Poincaré Section
0.00
5000
2.0
Points: 0
Phase Portrait
1000
Time Series θ(t)
3D Phase Space
System Parameters
0.50
1.15
0.667
0.10
0.0
Presets
Simulation Control
1.0x
20
Real-time Metrics
Time:
0.00
θ:
0.00
ω:
0.00
Drive Phase:
0.00
Kinetic Energy:
0.00
Potential Energy:
0.00
Theory
Equation of Motion:
θ̈ + βθ̇ + sin(θ) = γ·cos(ωt)
Physical Meaning
- θ: Pendulum angle from vertical downward
- β: Damping coefficient (friction)
- γ: Driving force amplitude
- ω: Driving force frequency
Understanding Poincaré Sections
A Poincaré section samples the system's state at regular intervals (once per drive period), creating a stroboscopic view of the dynamics. This reveals the underlying structure of the attractor:
- Single point: Period-1 motion
- Two points: Period-2 motion
- Finite set of points: Period-n motion
- Closed curve: Quasi-periodic motion
- Fractal structure: Chaotic motion
Route to Chaos
As the drive amplitude γ increases, the system undergoes a period-doubling cascade:
- Small γ: Simple periodic motion (Period-1)
- Medium γ: Period doubling (Period-2, 4, 8...)
- Large γ: Chaotic motion with fractal attractor
Observation Guide
- Start with Period-1 preset to see regular motion
- Increase γ gradually to observe period doubling
- Use Chaos preset to see fractal structure emerge
- Enable multiple initial conditions to test sensitivity
- Adjust sample phase to see different cross-sections
Applications
- Mechanical vibrations and structural engineering
- Electronic circuit oscillators
- Synchronization theory and coupled oscillators
- Climate dynamics and periodic forcing