Nonlinear dynamics · flows on the circle
Flows on the circle
See one periodic differential equation four linked ways: as motion around a circle, as an angular-velocity graph, as trajectories on a spacetime tube, and as equilibrium branches traced while a parameter varies. Follow the transition between phase locking and continual rotation without losing sight of the underlying equation.
One periodic equation · four linked views
Phase portrait, velocity graph, trajectory tube, and bifurcations
Circle phase portrait
Click to release a particle
Tangential arrows show the sign and relative speed of the angular velocity. Filled equilibrium markers are stable and hollow markers are unstable. Moving gold particles follow the differential equation. Use the left and right arrow keys to choose an angle, Enter to add a particle there, Space to pause, and 0 to restart.
Angular velocity f(θ)
θ̇ = ω − a sin θ
The curve plots f of theta from zero to two pi. Its zeros align with the equilibrium markers in the other views. Positive values move counterclockwise and negative values move clockwise. Click or drag to choose an initial angle.
Trajectory tube
Drag to rotate · scroll to zoom
Time runs from left to right along a transparent tube, and each elliptical cross-section is the phase circle. Bright solid curve segments are on the near side; faint dashed segments pass behind the tube. Rotating solutions wind around it, while locked solutions approach a longitudinal equilibrium line. The right rim is the current time and matches the particles in the circle portrait. Drag to rotate the tube around its long axis and scroll while over it to zoom the circular direction; these gestures change only the view. With the plot focused, use the left and right brackets to rotate, plus and minus to zoom, and R to reset the view.
Bifurcation diagram
Equilibrium angle and the selected parameter are shown either on a plane or on a transparent cylinder. In the cylindrical view, the parameter runs along the tube and equilibrium angle wraps around its circumference. Solid green traces are stable, dashed rust traces are unstable, and gold marks nonhyperbolic critical points and the current parameter slice. Click or drag horizontally to set the parameter. Use the left and right arrow keys to adjust it, Home and End for its bounds, and Space to start or pause its animation. In the cylindrical view, drag vertically to rotate, scroll to zoom, use the left and right brackets to rotate by keyboard, plus and minus to zoom, and R to reset the view.
Mathematical notes
Equilibria and topology. An equilibrium is an angle where f(θ*) = 0. Simple equilibria around a circle alternate stable and unstable. They can therefore be created or destroyed only in pairs. With no equilibria, every trajectory keeps circling in the same direction.
Rotation and mean frequency. If f never vanishes, one turn takes T = ∫₀²π dθ / |f(θ)|, and the signed mean angular frequency is Ω = ±2π/T. Uneven speed around the circle is therefore summarized by a single long-term rotation rate.
The locking threshold. In θ̇ = ω − a sin θ, equilibria exist when |ω| ≤ |a|. At equality, a stable–unstable pair collides. Just beyond the collision the orbit rotates, but lingers near the vanished pair: the infinite-period bottleneck.
The trajectory tube. The tube is time × S¹. A solution t ↦ θ(t) is a curve on its surface. Rotation winds around the tube, while phase locking approaches a longitudinal equilibrium line. When a parameter is animated, the tube records the resulting time-dependent motion.
Rolling the bifurcation diagram. The plane plots each equilibrium angle against one chosen parameter while holding the others fixed; its top and bottom edges represent the same circle seam 0 = 2π. Rolling the plane joins those edges into a cylinder. The gold slice is the parameter value used by the other three views.
Transcritical exchange. In the transcritical example, θ* = π is fixed while θ* = π + μ moves. At μ = 0 the branches cross and exchange stability rather than being created or destroyed. With x = θ − π, the vector field begins as ẋ = ½x(μ − x) + higher-order terms, the transcritical normal form.
Pitchforks and symmetry. In the symmetric pitchfork example, ε = 0 enforces a reflection symmetry. At μ = 0, one equilibrium loses stability while two stable equilibria emerge. A nonzero ε breaks the symmetry and unfolds the pitchfork into ordinary saddle-nodes.