Orbits in State Space
one describes the instantaneous configuration
of a system as a list of numbers; each number denoting the value of some
property of the system. In the case of a simple pendulum, the instantaneous
configuration is completely described by two numbers---the position of
the pendulum bob and its velocity. This list of numbers is called the system's
The collection of all possible configurations of a system is called
the state space.
The temporal behavior of a system is then viewed as the succession
of states in the system's state space. One imagines a line being traced
out as a system moves from one state to the next. These lines are called
either "trajectories" or "orbits"
The state the system starts in is called the "initial
condition". The system's behavior unfolds from the initial condition
by simply following the equation of motion from state to state.
One of the main goals of dynamics
is to detect and analyze
the different types of trajectories and other objects in the state space
that govern a system's behavior.
The long-time behavior of a (stable) system is called an attractor,
which is simply the list of states the system eventually moves towards.
In fact, most systems have several distinct attractors, so that depending
on which initial condition the system starts in, the long-time behavior
can be quite different and end up in different parts of the state space.
Dynamics classifies attractors into three rough categories---fixed
point attractors, limit cycle attractors, and
chaotic or strange attractors. These
describe three different kinds of temporal behavior---equilibrium, oscillation,
and unpredictable behavior, respectively.
The set of all initial conditions that go to a given attractor is called
the attractor's "basin". The boundaries between the basins are
In one sense, a analysis of a system is complete
when one lists all of the system's separatrices, basins, and attractors.
This collection is called the system's "attractor-basin portrait".
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