The State Space


The Dynamic

In qualitative 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 velecity. This list of numbers is called the system's state. For more complicated systems, say, like a chain of N pendula coupled together, the state of the system is much larger. It requires, in this case, 2N numbers to completely specify the state of the entire system.

The collection of all possible configurations of a system is called the state space. It has a dimension equal to the number of numbers required to specify the system's configurations.

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"---vocabulary that reminds us of the origins of qualitative dynamics in Poincare's study of planetary motion.

The rules that take a system in a given state to its next state over time are collected together in the "dynamic" or, equivalently, in the system's "equations of motion". Geometrically, we imagine that each state in the state space has a little arrow attached that indicates what state to move to next. The set of all these arrows is the dynamic, which determines the system's temporal behavior in an incremental or step-by-step manner.

The state the system starts in is called the "initial condition". The system's behavior unfolds from the initial condition by simply following the dynamic's arrows from state to state.

One of the main goals of qualitative 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.

Qualitative dynamics classifies attractors into three rough categories---fixed point attractors, limit cycle attractors, and chaotic 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 qualitative dynamics 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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