Parameter plane

The Parameter plane follows a bifurcation as two parameters change. The Branch view's parameter runs across, a second parameter runs up, and each fold, Hopf, branch point, flip or Neimark–Sacker point found in the Branch view becomes a curve. Choose Parameter plane in the View panel. It is offered for a system with two parameters or more whose equations don't contain t or n.

The desktop app follows the curves of any such system and marks the points where they meet. The browser preview draws the curves of the two gallery templates below only.

Example project: parameter-plane.sepx, CO oxidation after Search in two parameters, with k₂ across and K up. Its fold and Hopf curves meet at the two Bogdanov–Takens points, the fold curve has a cusp, and the Hopf curve two Bautin points.

Reach it

• In the Branch view, click Follow in two parameters… on an LP, H, BP, PD or NS row, pick the parameter to free in Free also, and click Follow.
• Or click Search in two parameters in the Special points header, which follows every such point on the branches.
• Or choose Parameter plane in the View control. With no curve yet, it starts from the Branch view's special points. With none, it says so and offers Go to the Branch view.

Read the plot

Each kind of curve has its own colour and pattern: the fold curve is solid, the Hopf and Neimark–Sacker curves dashed, the flip curve dotted dashes, and the branch-point curve, Branch point (BP) in the legend, a dash and two dots. On a branch-point curve another branch of fixed points crosses, as at a transcritical or pitchfork bifurcation. A faint dotted stretch is no bifurcation at all (a neutral saddle). Points where curves meet or change character are marked:

The crosshair is the current pair of parameter values, and the grey bar is the Branch view's range at the value of the parameter up that its branches were computed at. Click anywhere to set both parameters; the sliders and every other view follow. Click a row in Special points to set both parameters there.

Each row of Special points says in a sentence what happens at the point, using only what holds where it was found. A fold-flip found on a branch-point curve says there is no fold there, and a Bautin or Chenciner row names the side of the point where the bifurcation is subcritical, as the plot shows it: above, below, left or right. A Bogdanov–Takens row gives its normal-form coefficients a and b and the sign of ab, and a cusp row its coefficient c. Only the sign of ab is fixed: the row takes a positive. A row on a Neimark–Sacker curve gives the angle θ of the multipliers. In the desktop app each row also says how finely the point was located: "located to 1.5 × 10⁻¹³" is the width of the last bisection step along the curve, each parameter as a fraction of its range.

On a Neimark–Sacker curve, short ticks mark the weak resonances 1:5 to 1:9, where the multipliers turn by a fraction 1/q of a full turn. Each is the tip of a resonance tongue, where a period-q cycle appears.

The inset

A small plot in the emptiest corner shows the system at the crosshair. For a flow it is the Phase view as you last left it: its two states, its range and its start points, or five starts spread over the view if it has none, with the fixed points. For a map it is the orbit from each start point after at least 1,000 iterates. The caption under it gives the two parameters and what is there: "3 fixed points in view: 2 stable, 1 saddle." For a map it adds how the orbits end: "The orbit settles on a period-4 cycle." or "The orbit does not repeat within 3,000 iterates." While a new inset is computing, the last one stays, faded. Systems with one state have no inset.

Homoclinic curves

A homoclinic orbit leaves an equilibrium and comes back to it. In a flow, a curve of them starts at every Bogdanov–Takens point. Follow homoclinic on a BT row of Special points finds the orbit beside the point and follows the curve, drawn with a dash and a dot and listed as Homoclinic (Hom) in the legend. Each point is found by shooting along the saddle's one-dimensional unstable manifold (or its stable manifold, in reversed time), so a curve takes seconds, not milliseconds. Its row counts the points as they arrive, and Cancel stops the run and keeps nothing of it.

The row says how the curve ends: one end shrinks onto the saddle at the BT point. It also says where the saddle quantity σ, the sum of the real parts of the leading unstable and stable eigenvalues, changes sign, and where the orbit is a Shilnikov orbit with chaos nearby: the saddle is a saddle-focus and its real eigenvalue outweighs the complex pair, σ > 0 when the unstable manifold is the one-dimensional one.

The browser preview has the homoclinic curves of the CO oxidation template only.

Try it

• CO oxidation (Bykov, Yablonskii and Kim; MatCont Tutorial IV): the Branch view over k₂ finds H, LP, LP, H. Followed in K, the fold and Hopf curves meet at two Bogdanov–Takens points, the fold curve has a cusp, and the Hopf curve two Bautin points.
• Predator–prey map (MatContM Tutorial M-II): the Neimark–Sacker point at a = 5, followed in ε, passes the 1:4, 1:3 and 1:2 resonances and a Chenciner point. The branch point at a = 5/3 is where the coexistence fixed point meets the one without predators. Followed in ε, its curve passes a fold-flip at (a, ε) = (3, 2), where the predator-free fixed point flips. That flip happens at a = 3 whatever ε is, so the plane does not follow it in ε: it says ε doesn't move it. Its other flip, at a = 9/11, followed in ε, passes a generalized flip at (a, ε) = (0.646447, −1.671573).
• Freire circuit (AUTO-07p's HomCont demo cir): type x' = (-(beta + nu)*x + beta*y - 0.328578*x^3 + 0.933578*(y - x)^3)/0.6, y' = beta*x - beta*y - z - 0.933578*(y - x)^3 and z' = y, set β to 0.8, give ν's slider the range −1.2 to 0, and open the Branch view over ν. It finds a branch point at ν = −0.8 and a Hopf point at ν = −0.4173. Followed in β, the Hopf curve passes a Bogdanov–Takens point at (ν, β) = (−0.774597, 0.774597). Follow homoclinic there draws the curve of homoclinic orbits to the origin. σ changes sign near β = 0.5912: below it the orbit is a Shilnikov orbit.

Check the numbers

The codimension-2 gate follows these curves as the plane does and checks every point against the reference values. Two examples from its output:

• CO oxidation's first Bogdanov–Takens point is at (k₂, K) = (1.417628, 0.971398) on the fold curve, and the Hopf curve finds the same point to within 2 × 10⁻¹³.
• The predator–prey map's Neimark–Sacker curve passes the 1:4, 1:3 and 1:2 resonances at a = 7, 10.5 and 14, where ε = 7/6, 49/38 and 35/26, each to within 10⁻¹².
• The Freire circuit's homoclinic orbit is at ν = −0.721309 for β = 0.6 and at ν = −0.725693 for β = 0.4535645, where AUTO-07p has −0.7213093 and −0.7256925. σ is −0.0170 at the first and +0.2888 at the second, the Shilnikov case.

Save and open

Save while the plane is shown writes it into the project file: the two parameters, the box, Δs and the most points, the points each curve was followed from, the Bogdanov–Takens points each homoclinic curve was followed from, and the Branch view it came from, with the parameter values its branches were computed at. Opening the file follows every curve again, so it shows the same curves, special points and slice. Branch then shows the Branch view the plane came from.

A curve that no longer fits the system, because you changed the equations, is left out and the open banner says which. A plane whose parameter is gone opens as its Branch view, or as the default view. Separatrix 1.0.1 and older open a saved plane in the default view.

Export

Export… writes the plane as a pgfplots figure in the curves' colours and patterns, with the special points and the resonance ticks, and the special points as a booktabs table with their coefficients. It also writes the curves as CSV, with the special points in a second file, and the plot as PNG or SVG. The crosshair and the inset are left out. See Export figures and data.