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2D bifurcations

WHEN THE PHASE PORTRAIT CHANGES

Connections & bifurcations

4 equilibria

Move one parameter to reconnect saddle branches, form a loop, and create or destroy equilibria. Nullcline intersections locate the fixed points.

Heteroclinic connectiona ≈ −1.513437374

Phase portrait & nullclines

x′ = 0y′ = 0Stable branchUnstable branchConnectionYour trajectories

Click to add a trajectory · drag to pan · scroll to zoom. Arrows show forward time. ◆ saddle · ● attracting · ○ repelling.

Equilibria as a varies

Click or drag to choose a

Each curve gives the x-coordinate of an equilibrium. Solid green: attracting; dashed: saddle or repelling. HeC: heteroclinic · HC: homoclinic · H: Hopf · SN: saddle-node. Global connections do not change the equilibrium count.

Fixed points

J = [2x, −2y; −2x, 1]
(x, y)TypeEigenvalues

Reading the connections

A heteroclinic orbit leaves one saddle and approaches another. A homoclinic orbit leaves and returns to the same saddle.

The blue stable branches approach a saddle in forward time; the orange unstable branches leave it. A connection is highlighted in pink at its critical parameter.

x′ = 0: y = ±√(x² + 1)y′ = 0: y = x² + a

Explore planar bifurcations

This family changes with a single parameter a. The portrait, nullclines, fixed points, and equilibrium diagram update together.

The homoclinic value is found by matching the saddle’s stable and unstable branches on a transverse section. Its displayed loop joins these two computed halves to avoid long-time numerical drift. The exact heteroclinic line is y = mx − 1/(2m), where m³ − m − 1 = 0 and a = −m − 1/(4m).