# Plotting an image of a discrete dynamical system

I am trying to plot a discrete dynamical system of the form $$\vec{x}_{k+1} = A \vec{x}_k$$ where $$A$$ is a $$2\times 2$$ matrix in the form $$\begin{pmatrix}a&b\\c&d\end{pmatrix}$$ where $$a$$, $$b$$ and $$c$$ are real numbers. It has an initial value in the form $$\begin{pmatrix}e \\f\end{pmatrix}$$

I would like to create a plot similar to that in: Creating an image of a discrete dynamical system But am at a loss to get the function plotted as I have tried both VectorPlot and ListPlot with little success. Any advice would be much appreciated :-)

The exact problem I am working on is: \begin{align*} &\vec{x}_k = \begin{pmatrix}2ba-a-b&ba-a-b\\2(a+b-ab)&2(a+b)-ab\end{pmatrix}\vec{x},&\vec{x}_0 = \begin{pmatrix}2\\1/3\end{pmatrix}. \end{align*} I am looking at the plots created by different values for $$a$$ and $$b$$ such as $$1$$ and $$1/2$$.

I tried the following:

a = 1; b = 1/2;
A = {{2*b*a-a-b,b*a-a-b},{2(a+b-a*b),2(a+b)-ab}};
x0 = {1, 1/3};
pts = NestList[A.# &, x0, 15];
ListPlot[pts, Joined -> True, AspectRatio -> Automatic]


• More details please! Give us values for a,b,c,d,e,f. Show us the attempts you’ve made with VectorPlot and ListPlot. – MarcoB Jan 1 at 18:52
• You'd do well to look up MatrixPower[]. – J. M.'s torpor Jan 3 at 14:48

Use the sliders to modify matrix entries. Click and drag locators (small disks) to modify initial points; ALT+Click to add/remove locators.

Manipulate[ListLinePlot[Transpose @ NestList[#.{{a, b}, {c, d}} &, pt, 100],
PlotStyle -> PointSize[Medium], PlotRange -> 5 {{-1, 1}, {-1, 1}},
BaseStyle -> Arrowheads[{0., .05, 0.}], AspectRatio -> Automatic,
PlotLegends -> Placed[LineLegend[Defer /@ pt, LegendLabel -> "{x0,y0}",
LegendFunction -> Panel], Right],
Epilog -> {AbsolutePointSize[10],
{ColorData[97]@#, Point@pt[[#]]} & /@ Range[Length[pt]]},
ImageSize -> 400, Frame -> True] /. Line -> Arrow,
Spacer[10], Spacer[10], Spacer[10],
Grid[{{Item[Labeled[Control@{{a, .8, Style["a", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[a], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][a])],
Item[Labeled[Control@{{b, .0, Style["b", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[b], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][b])]},
{Item[Labeled[Control@{{c, .0, Style["c", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[c], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][c])],
Item[Labeled[Control@{{d, .4, Style["d", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[d], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][d])]}},
Alignment -> {Center, Center}, ItemSize -> {15, 15}, Dividers -> All],
{{pt, 3 {{1, 1}, {-1, 1}, {1, -1}}}, Locator,
Appearance -> None, LocatorAutoCreate -> {1, 10}},
Alignment -> Center, ControlPlacement -> Left]


An alternative implementation using Graphics:

Manipulate[Legended[Graphics[{AbsolutePointSize[10], ColorData[97]@#,
Arrow[Partition[NestList[{{a, b}, {c, d}}.# &, pt[[#]], t - 1], 2, 1]]} & /@
Range[Length[pt]],
ImageSize -> 400, Frame -> True, Axes -> True,
PlotRange -> 5 {{-1, 1}, {-1, 1}}],
Placed[LineLegend[ColorData[97] /@ Range[Length @ pt], Defer /@ pt,
LegendLabel -> "{x0,y0}", LegendFunction -> Panel], Right]],
Spacer[10], Spacer[10], Spacer[10],
Grid[{{Item[Labeled[Control @ {{a, .8, Style["a", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[a], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][a])],
Item[Labeled[Control @ {{b, .0, Style["b", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[b], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][b])]},
{Item[Labeled[Control @ {{c, .0, Style["c", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[c], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][c])],
Item[Labeled[Control @ {{d, .4, Style["d", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[d], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][d])]}},
Alignment -> {Center, Center}, ItemSize -> {16, 16}, Dividers -> All],
{{pt, 3 {{1, 1}, {-1, 1}, {1, -1}}}, Locator,
Appearance -> None, LocatorAutoCreate -> {1, 10}},
Spacer[10],
{{t, 1}, 1, 80, 1, Animator, AnimationRunning -> False, DisplayAllSteps -> True},
Alignment -> Center, ControlPlacement -> Left]


