Calculator Inputs
A fixed point is a value that remains unchanged by a transformation. In physics, fixed points describe equilibria of iterative maps and stationary states of dynamical models.
Example Data Table
The table below shows a common physics-inspired iterative map: the logistic map g(x)=r x(1−x) with r=2.8. It has fixed points near x*=0 and x*=1−1/r≈0.642857.
| r | g(x) polynomial coefficients | Suggested x₀ | Expected fixed point | Stability check |
|---|---|---|---|---|
| 2.8 | a₂ = −2.8, a₁ = 2.8, a₀ = 0 (others 0) | 0.6 | 0.642857… | |g′(x*)| = |r(1−2x*)| < 1 |
| 2.8 | a₂ = −2.8, a₁ = 2.8, a₀ = 0 (others 0) | 0.1 | 0 | |g′(0)| = |r| ≥ 1 (often unstable) |
Formula Used
Fixed-point iteration (map mode):
A fixed point satisfies x* = g(x*). The iteration updates x_{k+1} = (1−α)x_k + α g(x_k), where α is a relaxation factor.
Newton-Raphson (root mode):
A steady state of a 1D flow dx/dt = f(x) satisfies f(x*)=0. Newton’s update is x_{k+1} = x_k − f(x_k)/f′(x_k).
Stability (map mode): a fixed point is locally stable if |g′(x*)| < 1. This calculator evaluates g′ from the polynomial derivative.
How to Use This Calculator
- Choose a mode: fixed-point iteration for x=g(x), or Newton for f(x)=0.
- Enter the polynomial coefficients for g(x) or f(x).
- Set an initial guess x₀, tolerance, and maximum iterations.
- For difficult maps, reduce α to damp oscillations.
- Press Calculate. Results appear above the form with an iteration table.
- Use Download CSV or Download PDF to save the table.
Notes for Physical Interpretation
- In iterative models, fixed points represent steady responses after repeated updates.
- In dynamical systems, a fixed point is an equilibrium where time derivatives vanish.
- Convergence depends strongly on the initial guess and local stability conditions.
- If you know a physical scaling, use y=s·x+b to report results in physical units.
Article
1) Why fixed points matter in physics
Fixed points represent equilibria where an iterative update or a time evolution no longer changes the state. In nonlinear mechanics, electronics, and transport, these equilibria correspond to steady voltages, terminal velocities, or stable operating points. Detecting them numerically is often the first step before linearization and stability analysis.
2) Two practical problem statements
This calculator supports two equivalent views. In map form you solve x = g(x), common in Poincaré maps, renormalization updates, and discrete-time models. In root form you solve f(x)=0, typical for steady states of dx/dt=f(x). Both describe the same equilibrium when f(x)=x-g(x).
3) Convergence data you should watch
Each iteration records the current estimate, the function value, the derivative, the next estimate, and the step size. The stopping test uses the absolute step and a relative step scaled by max(1,|x|). For production work, tighter tolerances (e.g., 1e−10 to 1e−12) reduce drift in downstream calculations.
4) Stability criterion from derivatives
In map mode, local stability follows the contraction test |g′(x*)| < 1. Values near one indicate slow convergence and sensitivity to noise. The derivative column in the iteration table helps you diagnose borderline stability without extra computation.
5) Relaxation and damping for tough maps
The relaxation factor α applies under-relaxation: x_{k+1}=(1−α)x_k+αg(x_k). If the sequence oscillates, reducing α (for example 0.2–0.7) often restores convergence. This is a standard tactic in self-consistent field methods and iterative solvers.
6) Newton updates and expected speed
Newton-Raphson can converge quadratically near a simple root, meaning the number of correct digits can roughly double per step. However, it can fail when f′(x) is close to zero or when the initial guess is too far from the basin of attraction. The calculator warns when the derivative becomes numerically small.
7) Example data: logistic map equilibrium
For the logistic map g(x)=r x(1−x) with r=2.8, the nonzero fixed point is x*=1−1/r≈0.642857. The stability check evaluates |g′(x*)|=|r(1−2x*)|, which is below 1 here, so the equilibrium is stable and the iteration typically converges in a few dozen steps depending on α and tolerance.
8) Reporting results and exporting
Many physics workflows require mapping a dimensionless fixed point to a physical variable. Use y=s·x+b to apply calibration, nondimensionalization reversal, or unit conversion. Finally, export the iteration table as CSV for lab notebooks or as PDF for reports, keeping a reproducible record of settings and convergence behavior.
FAQs
1) What is a fixed point in this calculator?
A fixed point is a value x* that satisfies x* = g(x*) in map mode, or a value where f(x*) = 0 in Newton mode. Both represent equilibrium states in many models.
2) Why does my iteration diverge?
Divergence often means the initial guess is outside the attraction region or the stability condition fails. In map mode, if |g′(x*)| is near or above 1, convergence may be poor.
3) How should I choose the relaxation factor α?
Start with α = 1 for fast convergence. If the sequence oscillates or blows up, reduce α gradually (e.g., 0.7, 0.5, 0.3). Lower α damps updates and can stabilize difficult maps.
4) When is Newton-Raphson a better choice?
Use Newton when you have a steady-state equation f(x)=0 and a reasonable initial guess. Near a simple root it can converge very quickly, but it is sensitive to small derivatives and bad guesses.
5) What does the residual mean?
In map mode the residual is x − g(x), which should approach zero at a fixed point. In Newton mode the residual is f(x). A small residual indicates the equilibrium equation is satisfied.
6) Why do I see “Derivative near zero” warnings?
Newton steps divide by the derivative. If f′(x) is very small, the update can become huge and unstable. Try a different initial guess, rescale the problem, or switch to map mode with relaxation.
7) How do CSV and PDF exports help in practice?
Exports preserve the iteration trace: values, derivatives, and step sizes. This is useful for documentation, peer review, and reproducing numerical results. CSV supports analysis in spreadsheets, while PDF is convenient for reports.