Examples.nb

(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* Examples.nb - ChaosSim Example Simulations *)
(* Practical demonstrations of chaos simulation capabilities *)

(* ::Title:: *)
(*ChaosSim Examples and Demonstrations*)


(* ::Section:: *)
(*Setup*)


(* Load ChaosSim framework *)
Get[FileNameJoin[{NotebookDirectory[], "ChaosSim.nb"}]]
Get[FileNameJoin[{NotebookDirectory[], "Visualizations.nb"}]]

Print[Style["ChaosSim Examples Loaded", Bold, 16, Blue]]
Print["Run each section to see different chaos simulations in action.\n"]


(* ::Section:: *)
(*Example 1: Basic Bernoulli Chaos*)


(* ::Text:: *)
(*This example demonstrates how Bernoulli numbers create weighted chaos patterns.*)
(*The Bernoulli numbers provide non-uniform probability distributions that*)
(*generate interesting chaotic behaviors.*)


Print[Style["=== Example 1: Bernoulli Chaos Simulation ===", Bold, 14, Purple]]

(* Generate Bernoulli chaos with 500 iterations *)
bernoulliExample = SimulateBernoulliChaos[500, 12];

(* Display statistics *)
Print["Statistics:"]
Print["  Mean: ", Mean[bernoulliExample]]
Print["  Standard Deviation: ", StandardDeviation[bernoulliExample]]
Print["  Min: ", Min[bernoulliExample], " | Max: ", Max[bernoulliExample]]

(* Visualize *)
ListPlot[bernoulliExample,
  PlotStyle -> {Blue, PointSize[Small]},
  PlotLabel -> Style["Example 1: Bernoulli Chaos", Bold, 14],
  AxesLabel -> {"Iteration", "State"},
  ImageSize -> Large,
  GridLines -> Automatic
]


(* ::Section:: *)
(*Example 2: Fibonacci Golden Spiral*)


(* ::Text:: *)
(*This creates a 3D golden spiral based on Fibonacci numbers.*)
(*The spiral demonstrates the connection between Fibonacci sequences*)
(*and natural chaotic patterns found in nature.*)


Print[Style["\n=== Example 2: Fibonacci Golden Spiral ===", Bold, 14, Purple]]

(* Generate 3D Fibonacci spiral *)
fibonacciSpiralPoints = FibonacciSpiral3D[20, 100];

Print["Generated ", Length[fibonacciSpiralPoints], " points in golden spiral"]
Print["Golden Ratio ≈ ", N[GoldenRatio, 6]]

(* Visualize in 3D *)
ListPointPlot3D[fibonacciSpiralPoints,
  PlotStyle -> Directive[PointSize[Small]],
  ColorFunction -> Function[{x, y, z}, 
    ColorData["SunsetColors"][Norm[{x, y}]]
  ],
  BoxRatios -> {1, 1, 0.6},
  ImageSize -> Large,
  PlotLabel -> Style["Example 2: Fibonacci Golden Spiral", Bold, 14],
  AxesLabel -> {"X", "Y", "Z"},
  Background -> GrayLevel[0.95],
  ViewPoint -> {2, -2, 1.5}
]


(* ::Section:: *)
(*Example 3: Nash Equilibrium in Prisoner's Dilemma*)


(* ::Text:: *)
(*Classic game theory example: Finding Nash equilibrium in the Prisoner's Dilemma.*)
(*This demonstrates how game theory integrates with chaos simulation.*)


Print[Style["\n=== Example 3: Prisoner's Dilemma Nash Equilibrium ===", Bold, 14, Purple]]

(* Define Prisoner's Dilemma payoff matrices *)
(* (Cooperate, Defect) for each player *)
prisonerPayoff1 = {{-1, -3}, {0, -2}};  (* Player 1 payoffs *)
prisonerPayoff2 = {{-1, 0}, {-3, -2}};  (* Player 2 payoffs *)

Print["Player 1 Payoff Matrix (rows: P1 strategy, cols: P2 strategy):"]
Print[MatrixForm[prisonerPayoff1]]

Print["\nPlayer 2 Payoff Matrix:"]
Print[MatrixForm[prisonerPayoff2]]

