Complexicon: Dynamical System

Dynamical systems are all around you.

The back and forth motion of a pendulum

due to gravity is a very simple example.

More interesting examples are the motions of the planets in the solar system

in which the motion is again driven by gravity,

but at a much larger scale.

And the Earth’s weather system in which weather changes

driven by heating from the sun by the Earth’s rotation,

and by other geophysical interactions.

Even the number of animals in a population

can be represented as a dynamical system.

Take for example, the number of foxes and rabbits in Northern Canada

which shows surprisingly regular up and down oscillations over time.

What is a source of these oscillations?

Roughly speaking, rabbits multiply quickly if left to themselves,

but they are also killed and eaten by foxes,

If there are lots of rabbits to eat, then the fox population flourishes.

But when rabbits are scarce, the foxes starve.

The interplay between these forces which ultimately

causes the booms and busts in population.

To account for how variables like the size of a population,

or the position of a planet change with time,

you can write a system of equations that describes

how each variable changes over time

due to its interaction with all other variables.

This set of equations that accounts for all these changes

constitutes a dynamical system.

Once you can write down a dynamical system in mathematical form,

you can apply powerful tools from calculus to analyze the system

and predict how it develops in time.

Returning to our example of the one variable dynamical system,

the pendulum, its motion in time is simple to predict.

The pendulum undergoes periodic motion that would last forever

if there is no friction or air resistance.

However because of friction and air resistance,

the pendulum gradually slows down

and eventually comes to rest in a vertical position.

When there is more then one dynamical variable,

the resulting behavior can be much more interesting

and mathematics can be very helpful to understand

what happens in the long run.

In the case of the number of predators and prey

such as rabbits and foxes,

you might want to know if the population continue to oscillate

or whether one or both of the species can become extinct.

For the solar system you might want to know,

if the orbits of planets are stable

or whether orbits of the lighter planets

could be destabilized by the heavier planets.

Answering these kinds of questions about the fate of dynamical systems

represent some of the research that we perform at the Santa Fe Institute.