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Lorenz Attractor

Simulation of the famous Lorenz attractor, a chaotic dynamical system.

You can also find this example as a single file in the GitHub repository.

The Lorenz System

The system consists of three coupled nonlinear ODEs:

MATHDISPLAY0ENDMATH MATHDISPLAY1ENDMATH MATHDISPLAY2ENDMATH

Where the parameters are:

MATHINLINE3ENDMATH (Prandtl number)
MATHINLINE4ENDMATH (Rayleigh number)
MATHINLINE5ENDMATH (geometric factor)

Chaotic Behavior

For these parameters, the system exhibits sensitive dependence on initial conditions - tiny changes in starting values lead to completely different trajectories. Despite being deterministic, the system appears random and unpredictable over long timescales.

Building the System in PathSim

We'll construct the Lorenz system using basic blocks (integrators, amplifiers, multipliers, adders) to show PathSim's block-diagram approach to ODEs.

The Lorenz system is built using basic blocks like Integrator, Multiplier, and Adder to demonstrate PathSim\'s block-diagram approach.

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System Parameters and Initial Conditions

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Block Diagram

We create blocks for each equation:

For dx/dt = σ(y - x):

Compute (y - x) with an adder
Multiply by σ
Integrate to get x

For dy/dt = x(ρ - z) - y:

Compute (ρ - z)
Multiply by x
Subtract y
Integrate to get y

For dz/dt = xy - βz:

Multiply x and y
Subtract βz
Integrate to get z

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Connections

The connections wire up the differential equations according to the Lorenz system.

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Simulation Setup

We use an adaptive Runge-Kutta solver (RKBS32) with specified tolerances to handle the complex dynamics efficiently.

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12:43:49 - INFO - LOGGING (log: True)
12:43:49 - INFO - BLOCKS (total: 13, dynamic: 3, static: 10, eventful: 0)
12:43:49 - INFO - GRAPH (nodes: 13, edges: 20, alg. depth: 4, loop depth: 0, runtime: 0.141ms)
12:43:49 - INFO - STARTING -> TRANSIENT (Duration: 50.00s)
12:43:49 - INFO - --------------------   1% | 0.0s<3.8s | 1523.7 it/s
12:43:50 - INFO - ####----------------  20% | 0.4s<1.3s | 1684.8 it/s
12:43:50 - INFO - ########------------  40% | 1.0s<1.2s | 1491.0 it/s
12:43:51 - INFO - ############--------  60% | 1.7s<1.5s | 1572.8 it/s
12:43:52 - INFO - ################----  80% | 2.5s<0.6s | 1397.2 it/s
12:43:52 - INFO - #################### 100% | 3.0s<--:-- | 3011.4 it/s
12:43:52 - INFO - FINISHED -> TRANSIENT (total steps: 4669, successful: 4057, runtime: 2998.76 ms)

Results: Time Series

First, let's look at how x, y, and z evolve over time. Notice the irregular, non-repeating patterns characteristic of chaos.

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The Famous Butterfly Shape - 3D Attractor

The true beauty of the Lorenz attractor emerges in 3D phase space. The trajectory traces out a distinctive "butterfly" or "owl face" shape with two lobes. The system orbits chaotically around two unstable fixed points.

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2D Projections

We can also view 2D projections of the attractor to see its structure from different angles.

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