Update: Modification of the second method for the example in OP's update:

ClearAll[a, b, aA, x0] aA[a_, b_] := {{2 a b - a - b, a b - a - b}, {2 (a + b - a b), 2 ( a + b) - a b}} x0 = {1, 1/3};

Manipulate[Graphics[{AbsolutePointSize[10], ColorData[97]@1, Arrowheads[.03],
Point@x0,
Arrow[Partition[NestList[aA[a, b].# &, x0, t - 1], 2, 1]]},
AspectRatio -> 1, ImageSize -> 400, Frame -> True, Axes -> True,
PlotRange -> All], Spacer[10], Spacer[10], Spacer[10],
Grid[{{Item[Labeled[Control@{{a, 1, Style["a", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[a], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][a])],
Item[Labeled[Control@{{b, .5, Style["b", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[b], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][b])]}},
Alignment -> {Center, Center}, ItemSize -> {16, 16}, Dividers -> All],
Spacer[10],
{{t, 1}, 1, 15, 1, Animator, AnimationRunning -> False, DisplayAllSteps -> True},
Alignment -> Center, ControlPlacement -> Left]


If you want to control the starting point with a Locator:

Manipulate[Labeled[Graphics[{AbsolutePointSize[10], ColorData[97]@#,
Arrow[Partition[NestList[aA[a, b].# &, pt[[#]], t - 1], 2, 1]]} & /@
Range[Length[pt]], ImageSize -> 400, Frame -> True,
Axes -> True, PlotRange -> All, AspectRatio -> 1],
Dynamic[pt[[1]]], Top], Spacer[10], Spacer[10], Spacer[10],
Grid[{{Item[Labeled[Control@{{a, 1, Style["a", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[a], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][a])],
Item[Labeled[Control@{{b, .5, Style["b", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[b], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][b])]}},
Alignment -> {Center, Center}, ItemSize -> {16, 16}, Dividers -> All],
{{pt, {x0}}, Locator, Appearance -> None,  LocatorAutoCreate -> False},
Spacer[10],
{{t, 1}, 1, 15, 1, Animator, AnimationRunning -> False, DisplayAllSteps -> True},
Alignment -> Center, ControlPlacement -> Left]


• Very interesting code. The problem I am working on is the following \begin{align*} &\vec{x}_k = \left( \begin{array}{ccc}2ba-a-b&ba-a-b\\2(a+b-ab)&2(a+b)-ab\end{array} \right) \vec{x},&\vec{x}_0 = \left( \begin{array}{c}2\\1/3\end{array} \right). \end{align*} Where I will be inestigating the trajectories of different a and b values such as 1 and 1/2. I have tried evaluating some of these and putting them in to your code. I am just wondering how to change the x0 values? – FFerreira Jan 3 at 10:53
• @FFerreira, please see the update re modification of the second method for your example. – kglr Jan 3 at 15:39

Edit

We can change x0 by Locator and change {a,b} by Slide2D.

A[{a_, b_}] := {{2*b*a - a - b, b*a - a - b}, {2 (a + b - a*b),
2 (a + b) - a*b}};
Manipulate[
ListPlot[NestList[A[ab] . # &, x0, 15], Joined -> True,
PlotRange -> {{-10, 10}, {-10, 10}},
AspectRatio -> 1], {{ab, {1, 1/2},
Dynamic["{a,b}=" <>
ToString[ab, TraditionalForm]]}, {.8, .4}, {1.2, .6}},
Dynamic["x0=" <> ToString[x0, TraditionalForm]], {{x0, {2, 1/3}},
Locator}, ControlPlacement -> Right]


Original

A = {{Cos[π/3], -Sin[π/3] - .1}, {Sin[π/3], Cos[π/3]}};
x0 = {1, 1};
pts = NestList[A . # &, x0, 15];
ListPlot[pts, Joined -> True, AspectRatio -> Automatic]


Or

A = {{Cos[π/3], -Sin[π/3] - .1}, {Sin[π/3], Cos[π/3]}};
x0 = {1, 1};
pts = NestList[A . # &, x0, 15];
Graphics[Arrow[Partition[pts, 2, 1]]]