Function Repository Resource:

IteratedMap2D

Obtain the orbit of a 2D iterated map from given initial conditions

Contributed by: E. Chan-López, Erika Yuridia Flores Chan, Héctor Argote Morales & Víctor Castellanos

ResourceFunction["IteratedMap2D"][map,icv,depvar,trange,stepsize]

gives the orbit corresponding to the recurrence equations represented by map with initial conditions icv, dependent variables depvars, an iterator specification trange and a stepsize.

Details

The map must be a list of recurrence equations.

The icv must be a list of initial conditions.

The depvars must be a list containing the dependent variables.

The trange must be a list containing the independent variable and a matrix.

The stepsize corresponds to the steps taken to reach the maximum number of iterations.

ResourceFunction["IteratedMap2D"][map,icv,depvars,trange,stepsize,"TimeSeriesData"] can be used to obtain the time series data from orbit data.

Examples

Basic Examples (6)

Define the Gingerbreadman map:

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$Gingerbreadman[{x_[n_], y_[n_]}] := {x[n + 1] == 1 - y[n] + Abs[x[n]], y[n + 1] == x[n]}; X = {x[n], y[n]}$

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Initial condition:

In[3]:=

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Interval of the maximum number of iterations:

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Step size of iterations for initial condition X₀:

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Generate the orbit of the Gingerbreadman map:

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Visualize the orbit:

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Scope (2)

Define the following dynamical system:

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$eqns[{x_[n_], y_[n_]}] := {x[n + 1] == (4/5)*x[n] + y[n], y[n + 1] == 41/50 - x[n]^2}; X = {x[n], y[n]}$

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Initial condition:

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Interval of the maximum number of iterations:

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Step size of iterations for initial condition X₀:

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Orbit data:

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Visualize the orbit:

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Use IteratedMap2D for several initial conditions with the map:

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$eqns[{x_[k_], y_[k_]}] := With[{a = 1/13, b = 1/17, c = 1/23}, {x[k + 1] == y[k] - Sign[a*x[k]]*Sqrt[Abs[a b c x[k]*y[k] + c x[k]]], y[k + 1] == a/13 - x[k]}]; X = {x[k], y[k]}$

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Initial conditions:

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X0 = {{0.17, 0.17}, {0.099, 0.044}, {0.07, 0.14}, {-0.14, 0.067}, {-0.117, 0.039}, {-0.134, -0.029}, {-0.0705, -0.0298}, {-0.036, 0.00968}, {-0.0186, -0.0135}, {0.053, 0.0329}, {0.0237, -0.024}, {0.088, 0.091}, {0.1058, 0.193}, {0.0802, 0.143}, {-0.0294, -0.0182}, {0.15969, 0.1061}, {0.0856, 0.1671}, {-0.0799, -0.0288}, {-0.00859, -0.0052}, {0.1112, -0.035}, {0.214889, 0.0757}}

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Interval of the maximum number of iterations for each initial condition:

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Step size of iterations for each initial condition:

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Orbits data:

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Visualize the orbits:

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Applications (28)

Neimark-Sacker bifurcation (13)

Use IteratedMap2D considering three initial conditions with the following two-dimensional discrete-time dynamical system :

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Jacobian matrix:

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$J[{x_, y_}][r_] := Evaluate[D[vfield, {vars}]] MatrixForm@J[vars][r]$

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The non-trivial fixed point:

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The Neimark-Sacker critical bifurcation value:

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The transversality condition:

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The non-trivial fixed point is unstable for 𝓇>𝓇₀ and locally stable for 𝓇<r₀, with the stable closed invariant curve bifurcating from for 𝓇>𝓇₀. Define the system for 𝓇<r₀ and 𝓇>𝓇₀:

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$seqns[{x_[n_], y_[n_]}] := With[{r = 1.98}, {x[n + 1] == r x[n] (1 - y[n]), y[n + 1] == x[n]}]; ueqns[{x_[n_], y_[n_]}] := With[{r = 2.012}, {x[n + 1] == r x[n] (1 - y[n]), y[n + 1] == x[n]}]; X = {x[n], y[n]}$

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Initial conditions for 𝓇<r₀ and 𝓇>𝓇₀:

