How is a deterministic chaotic process different from chaos as ... | Filo
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Proof of Chaos: Measuring the Lyapunov Exponent 📈 The double pendulum is a classic example of a nonlinear dynamical system exhibiting deterministic chaos. To model its motion, we use Lagrangian Mechanics, focusing on energy states rather than force vectors. 1. Generalized Coordinates The system is defined by two angles, θ₁ and θ₂, measured from the vertical. The positions of the masses (m₁, m₂) are: • Mass 1: x₁ = L₁ sin(θ₁), y₁ = -L₁ cos(θ₁) • Mass 2: x₂ = x₁ L₂ sin(θ₂), y₂ = y₁ - L₂ cos(θ₂) 2
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Chaos Theory States: -------------------- Chaos theory explores how small changes in the initial conditions of a system can lead to vastly different and unpredictable outcomes, despite being governed by deterministic laws. --------------------------------------- ➡️ Explanation: ---------------- This theory reveals that even in systems following strict rules, predicting long-term outcomes is often impossible because of their extreme sensitivity to starting points. This unpredictability is famousl
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