Orbits in State Space


In dynamics, 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 state.

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 called separatrices.

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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