Nonlinear dynamics · planar flows
Bifurcations in the plane
Move the parameter r through the planar system ẋ = f(x, y; r), ẏ = g(x, y; r). The phase portrait, equilibria, periodic orbits, continuation diagram, and Jacobian eigenvalues update together so that local stability and global motion remain visible in one place.
One system · four linked views
Flow, branches, and local stability
Phase portrait at the selected r
r = 0.550
Arrows show the planar vector field. The two directly labeled nullclines show where x dot and y dot vanish; their intersections are equilibria. Filled circles mark attracting equilibria, open circles mark repelling equilibria, diamonds mark saddles, and squares mark nonhyperbolic equilibria. Solid closed curves mark attracting periodic orbits; dashed closed curves mark repelling periodic orbits. A gold loop marks a homoclinic saddle connection at its critical parameter. Click to add a trajectory. With the plot focused, use arrow keys to move the initial-condition cursor and Enter or Space to launch it. Scroll or use plus and minus to zoom, use Control with arrow keys to pan, and press 0 or F to restore the full view. Use the middle mouse button to drag the view.
Bifurcation diagram
Vertical axis: radial amplitude ρ
The horizontal axis is the parameter r. The vertical quantity is named in the panel heading and changes with the selected family. Solid branches are attracting, dashed branches are repelling, and dash-dot branches are saddles. A vertical line marks the current parameter. Click or drag to change r, scroll or use plus and minus to zoom, middle-drag or use Control with arrow keys to pan, and press 0 or F to fit the complete diagram.
Local stability
E1 · equilibrium
The imaginary axis separates local contraction from local expansion. Faint trails show how the selected object's eigenvalues move for nearby parameter values.
J = [ 0.550 −1.000 ; 1.000 0.550 ]
- Eigenvalues
- 0.550 ± 1.000i
- Trace
- 1.100
- Determinant
- 1.303
- Classification
- Repelling focus
Selected trajectory through time
x(t) and y(t)
The two curves show x and y as functions of time for the most recently selected or added trajectory. Their long-term behavior corresponds to the invariant objects shown in the phase portrait.
The selected trajectory begins at x = 1.20, y = 0.20.
Mathematical and numerical notes
Which events are generic? In a generic one-parameter family of planar vector fields, the local equilibrium bifurcations are saddle-node and Hopf. Transcritical and pitchfork crossings require an invariant structure or symmetry. Folds of periodic orbits and SNIC onset are global events: they depend on the organization of trajectories beyond a single equilibrium.
Equilibria, nullclines, and stability. An equilibrium is an intersection of the two nullclines f(x, y; r) = 0 and g(x, y; r) = 0. The eigenvalues of the Jacobian at that point describe local contraction, expansion, and rotation. A saddle has one eigenvalue on each side of the imaginary axis, so it is unstable overall, with one locally attracting and one locally repelling direction.
Periodic orbits. The supercritical Hopf family has an attracting periodic orbit of radius ρ = √r for r > 0. The bounded subcritical family adds a quintic term that confines the global flow without changing the local Hopf type. For general families, statements about periodic orbits are numerical detections in the displayed window, not proofs that every orbit has been found.
Parameter sweeps. Each fixed value of r defines an autonomous system. During a sweep, the display changes that system gradually to show the family. A moving particle during the sweep should therefore be read as a visual guide through nearby phase portraits, not as a solution of any one fixed autonomous equation.