Orbit - crowlogic/arb4j GitHub Wiki

In the context of dynamical systems, a trajectory and an orbit are closely related, but they are not exactly the same thing.

A trajectory refers to the path traced by a dynamical system in its state space as it evolves over time, given an initial condition and, possibly, a set of system parameters. The trajectory represents how the state of the system changes from one point in time to another, considering the underlying mathematical equations governing the system's behavior.

An orbit, on the other hand, refers specifically to the set of points visited by a dynamical system in its state space when it evolves over time for a given initial condition, without considering the order or time at which these points are visited. The orbit represents the collection of all possible states the system can be in, starting from the given initial condition.

While both trajectory and orbit are related to the evolution of a dynamical system in its state space, the trajectory emphasizes the path and the temporal order of the points visited, whereas the orbit focuses on the collection of points visited without regard to their order or timing.

The term "orbit" can imply a closed loop-like trajectory, but it doesn't always have to be the case. An orbit refers to the set of points visited by a dynamical system in its state space when it evolves over time for a given initial condition. Depending on the dynamical system and its parameters, the orbit can exhibit different types of behavior. Here are a few examples:

• Periodic orbit: This is a closed loop-like trajectory, where the system returns to its initial state after a certain period. This implies that the system's behavior repeats itself in a regular, periodic fashion.

• Quasi-periodic orbit: In this case, the system's behavior is still regular but not strictly periodic. The trajectory fills a torus (or a higher-dimensional analog) in the state space, and the system never exactly returns to its initial state but gets arbitrarily close.

• Chaotic orbit: For chaotic systems, the orbits are highly sensitive to the initial conditions and exhibit irregular, aperiodic behavior. These trajectories do not form closed loops in the state space, and their structure is complex and unpredictable.

• Fixed point (or equilibrium): A fixed point is a special case of an orbit where the dynamical system doesn't change its state over time. The trajectory remains at a single point in the state space.

So while some orbits can be closed loop-like trajectories, others might not be. The specific behavior of an orbit depends on the dynamical system, its parameters, and the initial condition.