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Understanding Chaos Theory: Patterns and the Butterfly Effect

Explore chaos theory, the logistic map, and the Lorenz system. Learn how deterministic math creates unpredictability and the famous butterfly effect.

#chaos-theory#mathematics#butterfly-effect#lorenz-system#logistic-map#nonlinear-dynamics#science-education
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Chaos Theory

Noah Ghobrial & Jeremy Wu

MHF4U Presentation of Learning | Friday Jan 16 2026

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Why can a simple rule make behavior impossible to predict?
How does deterministic math produce apparent randomness?
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Today we will cover

■What chaos is (Definition)
◣How chaos appears (Models)
⚲How to tell a system is chaotic (Diagnostics)
🌎Why it matters (Applications)
▶Activity: Logistic map demo
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Core Ideas

Deterministic rules: outcomes with no randomness

Sensitivity to initial conditions: tiny differences grow exponentially

Nonlinearity: outputs do not scale proportionally with inputs

Strange attractors: hidden structure in long term behavior
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The Logistic Map Equation

xn+1 = r xn(1 − xn)

Interpretation: xₙ is population fraction; r is growth parameter

Progression as r increases: Stable ➝ Periodic ➝ Chaotic

Example (r=3):

Let x₀ = 0.2
x₁ = 3(0.2)(1-0.2) = 0.48
x₂ = 3(0.48)(1-0.48) = 0.7488...

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Bifurcation Diagram

As r increases, the long term values of x split into cycles and then chaos.

xn Population
Stable Doubling Chaos
r Growth Parameter
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Activity — Logistic Map (GeoGebra)

Goal: See order turn to chaos.

1. Pick r between 2.5 and 4.0
2. We will switch to GeoGebra and iterate 50 times from x₀=0.2
3. Observe behavior: converges, cycles, or chaos.
r=2.8 (Stable)
r=3.5 (Periodic)
r=3.9 (Chaotic)
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Continuous Example: Lorenz System

ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz

Parameters: σ=10, ρ=28, β=8/3

Key Point: Simple ODEs used for weather show the butterfly effect.

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How to Detect Chaos

Δx(t) ≈ Δx₀ eλt

Lyapunov Exponent (λ): Positive value means exponential divergence (sensitivity).

Fractal Dimension: Non-integer dimension implies strange attractor structure.

Poincaré Section: Slicing the 3D flow reveals a structured 2D map.

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Routes to Chaos & Limits

Routes to Chaos

• Period Doubling (Feigenbaum) • Quasiperiodicity (Torus breakdown) • Intermittency (Bursts of chaos)

Practical Limits

• Noise & Measurement Error • Finite computer precision • Prediction horizon is always limited
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Applications

• Weather Forecasting (Ensemble methods)
• Ecology (Population dynamics)
• Engineering Control (Stabilizing systems)
• Secure Communications (Chaos masking)
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Conclusion

• Chaos = Deterministic rules + Practical unpredictability.
• Small causes can have huge effects (Butterfly Effect).
• Structure (Attractors) exists within the chaos.

References

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Understanding Chaos Theory: Patterns and the Butterfly Effect

Explore chaos theory, the logistic map, and the Lorenz system. Learn how deterministic math creates unpredictability and the famous butterfly effect.

Chaos Theory

Noah Ghobrial & Jeremy Wu

MHF4U Presentation of Learning | Friday Jan 16 2026

Why can a simple rule make behavior impossible to predict?

How does deterministic math produce apparent randomness?

Today we will cover

What chaos is (Definition)

How chaos appears (Models)

How to tell a system is chaotic (Diagnostics)

Why it matters (Applications)

Activity: Logistic map demo

Core Ideas

Deterministic rules: outcomes with no randomness<br><br>Sensitivity to initial conditions: tiny differences grow exponentially<br><br>Nonlinearity: outputs do not scale proportionally with inputs<br><br>Strange attractors: hidden structure in long term behavior

The Logistic Map Equation

<i>x</i><sub><i>n</i>+1</sub> = <i>r x<sub>n</sub></i>(1 − <i>x<sub>n</sub></i>)

Interpretation: xₙ is population fraction; r is growth parameter

Progression as r increases: Stable ➝ Periodic ➝ Chaotic

Let x₀ = 0.2<br>x₁ = 3(0.2)(1-0.2) = 0.48<br>x₂ = 3(0.48)(1-0.48) = 0.7488...

Bifurcation Diagram

As r increases, the long term values of x split into cycles and then chaos.

Activity — Logistic Map (GeoGebra)

1. Pick r between 2.5 and 4.0<br>2. We will switch to GeoGebra and iterate 50 times from x₀=0.2<br>3. Observe behavior: converges, cycles, or chaos.

NOW SWITCH TO GEOGEBRA

Continuous Example: Lorenz System

<span style='font-style:italic'>ẋ</span> = σ(<span style='font-style:italic'>y</span> − <span style='font-style:italic'>x</span>)<br><span style='font-style:italic'>ẏ</span> = <span style='font-style:italic'>x</span>(ρ − <span style='font-style:italic'>z</span>) − <span style='font-style:italic'>y</span><br><span style='font-style:italic'>ż</span> = <span style='font-style:italic'>xy</span> − β<span style='font-style:italic'>z</span>

Parameters: σ=10, ρ=28, β=8/3

Key Point: Simple ODEs used for weather show the butterfly effect.

How to Detect Chaos

Lyapunov Exponent (λ): Positive value means exponential divergence (sensitivity).

Fractal Dimension: Non-integer dimension implies strange attractor structure.

Poincaré Section: Slicing the 3D flow reveals a structured 2D map.

Routes to Chaos & Limits

• Period Doubling (Feigenbaum) • Quasiperiodicity (Torus breakdown) • Intermittency (Bursts of chaos)

• Noise & Measurement Error • Finite computer precision • Prediction horizon is always limited

Applications

• Weather Forecasting (Ensemble methods)<br>• Ecology (Population dynamics)<br>• Engineering Control (Stabilizing systems)<br>• Secure Communications (Chaos masking)

TAKEAWAY: Chaos reveals structure in apparent randomness and sets fundamental limits on prediction.

Conclusion

• Chaos = Deterministic rules + Practical unpredictability.<br>• Small causes can have huge effects (Butterfly Effect).<br>• Structure (Attractors) exists within the chaos.

References