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Intervals of the maximum number of iterations for 𝓇<r₀ and 𝓇>𝓇₀:

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Step size of iterations for each initial condition when 𝓇<r₀ and 𝓇>𝓇₀:

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Orbits data for 𝓇<r₀:

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Orbits data for 𝓇>r₀:

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Visualize the phase portrait graphics for the two conditions:

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${Graphics[ MapIndexed[{PointSize[0.006], ColorData["Rainbow"][#2[[1]]/Length[X0s]], Point[#1]} &, sdata], {Frame -> True, FrameTicksStyle -> GrayLevel[0], FrameLabel -> { TraditionalForm[r < Subscript[r, 0]], None}, Axes -> False, AspectRatio -> GoldenRatio^(-1)}], Graphics[ MapIndexed[{PointSize[0.006], ColorData["Rainbow"][#2[[1]]/Length[X0u]], Point[#1]} &, udata], {Frame -> True, FrameTicksStyle -> GrayLevel[0], FrameLabel -> { TraditionalForm[r > Subscript[r, 0]], None}, Axes -> False, AspectRatio -> GoldenRatio^(-1)}]} // GraphicsRow$

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Generate time series data for the discrete-time dynamical system:

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GraphicsGrid[
Partition[
Flatten[MapIndexed[{ListPlot[#1, PlotStyle -> {PointSize[0.01], ColorData["Rainbow"][#2[[1]]/Length[X0u]]}]} &, %]], 3]]

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Multiple Attractors in a Predator-Prey Model (5)

Define a predator-prey model using the Crowley-Martin functional response and initial conditions:

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$eqns[{x_[n_], y_[n_]}] := With[{a = 24647595988461150985968273113/665860725091715916800000000, b = 247579691107/12800000000, c = 6400000000/80393230369, d = 323537/40000, k = 2828296/1090611, r = 1414148/323537, \[Gamma] = 323537/60000}, {x[n + 1] == r x[n] (1 - x[n]/k) - ( a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])), y[n + 1] == (\[Gamma] a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])) - d y[n]}]; X0 = {{0.995, 0.995}, {0.995, 0.995}, {0.77, 0.82}, {0.9, 0.77}}$

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Interval of the maximum number of iterations for each initial condition:

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Step size of iterations for each initial condition:

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Orbits data:

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Visualize the orbits:

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A chaotic attractor:

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$eqns[{x_[n_], y_[n_]}] := With[{a = 17275862227/622054400, b = 6883/512, c = 256/2209, d = 55/8, k = 488/189, r = 244/55, \[Gamma] = 55/12}, {x[n + 1] == r x[n] (1 - x[n]/k) - ( a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])), y[n + 1] == (\[Gamma] a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])) - d y[n]}]; X = {x[n], y[n]}$

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Interval of the maximum number of iterations:

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Step size of iterations for the initial condition X₀:

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Orbit data:

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Visualize the orbit:

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First strange attractor:

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$eqns[{x_[n_], y_[n_]}] := With[{a = 4592791259697/181297792000, b = 152923/12800, c = 6400/48841, d = 261/40, k = 776/301, r = 388/87, \[Gamma] = 87/20}, {x[n + 1] == r x[n] (1 - x[n]/k) - (a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])), y[n + 1] == (\[Gamma] a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])) - d y[n]}]$

Orbit data:

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Visualize the orbit:

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Second strange attractor:

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$eqns[{x_[n_], y_[n_]}] := With[{a = 8592991153063/334815411200, b = 155587/12800, c = 6400/49729, d = 263/40, k = 2344/909, r = 1172/263, \[Gamma] = 263/60}, {x[n + 1] == r x[n] (1 - x[n]/k) - (a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])), y[n + 1] == (\[Gamma] a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])) - d y[n]}]$

Orbit data:

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Visualize the orbit:

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Third strange attractor with multiple periodic orbits:

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$eqns[{x_[n_], y_[n_]}] := With[{a = 24686881771241562225059473113/665860725091715916800000000, b = 247579691107/12800000000, c = 6400000000/80393230369, d = 323537/40000, k = 2828296/1090611, r = 1414148/323537, \[Gamma] = 323537/60000}, {x[n + 1] == r x[n] (1 - x[n]/k) - ( a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])), y[n + 1] == (\[Gamma] a x[n]*y[n])/((1 + b x[n]) (1 + c y[n])) - d y[n]}]; X = {x[n], y[n]}$

