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In the Applet we display the behaviour of several examples of dynamical systems, i.e. systems of first-order time-dependent non-linear ordinary differential equations. Such systems arise when one considers the time evolution of physical, biological, economic systems. At each time, the states of the various components of the system are specified by a number of state variables; the differential equations describe how the variables of the state change per unit of time.
Depending on the structure of the equations and the number of the variables, one can have a wide range of behaviour:
The Volterra equations provide a very simple description of what happens when two (biological) species are in competition: One assumes that the number of sheep grows by natural reproduction in proportion to the number already there, but deceases when sheep and wolves meet. The number of wolves is increased by that process, as their reproduction is enhanced when they have food, but they die out if no food is available:
This results in the following cyclic behaviour: When the wolves are numerous, they decimate the population of the sheep, but if they are too numerous, they die of hunger. Being less disturbed by the few wolves, the sheep population can recover by normal growth, so the small number of wolves thrives better and they become more numerous.... There exists an equilibrium point (s=w=1 in our schematised equations) where the two populations could co-exist. But if one or the other population become larger or smaller - say by external effects - oscillations will commence about the equilibrium point. The very simplified Volterra-system neglects many effects that are present in nature. So the oscillations occur always with constant amplitude.
The VanderPol equations
describe an electronic oscillator, which is unstable against small perturbations about the zero-amplitude state, so the amplitude of oscillations quickly grow until they are limited by the nonlinearity in the electronic circuit (essentially the vacuum tube which is the essential active but nonlinear element). In the simulations, the limiting solution of a constant amplitude (but) nonlinear oscillation is represented by a closed-loop in the x-y plot. Such as feature is called a limit cycle.
Rössler's system is also a (nonlinear) oscillator, but in three dimensions:
This permits the chaotic flipping between a simple oscillation - represented by a circle in the X-Y-plane - and an excursion to the right. This system is the simplest to show genuinely chaotic solutions
A drastically simplified model for the flow of air and heat in the terrestial atmosphere is Lorentz 's system:
It forms two coupled oscillators, and the system chaotically jumps between them. Changing the parameters from the default values may completely cut-off the chaotic behaviour.
The movement of stars in the combined gravitational field of all the other stars in a galaxy lead to the equations of Hénon-Heiles. (under construction)
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