Learning objectives
By the end of this lesson, you should be able to:
- Interpret a recurrence plot.
- Explain why recurrence is useful for nonlinear trajectories.
- Describe how a Poincaré section reduces continuous flow to repeated intersections.
Prerequisites: The Lorenz Attractor
EXPERIMENT Interactive dynamics laboratory
Perturb → Evolve → CompareChange one parameter at a time. Predict the qualitative behavior before running or moving the control.
Recurrence
A recurrence plot marks pairs of times whose states are close in phase space. A common definition is \(R_{ij}=\Theta(\varepsilon-\|\mathbf{x}_i-\mathbf{x}_j\|)\).
Patterns in recurrence plots
Diagonal lines indicate segments with similar temporal evolution, vertical structures can indicate laminar behavior, and changes in texture can reveal dynamical transitions.
Poincaré sections
A Poincaré section records repeated intersections of a continuous trajectory with a chosen surface. This converts a flow into a lower-dimensional return map and can expose periodic, quasiperiodic, or chaotic structure.
Key takeaways
What should stay with you
- Recurrence visualizes when a system revisits similar states.
- Poincaré sections convert a flow into a lower-dimensional return map.
- These tools reveal temporal and geometric structure that may be hidden in raw signals.
Knowledge check
What does a dark point at coordinates (i,j) in a recurrence plot typically indicate?
Recurrence plots mark pairs of states that lie within a chosen phase-space distance threshold.