The module starts from the thing a mathematician actually writes down: the equations. Type x' = s*(y-x) or x = r*x*(1-x), and Magic Stat detects whether it is an ODE or a map, typesets the equation as you write, and gives you the attractor, the bifurcation diagram, the Lyapunov spectrum and the rest.
There are two ways to study a dynamical system. The first is to know the equations: write them, integrate them, and explore the attractor, the bifurcation structure, the fixed points and their stability, the Poincaré sections and the Lyapunov exponents. Magic Stat takes the equations in the notation a person would use on paper — derivatives x', maps x_{t+1}, sums, roots and fractions — parses them symbolically and integrates them.
The second is to have only a time series and ask whether it is chaotic. The module implements the methods used for that question in ecology — s-map, Jacobian and direct Lyapunov exponents, the 0–1 test, recurrence quantification, permutation entropy and horizontal visibility graphs — as functions over any real series, not only over simulated ones.
It is not a computer algebra system. sympy is used for parsing, the symbolic Jacobian and the LaTeX preview, but the module is numerical. If you need closed-form solutions, stability proofs or symbolic manipulation, use a computer algebra system; Magic Stat is for numerical exploration and for the chaos tests on data.
Integration: Euler, RK4 or RK45, with the time span and step set in the dialog; maps are iterated for the number of generations you specify. Bifurcation: iterations per point, transient to discard and number of points.
Real-series methods: s-map lag and embedding by cross-validation, JLE/DLE, the 0–1 test, RQA, permutation entropy and HVG. Symbols: a palette of structures, functions, operators and Greek letters, or typed directly.
For the first half of the module, yes — it is an equation-first tool. For the second half, no: if you have only a measured series, the chaos-detection methods (s-map, Lyapunov exponents, 0–1 test, RQA) work on the data directly.
RK4 is the sensible default for smooth ODEs, with RK45 when the system is stiff or the time span is long; Euler is there for teaching and for maps. The step size matters — a chaotic system integrated with too large a step produces numerical chaos, not the system's.
It means nearby trajectories separate exponentially, which is the defining feature of chaos. The module reports the spectrum, not only the leading exponent; but a positive exponent from a short or noisy series should be treated as an indication, not a proof.
No. sympy is used for parsing, the symbolic Jacobian and the LaTeX preview, but the module is numerical. For closed-form solutions and symbolic work, use a CAS such as Mathematica, Maple or sympy directly.
No. Everything runs locally; the only automatic signal is an anonymous installation counter.