Nonlinear dynamics · one-dimensional flows
Bifurcations in motion
Move the parameter r through an autonomous differential equation and watch its bifurcation diagram, slope field, and trajectories change together. Start with a classical normal form or generate an analytic family, then inspect any marked bifurcation point through its local Taylor model.
One equation · three linked views
Bifurcation diagram, slope field, and phase line
Bifurcation diagram
Scroll: zoom · left-drag: set r · middle-drag: pan
The three-dimensional equilibrium surface in x, r, and alpha. The highlighted alpha plane intersects the surface in the branches shown in the bifurcation diagram. Drag to rotate, scroll or pinch to zoom, use the arrow keys to rotate, plus and minus to zoom, and 0 to reset.
Solid branches are stable equilibria. Dashed branches are unstable. Numbered diamonds mark detected bifurcation points. Moving circles on the current-parameter line mirror the same trajectories shown on the phase line below. The slope field and phase line use this diagram's visible x range. For the pitchfork unfolding, the 3D dock in this panel's heading shows the diagram as a fixed-alpha slice through the equilibrium surface. Hover or focus it to enlarge it, drag to rotate it, and scroll or pinch to zoom it. Scroll over Diagram 01 to zoom, left-click or left-drag to change r, and middle-drag to pan. With the diagram focused, plus and minus zoom, Control plus arrow keys pan, arrow keys change r, and F or 0 restores the complete view.
Slope field at the selected r
r = 0.000
Short line segments show the direction of solutions in the time-state plane. Horizontal guide lines mark equilibria. Gold dots traveling along those lines highlight the corresponding constant solutions, while the other solution curves show where several initial states move. The gold arrow beside the state axis marks the keyboard-selected initial state. With the plot focused, use arrow keys to move that arrow and press Enter or Space to add its trajectory.
Phase line and trajectories
Finding equilibria…
Arrows show the sign of f. Filled circles mark stable equilibria, open circles mark unstable equilibria, and moving gold points follow the differential equation forward in time. Stable equilibria attract nearby points; unstable equilibria repel them. With the plot focused, use arrow keys to choose an initial state and press Enter or Space to add it.
04 · Memory in a scalar flow
Hysteresis: the path depends on the past
Ready to run a parameter loop
In the bistable interval, the same parameter value supports two stable equilibria separated by one unstable equilibrium. As r increases, the tracked state stays near the lower branch and tips upward after the right fold; as r decreases, it stays near the upper branch and tips downward after the left fold. The static folds are the quasi-static thresholds. At finite sweep speed, tipping can lag beyond them; the speed control changes that lag and the recorded loop width.
Mathematical notes
Random analytic families. Each seeded family combines two or three separated analytic event blocks chosen from saddle-node, transcritical, and pitchfork forms. Positive trigonometric and logarithmic factors vary the flow speed without changing the equilibria. This generator is deliberately curated rather than statistically generic: exact transcritical and pitchfork events require special structure, but including them makes the generated diagrams more useful for comparison.
N-branch crossings and pitchfork unfoldings. The adjustable n-branch family (including n = 4) is a genuine analytic scalar differential equation with several equilibrium branches meeting at one parameter value. These deliberately constructed crossings are high-codimension events, not structurally stable one-parameter bifurcations. A small generic perturbation separates them into simpler events. The pitchfork panel shows the complete local cubic unfolding α + βx² + εrx + σx³: perfect, transcritical-plus-saddle-node, three-saddle-node, triple-root-boundary, and one-saddle-node regimes. This is locally exhaustive near an ordinary cubic pitchfork, not a classification of higher-order or global degeneracies. In that family, the 3D dock shows the equilibrium surface in (x, r, α)-space. Moving α translates the slicing plane whose intersection is Diagram 01; moving β deforms the surface itself.