Pi/3]}},
 {t,0,4Pi},
 PlotStyle->
  {{Dashing[{0.01,0.01}]},
   {Dashing[{0.01,0.01}]}},
 DisplayFunction->Identity,
 AspectRatio->Automatic]

Display the result together with the two centres and the starting point.  Note how both paths spiral inwards.

p1 =
Show[basic,stab1,centre1,centre2,start,
 PlotRange->{{-4.2,4.2},{-3.5,3.5}},
 DisplayFunction->$DisplayFunction,
 Axes->None,
 AspectRatio->Automatic]

Second we have instability in each of the two modes.

stab2 =
ParametricPlot[
 {{x0,y0} + 
  {{c ,s}, {-s, c}}.
   (Exp[+t/40]*{1.5*Cos[-t+Pi/3], 0.75*Sin[-t+Pi/3]}),
  Exp[+t/40]*{3*Cos[-t+Pi/3], 2*Sin[-t+Pi/3]}},
 {t,0,4Pi},
 PlotStyle->
  {{Dashing[{0.01,0.01}]},
   {Dashing[{0.01,0.01}]}},
 DisplayFunction->Identity,
 AspectRatio->Automatic]

Display the result together with the two centres and the starting point.  Note how both paths spiral outwards.

p2 =
Show[basic,stab2,centre1,centre2,start,
 PlotRange->{{-4.2,4.2},{-3.5,3.5}},
 DisplayFunction->$DisplayFunction,
 Axes->None,
 AspectRatio->Automatic]

Third we have instability in the epicyclic mode and asymptotic stability in the librational mode.

stab3 =
ParametricPlot[
 {{x0,y0} + 
  {{c ,s}, {-s, c}}.
   (Exp[+t/40]*{1.5*Cos[-t+Pi/3], 0.75*Sin[-t+Pi/3]}),
  Exp[-t/8]*{3*Cos[-t+Pi/3], 2*Sin[-t+Pi/3]}},
 {t,0,4Pi},
 PlotStyle->
  {{Dashing[{0.01,0.01}]},
   {Dashing[{0.01,0.01}]}},
DisplayFunction->Identity,
AspectRatio->Automatic]

Display the result together with the two centres and the starting point.  Note how the eipcyclic path spirals outwards (unstable) while the librational path spirals inwards (asymptotically stable).

p3 =
Show[basic,stab3,centre1,centre2,start,
 PlotRange->{{-4.2,4.2},{-3.5,3.5}},
 DisplayFunction->$DisplayFunction,
 Axes->None,
 AspectRatio->Automatic]

Fourth and last we have asymptotic stability in the epicyclic mode and instability in the librational mode.

stab4 =
ParametricPlot[
 {{x0,y0} + 
  {{c ,s}, {-s, c}}.
   (Exp[-t/8]*{1.5*Cos[-t+Pi/3], 0.75*Sin[-t+Pi/3]}),
  Exp[+t/40]*{3*Cos[-t+Pi/3], 2*Sin[-t+Pi/3]}},
 {t,0,4Pi},
 PlotStyle->
  {{Dashing[{0.01,0.01}]},
   {Dashing[{0.01,0.01}]}},
DisplayFunction->Identity,
AspectRatio->Automatic]

Display the result together with the two centres and the starting point.  Note how the eipcyclic path spirals inwards (asymptotically stable) while the librational path spirals outwards (unstable).

p4 =
Show[basic,stab4,centre1,centre2,start,
 PlotRange->{{-4.2,4.2},{-3.5,3.5}},
 DisplayFunction->$DisplayFunction,
 Axes->None,
 AspectRatio->Automatic]

Display all of the plots together.

Show[GraphicsArray[{p1,p3,p4,p2}]]