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The following image contains solution trajectories of the system:

$$\vec x_{k+1}= \begin{bmatrix} 0.80 & 0\\ 0 & 0.64 \end{bmatrix}\,\vec x_k $$ I've managed to do this:

Clear[A, pts1, pts2, pts]
A = {{.80, 0}, {0, .64}}; 
pts1 = Table[NestList[A.# &, {k, 3}, 20], {k, -3, 3}];
pts2 = Table[NestList[A.# &, {k, -3}, 20], {k, -3, 3}];
pts = Flatten[Join[pts1, pts2], 1];
ListPlot[pts,
 PlotStyle -> PointSize[Medium],
 PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}},
 AspectRatio -> Automatic]

Which gives this image:

enter image description here

What I would like to learn how to do is to include a smooth curve for each trajectory, points of the trajectory lying on the curve, and a little arrow on each curve pointing in the direction of the origin.

I'd appreciate any help.

Thanks.

Update: Thanks to some nice help from cyrille.piatecki and Sumit, I was able to produce this:

Clear[A, pts1, pts2, pts]
A = {{.80, 0}, {0, .64}};
pts1 = Table[NestList[A.# &, {k, 3}, 10], {k, -3, 3}];
pts2 = Table[NestList[A.# &, {k, -3}, 10], {k, -3, 3}];
pts = Join[pts1, pts2];
Show[
 ListLinePlot[pts,
   InterpolationOrder -> 2,
   AxesLabel -> {"\!\(\*SubscriptBox[\(x\), \(1\)]\)", 
     "\!\(\*SubscriptBox[\(x\), \(2\)]\)"},
   PlotRange -> {{-3.3, 3.3}, {-3.1, 3.1}},
   AspectRatio -> Automatic,
   Epilog -> {
     Text[
      Style["\!\(\*SubscriptBox[\(x\), \(0\)]\)", 12, Black], {3, 
       3}, {-1, 1}],
     Text[
      Style["\!\(\*SubscriptBox[\(x\), \(1\)]\)", 12, Black], {2.4, 
       1.92}, {-1, 1}],
     Text[
      Style["\!\(\*SubscriptBox[\(x\), \(2\)]\)", 12, Black], {1.92, 
       1.23}, {-1, 1}],
     Text[
      Style["\!\(\*SubscriptBox[\(x\), \(0\)]\)", 12, Black], {-3, 
       3}, {1, 1}],
     Text[
      Style["\!\(\*SubscriptBox[\(x\), \(1\)]\)", 12, Black], {-2.4, 
       1.92}, {1, 1}],
     Text[
      Style["\!\(\*SubscriptBox[\(x\), \(2\)]\)", 12, Black], {-1.92, 
       1.23}, {1, 1}]
     }] /. Line[x_] -> {Arrowheads[{0., 0.04, 0.}], Arrow[x]},
 ListPlot[pts,
  PlotStyle -> PointSize[0.015]]
 ]

Which produced this image:

enter image description here

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  • 1
    $\begingroup$ What is the initial conditions of the system? i.e. what is x at time zero? $\endgroup$ – Nasser Jul 9 '16 at 7:03
  • $\begingroup$ @nasser I believe so. This is from Lay's Linear Algebra and its applications book. I started 7 trajectories at the top edge and 7 I started from the bottom edge. $\endgroup$ – David Jul 9 '16 at 16:43
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Avoid the Flatten. It will preserve the distinct lines.

pts = Join[pts1, pts2];
ListLinePlot[Evaluate@pts] /. Line[x__] :> {Arrowheads[{0., 0.05, 0.}], Arrow[x]}

enter image description here

For smootness you can use InterpolationOrder with ListLinePlot.

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6
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I hope this

Clear[A, pts1, pts2, pts]
A = {{.80, 0}, {0, .64}};
pts1 = Table[NestList[A.# &, {k, 3}, 20], {k, -3, 3}];
pts2 = Table[NestList[A.# &, {k, -3}, 20], {k, -3, 3}];
p1 = ListLinePlot[pts1, PlotStyle -> PointSize[Medium], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
AspectRatio -> Automatic] /. 
Line[x_] :> {Arrowheads[{0, 0.04, 0}], Arrow[x]};
p2 = ListLinePlot[pts2, PlotStyle -> PointSize[Medium], 
PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, 
AspectRatio -> Automatic] /. 
Line[x_] :> {Arrowheads[{0, 0.04, 0}], Arrow[x]};
Show[p1, p2]

Which gives this

enter image description here

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