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Initial condition:

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Orbit data:

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Visualize the orbit:

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Zoom into an "interesting" region:

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Standard Map (5)

The standard map is defined by the equation 𝓍_n+1=𝓍_n+ε sin(𝓍_n) , 𝓎_n+1=𝓎_n+𝓍_n+1 where 𝓍_n and 𝓎_n are taken modulo 2π:

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$StandardMap[{x_[n_], y_[n_]}][\[CurlyEpsilon]_] := {x[n + 1] == Mod[x[n] + y[n] + \[CurlyEpsilon] Sin[x[n]], 2 \[Pi]], y[n + 1] == Mod[y[n] + \[CurlyEpsilon] Sin[x[n]], 2 \[Pi]]}; X0 = {{0.0952, 0.2801}, {2.9067, 4.5882}, {0.6671, 5.8718}, {0.8733, 4.5413}, {0.9015, 1.03502}, {5.8875, 5.6839}, {5.3908, 0.2054}, {1.48266, 0.1584}, {1.6982, 0.1427}, {1.9887, 0.0645}, {2.3261, 0.0801}, {2.6635, 0.0958}, {3.0009, 0.0801}, {5.9812, 5.6842}, {0.1986, 4.0092}, {0.1612, 2.0112}, {1.4451, 3.1748}, {0.1237, 1.646}, {0.4704, 1.6512}, {0.0581, 2.5017}, {1.8856, 2.6791}, {2.148, 2.7313}, {2.551, 2.893}, {0.0581, 4.448}, {0.3673, 4.3436}, {5.3346, 4.6776}, {1.642, 4.8237}, {1.164, 5.8307}, {1.2296, 5.4759}, {0.208, 1.7242}, {0.3673, 2.2147}, {2.9166, 3.0809}, {0.6766, 1.6303}};$

Interval of the maximum number of iterations for each initial condition:

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Step size of iterations for each initial condition:

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Orbits data:

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Visualize the orbits:

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Gumowski-Mira Attractor (5)

The Gumowski-Mira map is defined by 𝓍_n+1=𝒷 𝓎_n+𝒻(𝓍_n) , 𝓎_n+1=𝒻(𝓍_n+1)-𝓍_n, where :

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$f[x_[n_]][a_] := a x[n] + (2 (1 - a) x[n]^2)/(1 + x[n]^2)^2; GMAttractor[{x_[n_], y_[n_]}][{a_, b_}] := {x[n + 1] == b y[n] + a x[n] + (2 (1 - a) x[n]^2)/(1 + x[n]^2)^2, y[n + 1] == ReplaceAll[ x[n] -> b y[n] + a x[n] + (2 (1 - a) x[n]^2)/(1 + x[n]^2)^2][ f[x[n]][a]] - x[n]}; X0 = CompressedData[" 1:eJwBwQE+/iFib1JlAgAAABsAAAACAAAAmpmZmZmZ2T+amZmZmZnpP3im89iz m+Q/pJU/eXFZ8D8A0nv1WufkPzbXUirsFPM/aMFoEnKH5j/z+xHTiVTxP6Bd al/YS9G/hXUPYVefxj9gSs8tzSLJvxRij9pU8rc/uO105ZUw1z/qfuPiMW68 P8CI2ReBAsI/1DYRTok4sT/AQggq7N6ZP7wI/P6Mv6I/gO5KipLBqD/eWnqY XdO2PyAOF/XqkuM/Gmfl3GhhxD9wPESZBozUv6A6cfTdIek/BE+hRKfV8z9t fPbEm2npP97QD+C9EvU/VFGZd+Ph6D+2NLWPaS/3P4gIkRSQbug/8InNFKbF 5z/U57zzY4H5P0R5mB7VtgBAonow32bU+D90IQtGnBfyvxjyAfT56PI/WP8E YwqD4z9k3ou/CqsBQI5FyTBZpgRAfUZjNnez+7+K2HL/SE8FQKih69vvk/q/ EFXx6uZj7z9Mg/MnlfkCwEJI7EwxPvI/8pVS2/rTA8BesHjJA27zPw2ppQJV CwXAQqcGdOBe9T+CS379GG4GwKrpYsPETv0/gjM36v2cBcDCSx2XAvf/vzRg qa74owJAZtDkwQ== "]$