(* Find Nash equilibrium *)
equilibria = FindNashEquilibrium[prisonerPayoff1, prisonerPayoff2];

Print["\nNash Equilibrium found at: ", equilibria]
Print["Interpretation: Both players choosing Defect (strategy 2, 2)"]

(* Visualize payoff matrices *)
GraphicsRow[{
  ArrayPlot[prisonerPayoff1,
    ColorFunction -> "Rainbow",
    PlotLabel -> "Player 1 Payoffs",
    FrameLabel -> {"P2 Strategy", "P1 Strategy"},
    PlotLegends -> Automatic
  ],
  ArrayPlot[prisonerPayoff2,
    ColorFunction -> "Rainbow",
    PlotLabel -> "Player 2 Payoffs",
    FrameLabel -> {"P2 Strategy", "P1 Strategy"},
    PlotLegends -> Automatic
  ]
}, ImageSize -> Large]


(* ::Section:: *)
(*Example 4: Chaotic Game Evolution*)


(* ::Text:: *)
(*Simulate a game where payoff matrices evolve chaotically over time.*)
(*This shows how Nash equilibria shift in dynamic, chaotic environments.*)


Print[Style["\n=== Example 4: Chaotic Game Evolution ===", Bold, 14, Purple]]

(* Run chaotic game simulation *)
chaosGameHistory = ChaosGameSimulation[100, 2, 0.3];

(* Extract data *)
rounds = chaosGameHistory[[All, 1]];
strategies = chaosGameHistory[[All, 2]];
equilibriaCounts = Length /@ chaosGameHistory[[All, 3]];

Print["Simulation complete: ", Length[rounds], " rounds"]
Print["Average equilibria per round: ", Mean[equilibriaCounts]]

(* Visualize strategy evolution *)
ListLinePlot[
  {strategies[[All, 1]], strategies[[All, 2]]},
  PlotStyle -> {{Blue, Thick}, {Red, Thick}},
  PlotLabel -> Style["Example 4: Strategy Evolution in Chaotic Game", Bold, 14],
  AxesLabel -> {"Round", "Strategy Choice"},
  PlotLegends -> {"Player 1", "Player 2"},
  ImageSize -> Large,
  GridLines -> Automatic
]


(* ::Section:: *)
(*Example 5: Multi-Agent Chaos System*)


(* ::Text:: *)
(*Simulate 5 agents seeking Nash equilibrium in a chaotic environment.*)
(*Agents adjust their strategies to minimize conflicts, creating emergent patterns.*)


Print[Style["\n=== Example 5: Multi-Agent Chaos Equilibrium ===", Bold, 14, Purple]]

(* Run multi-agent simulation *)
multiAgentChaos = MultiAgentChaosEquilibrium[5, 200];

(* Extract agent states over time *)
agentStates = multiAgentChaos[[All, 2]];

Print["Number of agents: 5"]
Print["Iterations: 200"]
Print["Final agent states: ", agentStates[[-1]]]
Print["State convergence: ", StandardDeviation[agentStates[[-1]]]]

(* Visualize all agents *)
ListLinePlot[
  Table[agentStates[[All, i]], {i, 1, 5}],
  PlotStyle -> Table[Directive[Thick, ColorData[97][i]], {i, 1, 5}],
  PlotLabel -> Style["Example 5: Multi-Agent State Evolution", Bold, 14],
  AxesLabel -> {"Iteration", "Agent State"},
  PlotLegends -> Table[StringTemplate["Agent ``"][i], {i, 1, 5}],
  ImageSize -> Large,
  GridLines -> Automatic
]


(* ::Section:: *)
(*Example 6: Unified Chaos Simulation*)


(* ::Text:: *)
(*Combines all three chaos types (Bernoulli, Fibonacci, Nash) into one system.*)
(*This demonstrates the full power of ChaosSim's integrated approach.*)


Print[Style["\n=== Example 6: Unified Chaos System ===", Bold, 14, Purple]]

(* Run unified simulation *)
unifiedChaos = UnifiedChaosSimulation[400];

(* Analyze correlations *)
correlations = ChaosCorrelationAnalysis[unifiedChaos];

Print["Unified chaos simulation complete"]
Print["Data points: ", Length[unifiedChaos]]
Print["\nCorrelation Analysis:"]
Print[Grid[
  Prepend[correlations, {"Component Pair", "Correlation"}],
  Frame -> All,
  Background -> {None, {LightBlue, White}}
]]

