IntroductionMathgrapher version 2 has just been released. You are invited to download a fully functional trial version. Install Mathgrapher, start the Demonstrations and see what Mathgrapher can do for you. Functions in 2D and 3D MathGrapher is a graphical calculator for functions of the form F(x) and F(x,y) containing up to 20 subfunctions and 150 numerical and 100 named constants. Cartesian as well as polar coordinates can be chosen and functions can be represented in patametrized form (2D). F(x,y) can be represented in 2D and 3D by Shaded surfaces, Contour plots and Cross-sections through Contourplots. In the 3D viewer you may rapidly vary the viewing angle, distance and shading of the 3D surface using your mouse. Data in 2D and 3D Edit and draw graphs of your 2D or 3D Data. 3D surfaces can be previewed in the 3D viewer (OpenGL). Shaded surfaces, Contour plots and Cross-sections through Contour plots can be drawn in same way as 3D Functions. Curve fitting (linear and non-linear) A number of least squares curve fitting methods can be selected: e.g. linear regression, polynomials, trigonometric polynomials and cubic splines. An important feature of this program is that you can use the general and powerfull (non-linear) Levenberg-Marquardt method to fit your data to any continuous function you define. Iterations Calculate algebraic series or study iterative multi-dimensional maps. Several mathematical tools are available to analyse the results (zie ODE's below). Look at the examples to see how you may use Mathgrapher to study the route to chaos via period doublings in the simple logistic mapOrdinary Differential Equations (ODE's) The evolution of dynamical systems in physics, chemistry, electronics, economics and population dynamics can often be described with a set of coupled ODE's. Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector algorithm to integrate systems of up to 20 coupled ODE's. Several tools are available to analyse the results of the integrations (and iterations) such as: Graph of the time evolution, Projections in 2 or 3 dimensions, Surfaces of Section and Power spectrum analysis.

MathGrapher | The Mathematical Graphing Tool for Students, Scientists and Engineers

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Iterations

Calculate algebraic series such as e = 1+ 1/2! + 1/3! + ..., a square wave, Fibonacci numbers. Study iterative maps, e.g. the (one-dimensional) Logistic map: (see below) or more complicated multi-dimensional maps. The logistic map is perhaps one of the simples mathematical system showing many characteristics of the development of chaotic behaviour.
Several analytical tools are valailable to study the results of Iterations and ODE's such as:
time series | Power spectra | 2D projections | Fixed points | Lyapunov exponents
A special window has been added in Mathgrapher v2 to allow detailed presentation of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.

Examples:

Logistic map: Sensitivity to initial conditions Projection in 2D Power spectrum Bifurcation diagram Lyapunov exponents

Henon map: Definition 2D orbit Region of Stability

Mandelbrot and Julia sets: Definition Mandelbrot: vary parameters Julia: vary initial conditions

Series representation of e

Define F1=F2 and F2=F1+1/n! with initial values 0 and 1 respectively and do an iteration to see how fast the sum given below approaches e= 2.71828175