Here I implemented basic functions which are well known such as chaotic

In examples directory contains comprehensive example scripts demonstrating the analysis capabilities of DynamicalModels.jl.

Each example can be run independently:

julia --project=.. examples/lorenz_analysis.jl
julia --project=.. examples/rossler_analysis.jl
julia --project=.. examples/vanderpol_analysis.jl
julia --project=.. examples/hodgkin_huxley_analysis.jl

Or from within Julia:

using Pkg
Pkg.activate("..")  # Activate the DynamicalModels environment
include("examples/lorenz_analysis.jl")

Comprehensive analysis of the Lorenz attractor including:

• Lyapunov Exponents: Calculation of largest exponent and full spectrum
• Poincaré Sections: Visualization of the strange attractor
• Fractal Dimensions: Kaplan-Yorke dimension estimation
• Butterfly Effect: Sensitivity to initial conditions

Key Concepts:

• Chaotic dynamics
• Strange attractors
• Dissipative systems

Analysis of the Rössler attractor focusing on:

• Poincaré Sections: Classic bun-shaped attractor
• Return Maps: One-dimensional maps from continuous flow
• Period-Doubling: Route to chaos as parameters vary
• Comparison with Lorenz: Simpler chaos example

Key Concepts:

• Period-doubling bifurcations
• Return maps
• Routes to chaos

Analysis of self-sustained oscillations:

• Limit Cycles: Stable periodic orbits
• Damping Effects: How ϵ parameter affects oscillations
• Forced Oscillations: Response to periodic driving
• Applications: Electronic and biological oscillators

Key Concepts:

• Limit cycles
• Self-sustained oscillations
• Relaxation oscillations

Neural excitability and action potentials:

• Resting State: Equilibrium analysis
• Excitability: Threshold behavior
• Action Potentials: Spike generation mechanism
• F-I Curves: Firing rate vs current
• Phase Space: 4D dynamical system

Key Concepts:

• Excitable systems
• All-or-none response
• Neural computation
• Nobel Prize winning model

All examples demonstrate the following analysis techniques:

λ_max = lyapunov_exponent(model, x0, time_step)
λs = lyapunov_spectrum(model, x0, time_step)

section = poincare_section(model, x0, plane_normal, plane_point, t_max)
x_coords, y_coords = poincare_map_2d(model, x0, (1, 2), 
                                      plane_normal, plane_point, t_max)

D_KY = kaplan_yorke_dimension(lyapunov_exponents)
radii, C, dim = correlation_dimension(trajectory_points)

• Largest Lyapunov exponent: ~0.9
• Kaplan-Yorke dimension: ~2.06
• Behavior: Chaotic

• Largest Lyapunov exponent: ~0.07
• Behavior: Chaotic
• Clear period-doubling route

• Largest Lyapunov exponent: ~0 (periodic)
• Behavior: Stable limit cycle

• Behavior: Excitable (all-or-none)
• Action potential duration: ~2-3 ms
• Refractory period: ~2-5 ms

1. Vary Parameters: Try different parameter values to see how behavior changes
2. Initial Conditions: Experiment with different starting points
3. Integration Time: Longer integration reveals more structure
4. Convergence: Increase iterations for more accurate Lyapunov exponents

While these examples don't generate plots (to avoid dependencies), you can visualize results using:

using Plots

# Poincaré section
scatter(x_coords, y_coords, label="Poincaré Section")

# Return map
scatter(x_n, x_n1, label="Return Map")

# Lyapunov spectrum
bar(1:length(λs), λs, label="Lyapunov Exponents")

• See the documentation for detailed model descriptions
• Check the analysis tools documentation for method details
• Refer to test files in ../test/analysis/ for more examples

Feel free to add more example scripts demonstrating:

• Other models in the package
• Advanced analysis techniques
• Comparison studies
• Real-world applications