(* Visualize in 3D phase space *)
ListPointPlot3D[unifiedChaos,
  PlotStyle -> Directive[PointSize[Small]],
  ColorFunction -> Function[{x, y, z}, 
    ColorData["Rainbow"][z]
  ],
  BoxRatios -> {1, 1, 1},
  ImageSize -> Large,
  PlotLabel -> Style["Example 6: Unified Chaos Phase Space", Bold, 14],
  AxesLabel -> {"Bernoulli", "Fibonacci", "Nash"},
  Background -> Black,
  ViewPoint -> {1.5, -2.5, 1.2}
]


(* ::Section:: *)
(*Example 7: Bernoulli 3D Attractor*)


(* ::Text:: *)
(*A beautiful 3D attractor generated using Bernoulli number weights.*)
(*This creates complex, non-repeating patterns in 3D space.*)


Print[Style["\n=== Example 7: Bernoulli 3D Attractor ===", Bold, 14, Purple]]

(* Generate attractor *)
attractorPoints = BernoulliAttractor[3000, 3];

Print["Generated ", Length[attractorPoints], " points"]
Print["Attractor bounds:"]
Print["  X: [", Min[attractorPoints[[All, 1]]], ", ", Max[attractorPoints[[All, 1]]], "]"]
Print["  Y: [", Min[attractorPoints[[All, 2]]], ", ", Max[attractorPoints[[All, 2]]], "]"]
Print["  Z: [", Min[attractorPoints[[All, 3]]], ", ", Max[attractorPoints[[All, 3]]], "]"]

(* Create stunning visualization *)
ListPointPlot3D[attractorPoints,
  PlotStyle -> Directive[PointSize[Tiny]],
  ColorFunction -> Function[{x, y, z}, 
    ColorData["DarkRainbow"][Norm[{x, y, z}]/2]
  ],
  BoxRatios -> {1, 1, 1},
  ImageSize -> Large,
  PlotLabel -> Style["Example 7: Bernoulli Attractor", Bold, 14],
  AxesLabel -> {"X", "Y", "Z"},
  Background -> Black,
  Boxed -> False,
  ViewPoint -> {2, -2, 1}
]


(* ::Section:: *)
(*Example 8: Fibonacci Chaos Map*)


(* ::Text:: *)
(*Creates a chaotic map using Fibonacci ratios as the driving parameter.*)
(*Shows bifurcations and sensitivity to initial conditions.*)


Print[Style["\n=== Example 8: Fibonacci Chaos Map ===", Bold, 14, Purple]]

(* Generate Fibonacci chaos map *)
fibChaosMap = FibonacciChaosMap[800];

Print["Generated ", Length[fibChaosMap], " iterations"]
Print["Mean value: ", Mean[fibChaosMap]]
Print["Chaos entropy: ", ChaosEntropy[fibChaosMap]]

(* Create phase space plot *)
GraphicsRow[{
  ListPlot[fibChaosMap,
    PlotStyle -> {Orange, PointSize[Small]},
    PlotLabel -> "Time Series",
    AxesLabel -> {"Iteration", "Value"},
    ImageSize -> Medium
  ],
  ListPlot[Partition[fibChaosMap, 2, 1],
    PlotStyle -> {Red, PointSize[Tiny]},
    PlotLabel -> "Phase Space",
    AxesLabel -> {"x(t)", "x(t+1)"},
    ImageSize -> Medium,
    AspectRatio -> 1
  ]
}, ImageSize -> Large, 
  PlotLabel -> Style["Example 8: Fibonacci Chaos Map Analysis", Bold, 14]]


(* ::Section:: *)
(*Example 9: Comparative Chaos Analysis*)


(* ::Text:: *)
(*Compare all three types of chaos side-by-side.*)
(*Reveals unique characteristics of each approach.*)


Print[Style["\n=== Example 9: Comparative Chaos Analysis ===", Bold, 14, Purple]]

(* Generate all three types *)
iterations = 400;
bernoulliChaos = SimulateBernoulliChaos[iterations, 12];
fibonacciChaos = FibonacciChaosMap[iterations];
nashChaos = MultiAgentChaosEquilibrium[5, iterations][[All, 2, 1]];