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Interval of the maximum number of iterations for each initial condition:

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Step size of iterations for each initial condition:

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Orbits data:

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Visualize the orbits:

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Properties and Relations (4)

The map exhibit a Neimark-Sacker bifurcation for 𝓇>𝓇₀ at . IteratedMap2D can be used with the resource function PlotGrid to exhibit in a clear fashion a simple bifurcation diagram for the above map:

Orbits data for 𝓇<r₀:

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Orbits data for 𝓇>r₀:

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Visualize the bifurcation diagram:

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$ResourceFunction[ "PlotGrid"][{{Graphics[ MapIndexed[{PointSize[0.006], ColorData["Rainbow"][#2[[1]]/Length[X1s]], Point[#1]} &, sdata1], {Frame -> True, FrameTicksStyle -> GrayLevel[0], FrameLabel -> { TraditionalForm[r < Subscript[r, 0]], None}, Axes -> False, AspectRatio -> GoldenRatio^(-1)}], Graphics[ MapIndexed[{PointSize[0.006], ColorData["Rainbow"][#2[[1]]/Length[X1u]], Point[#1]} &, udata1], {Frame -> True, FrameTicksStyle -> GrayLevel[0], FrameLabel -> { TraditionalForm[r > Subscript[r, 0]], None}, Axes -> False, AspectRatio -> GoldenRatio^(-1)}]}}, ImageSize -> 700]$

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Publisher

Ramón Eduardo Chan López

Version History

1.0.0 – 16 September 2022

Source Metadata

Citation:
- Alligood, K.T., Sauer, T.D., Yorke, J.A. (1997). Chaos in Two-Dimensional Maps. In: Chaos. Textbooks in Mathematical Sciences. Springer, Berlin, Heidelberg. DOI: 10.1007/978-3-642-59281-2_5
- Schuster, H. G., & Just, W. (2006). Deterministic chaos: an introduction. John Wiley & Sons. DOI:10.1002/3527604804
- Moon, F. C. (2008). Chaotic and fractal dynamics: introduction for applied scientists and engineers. John Wiley & Sons. DOI:10.1002/9783527617500
- Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory (Vol. 112). Springer Science & Business Media.
- Kuznetsov, Y. A., & Meijer, H. G. (2019). Numerical Bifurcation Analysis of Maps (Vol. 34). Cambridge University Press. DOI: 10.1017/9781108585804
- Govaerts, W. J. (2000). Numerical methods for bifurcations of dynamical equilibria. Society for Industrial and Applied Mathematics.
- Jury, E., & Gutman, S. (1975). On the stability of the A matrix inside the unit circle. IEEE Transactions on Automatic Control, 20(4), 533-535. DOI: 10.1109/TAC.1975.1100995
- Skalski, G. T., & Gilliam, J. F. (2001). Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology, 82(11), 3083-3092. DOI: 10.2307/2679836
- Parshad, R. D., Basheer, A., Jana, D., & Tripathi, J. P. (2017). Do prey handling predators really matter: Subtle effects of a Crowley-Martin functional response. Chaos, Solitons & Fractals, 103, 410-421. DOI: 10.1016/j.chaos.2017.06.027

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

IteratedMap2D

Details

Examples

Basic Examples (6)

Scope (2)

Applications (28)

Neimark-Sacker bifurcation (13)

Multiple Attractors in a Predator-Prey Model (5)

Standard Map (5)

Gumowski-Mira Attractor (5)

Properties and Relations (4)

Publisher

Related Links

Version History

Source Metadata

Related Resources

License Information

IteratedMap2D

Details

Examples

Basic Examples (6)

Scope (2)

Applications (28)

Neimark-Sacker bifurcation (13)

Multiple Attractors in a Predator-Prey Model (5)

Standard Map (5)

Gumowski-Mira Attractor (5)

Properties and Relations (4)

Publisher

Related Links

Version History

Source Metadata

Related Resources

Related Symbols

License Information