Many dynamical systems in physics, astronomy, chemistry, physiology, meteorology, economics, population dynamics can be described by Ordinary Differential Equations.
In the second half of the 20th century much attention has been focused on the often chaotic, i.e. unpredictable behavior of (non-linear) ODE’s. A well-known example is the Lorenz atractor, illustrating the “Butterfly effect”: small causes can have large effects. Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector algorithm to integrate systems of up to 20 coupled differential equations.

Several analytical tools are available for ODE’s such as: Time series | Power spectra | 2D and 3D projection | Phase portraits | Poincare section | Equilibrium points | Lyapunov exponents

Examples: Lorenz Equations | Hénon-Heiles potential | Rössler Equation | van der Pol oscillator | Duffing oscillator | Predator-Prey equation (Voltera)

Lorenz Attractor – the “butterfly effect”

Lorenz attractor (the “Butterfly effect”) Lorenz studied a set of differential equations that are well known in hydrodynamics (the Navier-Stokes quations in combination with heat conduction equation and the continuity equation) which are used for example in numerical weather simulation programs. He came up with a simple (though non-linear) set of ODE’s which turned out to exhibit very remarkable behavior. One of the demonstrations in Mathgrapher shows the integration of this system and some analysis of the results. The 3D orbit looks like a butterfly. The wings of the butterfly may represent two quite different states of the system like a thunderstorm and calm sunny weather. When the evolution of the system (the orbit) is studied it turns out that the system switches from one wing of the butterfly to the other in a chaotic way. The behavior turns out to be highly sensitive to the initial conditions and therefore becomes highly unpredictable or chaotic. Lorenz remarked that “the flapping of a butterfly’s wings could change the weather”. The orbit in the Lorenz system of ODE’s remains confined to a definite volume in space. It appears as though the trajectory is attracted to a certain region in space. This is called the “Lorenz attractor”. The Lorenz attractor even belongs to the category of “strange attractors” which exhibit chaotic behavior.

Y-Z projection of the Lorenz attractor

The chaotic behavior becomes clear when one looks at the power spectrum analysis of the orbit.

Power spectrum of the Lorenz attractor