What is this picture?
Every pixel is one parameter: the two coordinates chosen under Axes, with all other parameters fixed. For each pixel the app iterates the equation as a map, zn+1 = F(zn, θn) with the angle θn+1 = θn + T, starting from f(0) (a critical point of F, or z₀). It discards a transient, then measures two things along the orbit:
- the fibre Lyapunov exponent λ, the mean of log|∂F/∂f|: λ < 0 means nearby orbits are pulled together onto an attractor, λ > 0 means they separate (chaos);
- the number of sheets, by comparing zn with zn+mq at the near-return times q of the rotation (shown under T): if the attractor is a smooth graph over the circle made of k sheets, the orbit comes back to the same sheet every k returns.
With Dynamics colouring, dark means the orbit escapes to infinity; blue is one smooth invariant graph; orange, green, pink, purple are 2, 3, 4, ≥5 sheets; mauve is λ < 0 with no finite sheet structure (a candidate strange nonchaotic attractor, SNA); teal is λ ≈ 0 (neutral, quasi-periodic); red is chaos. Brightness grows with eλ, so the strongly attracting centres of components are dark and their edges, where λ → 0, are bright. Lyapunov colours λ itself; Escape colours how fast the orbit escapes.
The bounded region is the forced analogue of the Mandelbrot set: the parameters for which the critical orbit stays bounded although the map is shaken by the rotation. Its components are classified by the shape of the attractor. When F is linear in f (a cocycle) the picture is computed exactly from closed forms instead of orbits, and the status bar says exact. The plane is drawn progressively, coarse first, using all processor cores but one.
Click a point to analyse it on the right; drag to pan, scroll or pinch to zoom; hold Shift to draw slices and borders (Curves on the plane).
Pick a parameter
What is this diagram?
The attractor of the analysed parameter as a function of the angle θ: the solution f(θ) of f(θ + T) = F(f(θ), θ), with Re f, Im f, |f| or arg f on the vertical axis.
Grey dots are the orbit itself, the points (θn mod 2π, zn) after the transient (the latest Orbit dots of them). Coloured lines are the invariant graph computed independently by Newton’s method on Grid per 2π points per circle: the equation is solved exactly on the grid (the residual is in the Summary), not fitted to the dots. When the dots lie on the lines, the orbit has converged onto that graph.
A structure with k sheets has k lines: either c separate graphs that the map permutes, or one graph that needs d turns of θ to close (a d-fold cover), or both (k = c·d). Each sheet has its own colour, and the dashed line is its mean (for Re and Im). The amber band is the section θ* ± w set above; its newest Points orbit points are highlighted, one cluster per sheet. arg f is taken in (−π, π], so a sheet that winds around 0 is drawn with a break where it crosses ±π.
Near the edge of a component the lines wrinkle (watch the Spectrum tab): this is how a smooth graph breaks up into a strange nonchaotic attractor.
What is this diagram?
The same attractor in the complex plane, forgetting the angle: each orbit point zn is a grey dot, and each Newton solution is drawn as a closed curve (one per component). A smooth invariant graph shows up as a closed curve, k sheets as k curves or a curve that winds k times; a strange nonchaotic attractor is a fuzzy set that is not a curve, and chaos fills an area. The amber points are the orbit points with θ ≈ θ*: a k-sheet attractor shows k clusters, which move along the curves as θ* turns.
What is this diagram?
The Fourier coefficients |f̂(ω)| of the Newton solution on a log scale, against the frequency ω (per radian of θ; multiples of 1/d for a d-fold cover). A straight decline means exponential decay, |f̂(ω)| ≈ e−ρ|ω|: then f extends analytically into the complex strip |Im θ| < ρ, and ρ, the analyticity width, is fitted and shown. It measures how smooth the graph is.
As a parameter approaches the edge of a smooth component, ρ shrinks to 0: higher and higher frequencies are needed, the graph wrinkles (fractalization) and can become a strange nonchaotic attractor. The floor near 10−16 is round-off; if the tail stays well above it, the grid is too coarse (the Summary's spectral tail warns about it; raise Grid per 2π).
No slice yet: draw one on the plane (Curves on the plane, left).
What is this diagram?
An attractor of f(x+T) = F(f(x), x) lives over the circle θ = x mod 2π, so it cannot be drawn as a single number per parameter. The slice cuts it with the vertical line θ = θ* (the shaded band in “Graph f(θ)”, the highlighted points in “Attractor in ℂ”) and plots what lies on that line, for every parameter along the segment — the analogue of the logistic bifurcation diagram.