(* Calculate statistics *)
Print["Chaos Type Statistics:"]
Print[Grid[{
  {"Type", "Mean", "Std Dev", "Entropy"},
  {"Bernoulli", Mean[bernoulliChaos], StandardDeviation[bernoulliChaos], 
   ChaosEntropy[bernoulliChaos]},
  {"Fibonacci", Mean[fibonacciChaos], StandardDeviation[fibonacciChaos], 
   ChaosEntropy[fibonacciChaos]},
  {"Nash", Mean[nashChaos], StandardDeviation[nashChaos], 
   ChaosEntropy[nashChaos]}
}, Frame -> All, Background -> {None, {LightYellow, White}}]]

(* Visualize comparison *)
ListLinePlot[
  {bernoulliChaos, fibonacciChaos, nashChaos},
  PlotStyle -> {
    {Blue, Thick, Opacity[0.7]},
    {Orange, Thick, Opacity[0.7]},
    {Green, Thick, Opacity[0.7]}
  },
  PlotLabel -> Style["Example 9: Chaos Type Comparison", Bold, 14],
  AxesLabel -> {"Iteration", "State Value"},
  PlotLegends -> {"Bernoulli", "Fibonacci", "Nash Equilibrium"},
  ImageSize -> Large,
  GridLines -> Automatic
]


(* ::Section:: *)
(*Example 10: Custom Chaos Application*)


(* ::Text:: *)
(*A practical example: Using chaos for random sampling in a market model.*)
(*Simulates price fluctuations driven by chaotic dynamics.*)


Print[Style["\n=== Example 10: Chaotic Market Simulation ===", Bold, 14, Purple]]

SimulateChaoticMarket[days_Integer, initialPrice_Real: 100.0] := Module[
  {chaos, prices, returns},
  
  (* Generate Bernoulli chaos for market volatility *)
  chaos = SimulateBernoulliChaos[days, 10];
  
  (* Convert chaos to price movements *)
  returns = (chaos - 0.5) * 0.1; (* Scale to ±5% daily returns *)
  
  (* Calculate cumulative prices *)
  prices = FoldList[
    #1 * (1 + #2) &,
    initialPrice,
    returns
  ];
  
  prices
]

(* Run market simulation *)
marketPrices = SimulateChaoticMarket[250, 100.0];

Print["Market simulation: 250 trading days"]
Print["Initial price: $100.00"]
Print["Final price: $", marketPrices[[-1]]]
Print["Max price: $", Max[marketPrices]]
Print["Min price: $", Min[marketPrices]]
Print["Total return: ", (marketPrices[[-1]] - 100.0), "%"]

(* Visualize market *)
ListLinePlot[marketPrices,
  PlotStyle -> {Darker[Green], Thick},
  PlotLabel -> Style["Example 10: Chaotic Market Prices", Bold, 14],
  AxesLabel -> {"Trading Day", "Price ($)"},
  ImageSize -> Large,
  Filling -> Axis,
  FillingStyle -> Opacity[0.2, Green],
  GridLines -> Automatic
]


(* ::Section:: *)
(*Summary*)


Print[Style["\n\n" <> StringRepeat["=", 60], Bold, Blue]]
Print[Style["ChaosSim Examples Complete!", Bold, 18, Blue]]
Print[Style[StringRepeat["=", 60] <> "\n", Bold, Blue]]

Print["You have explored:"]
Print["  ✓ Bernoulli number-based chaos generation"]
Print["  ✓ Fibonacci sequence chaotic patterns"]
Print["  ✓ Nash equilibrium in game theory"]
Print["  ✓ Multi-agent chaos systems"]
Print["  ✓ Unified chaos simulations"]
Print["  ✓ 3D attractors and visualizations"]
Print["  ✓ Practical applications (market simulation)"]

Print["\nNext steps:"]
Print["  • Modify parameters in any example to explore variations"]
Print["  • Create your own chaos combinations"]
Print["  • Use Visualizations.nb for advanced plotting"]
Print["  • Refer to README.md for detailed documentation"]

Print[Style["\nHappy chaos simulation! 🌀", Bold, 16, Purple]]