Day 27 - Hallmarks of Chaos

Conceptualizing the Lyapunov Exponent

Trajectories diverge exponentially in time

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Hallmarks of a Classically Chaotic System

1. Deterministic

2. Sensitive to Initial Conditions

3. Non-periodic Behavior

4. Strange Attractors

5. Parameter Sensitivity

6. (Sometimes) Periodic Behavior

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Limit Cycle

A limit cycle is a closed trajectory in phase space that is an attractor for a dynamical system.

The Van der Pol Oscillator exhibits a limit cycle.

\[\ddot{x} - \mu (1 - x^2) \dot{x} + x = 0\]

Random initial conditions converge to a limit cycle. Modeled with \(\mu=2\).

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The Lyapunov Exponent

\(\vec{\delta}(t)\) is the separation vector between two trajectories in phase space \(\vec{\delta}(t) = \vec{x}_2(t) - \vec{x}_1(t)\).

Do trajectories diverge exponentially in time, \(|\vec{\delta}(t)| \approx |\vec{\delta}(0)| e^{\lambda t}\)?

Each phase coordinate can change at a different rate: \(\vec{\lambda} = \langle \lambda_1, \lambda_2, \dots, \lambda_n \rangle\).

Largest \(\lambda_i > 0\)? Chaotic system.

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Strange Attractors

A strange attractor is a set of points in phase space that a chaotic system approaches.

Chen Attractor

\[\dot{x} = \alpha x-yz\]

\[\dot{y} = \beta y + xz\]

\[\dot{z} = \gamma z + xy/3\]

\(\alpha=5\), \(\beta=-10\), \(\gamma=-0.38\).

Interactive 3D Model

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Example 1: Duffing Equation

\[\ddot{x} + \beta \dot{x} + \alpha x + \gamma x^3 = F_0 \cos(\omega t)\]

Exhibits Periodic and Chaotic Behavior

Illustrates period doubling bifurcations as route to chaos

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Example 2: Lorenz System

\[\dot{x} = \sigma (y - x)\]

\[\dot{y} = x (\rho - z) - y\]

\[\dot{z} = x y - \beta z\]

Exhibits sensitive dependence on initial conditions Demonstrates the concept of a strange attractor