Sprott Explorer
Understand how compact Sprott-style codes are decoded into maps or flows, simulated, filtered, searched, and rendered as candidate chaotic dynamics.
What a Sprott Code Means
A Sprott-style code is a compact recipe for a dynamical system. It is not an image and it is not a label of chaos. The code is decoded into algebraic terms, coefficients, dimension, and update type. Only after that conversion does the toolbox simulate the resulting map or flow.
The educational value is that a short symbolic record can generate many different phase portraits. Students can see how simple deterministic functions produce fixed points, cycles, dense clouds, folded bands, or apparently chaotic attractors.
1. Family
The first symbol selects a family. In the toolbox grammar, families A-D are 1D maps, E-H are 2D maps, I-L are 3D maps, M-P are 4D maps, Q-T are 3D flows, and U-X are 4D flows.
2. Dimension and order
The family implies the number of variables and the polynomial terms that may appear. Maps are iterated directly; flows are integrated as differential equations.
3. Coefficients
The remaining characters encode coefficients. The decoder maps characters into numeric weights and attaches those weights to the monomials allowed by the selected family.
4. Function
The decoded record becomes an explicit function: x(n+1)=F(x(n)) for maps, or dx/dt=f(x) for flows. This function is what the toolbox actually simulates.
Search Logic
The searcher is a triage workflow. It does not prove chaos. It tries many compact recipes, simulates them with bounded numerical settings, rejects obvious failures, and keeps visually structured candidates for deeper analysis.
- Generate or load a compact code.
- Decode the code into equations, dimension, order, coefficients, and update type.
- Simulate the map or flow from a chosen initial condition.
- Discard transient points so startup motion does not dominate the figure.
- Reject obvious failures such as divergence, collapse, NaN values, or empty ranges.
- Render several projections and color modes to see whether the structure survives visual changes.
- Promote promising candidates to longer simulations and independent diagnostics.
From Function to Graphic
- Decode: split the compact code into family, dimension, polynomial order, and coefficient list.
- Build the function: combine coefficients with allowed monomials to form F(x) for maps or f(x) for flows.
- Simulate: iterate maps directly or integrate flows with Euler/RK4 using a chosen step size.
- Discard transient: remove the initial segment so the figure reflects long-term behavior rather than startup motion.
- Project: choose x-y, x-z, y-z, 3D, or a 4D projection where the hidden coordinate appears as color.
- Filter: reject trajectories that diverge, collapse to a point, leave the plotting range, or show obvious numerical failure.
How Parameters Change This Figure
Iterations or total time
Too few points can miss the attractor. Too many points can overplot into a solid mass unless point size and alpha are reduced.
Transient length
A startup spiral or jump can look interesting but disappear after enough transient points are removed.
Step size for flows
Euler or RK4 step size changes numerical stability. A large step can create false bounded shapes or divergence.
Initial condition
Changing the start tests whether a candidate is robust, multistable, hidden, or simply a local transient.
Projection
2D, 3D, and 4D color projections reveal different aspects of the same data. A hidden coordinate can be shown through color.
Alpha and point size
Dense maps need smaller points and lower alpha so returns remain visible instead of forming an opaque block.
How to Interpret a Candidate
A candidate image is a hypothesis. If it is bounded, non-collapsed, and visually structured, it deserves further inspection. The next steps are to run longer simulations, change the initial condition, reduce the integration step for flows, compare projections, estimate Lyapunov exponents, and check whether the apparent structure survives.
This is why the toolbox labels quick results conservatively. A visual candidate can guide exploration, but chaos claims require numerical diagnostics and reproducibility.
Common Errors
- Calling every dense cloud chaotic: numerical noise, bad projection, or too many overplotted points can mimic complexity.
- Skipping transients: a startup spiral can look interesting but disappear after enough iterations.
- Using only one initial condition: multistability and hidden attractors require varying starts, not trusting a single default point.
- Overreading candidate_chaotic: this is a quick filter outcome, not a formal mathematical classification.