Each dot is an orbit value zn whose angle θn = nT mod 2π falls within ±w of θ* — the spread. The line under the sliders says how many of the recorded orbit points fall in the window (the angles are the same for every parameter, about N·w/π of the last N), and how many of them, newest first, are drawn. An invariant graph with k sheets meets the line in exactly k points, so the diagram shows k curves: the number of lines is the number of sheets. Doublings appear as a curve splitting in two; a vertical smear means the section is no longer a finite set — a strange nonchaotic attractor (λ < 0) or chaos (λ > 0). Moving θ* changes where the curves sit, never how many there are.
A narrow spread gives sharp curves from few points; widening it thickens each curve into a band of height ≈ |f′|·2w (at w = π every orbit point is drawn, and the column is the whole range of f). The angles inside a window are not evenly spread: when T/2π is close to p/q, the θn stack into q columns 2π/q apart, so a window narrower than π/q may hold a single column visited many times.
Colours are the plane's categories; λ is drawn underneath. A slice can be a straight line, a polyline (its corners are dotted vertical lines in the diagram), a traced border or a smoothed freehand trace; the horizontal axis is the plane coordinate for a horizontal or vertical line, and otherwise the position along the path, by arc length from 0 to 1. Click the diagram, or use Move along under the plane, to analyse the parameter at that position.
What is this table?
How the analysis decides which structure the orbit has. Every candidate (c components, each a d-fold cover of the circle, glued with shift s; k = c·d sheets, up to Max sheets k) is tested: the orbit points are placed on that cover, and inside short bins the values are fitted by straight lines. The score is the fraction of the variance the fit leaves unexplained: near 0 when the orbit really is a smooth function on that cover, near 1 when it is not.
The smallest k whose score is below 10−3 wins (highlighted), and its binned values are the starting guess for Newton's method, which then solves the equation exactly on that structure. If no candidate passes, there is no finite sheet structure: a strange nonchaotic attractor (λ < 0), neutral motion (λ ≈ 0) or chaos (λ > 0).
What are these diagrams?
A cocycle is a rule that multiplies (or adds) something by a function of the angle at every step: zn+1 = g(θn)·zn. This tab shows the one that matters for the analysed point.
When F is linear in f, F = g(x)·f or F = f + L(x), the equation itself is a cocycle and the orbit has a closed form. Write log g = λ + iβ + L̃(θ) + iwθ: a mean growth rate λ, a mean rotation β, a mean-zero periodic part L̃ and a winding w. Solve ψ(θ + T) − ψ(θ) = L̃(θ) in Fourier, ψm = L̃m / (eimT − 1); then zn = z0·e(λ+iβ)n·eψ(θn) − ψ(0). The lower plot compares the computed orbit (grey) with this prediction (coloured).
Otherwise the tab shows the tangent cocycle: how a small perturbation of the invariant graph evolves, δ(θ + T) = w(θ)·δ(θ) with w = ∂F/∂f along the Newton solution (over one pass through all sheets, if there are several). Its mean log|w| is the Lyapunov exponent λ. When w reduces to a constant (winding 0, ψ resolved), the perturbation is simply multiplied by the fibred multiplier μ = eλ+iβ at every step, the forced analogue of a fixed point's multiplier. The lower plot shows the curve w(θ) and μ inside the unit circle. As the parameter moves to the edge of the component, μ reaches the circle, and its angle β/2π says what happens there: 0 a fold, 1/2 a doubling into 2 sheets, p/q a bulb with q sheets (the rays mark these angles). For analytic forcing f² + c + r·eix, μ equals the unforced multiplier 2a₀ exactly, which is why the bulbs sit where the Mandelbrot bulbs do. For F = g(x)·f the tangent cocycle is g itself, so both views agree.
Upper plot: per frequency m, the forcing (|L̃m|, or the Fourier coefficients of log w), the solution |ψm| and the amplification 1/|eimT − 1|. The amplification peaks at the continued-fraction denominators of T/2π (for T = 1: 333, 710, …), the small divisors; ψ exists when the forcing decays faster than they grow. The last metric checks the Newton solution: g = f(θ + T)/f(θ) is always a coboundary, so its mean log|g| must vanish.