The most used way to graphically show the evolution of a dynamic system is to draw its trajectory in the phase space: a space with as many dimensions as state variables (or even degrees of freedom) in the system. Each point of this space represents a very specific state in which the system can come to find itself. The phase space of the pendulum - for instance - consists of the Cartesian coordinates system identified by two variables: the angle from the vertical and the angular speed of the pendulum.
Not all points of the infinite phase space (also known as state space) are possible points for the system: indeed, most of them are impossible points and the system will never reach them. Mathematically this means that those points (combination of status variables) are not a solution of the differential equations governing the dynamics.
If we consider all possible points of the system in a chronological order, we will find that it draws a trajectory line in the phase space, which looks like the flight of a fly. The study of these trajectories tells much about systems.
Generally speaking the development of a trajectory depends on the initial state and thus by the initial conditions.
Left free to evolve a system displays normally a short-term dynamic, temporary, known as transient stage, and a long-term dynamic conditions. The latter can be a null state (think of a spring that is discharged, or the swings of a pendulum at the end of its oscillations, when it doesn't receive additional forces) or a steady state. The current in a transformer is periodic indefinitely, for example.
The state we are interested to is the chaotic one. In this case, different initial conditions cause different trajectories, which never overlap nor cross both a given point. The chaotic trajectories are an infinite development of a curve in the phase space that do not ever repeat twice the same even for a tiny tract.
Should we draw it, we would see that this curve never intersects itself. If you think about it, this fact is closely related to the impredicibility of chaos: should an overlap ever occur periodicity would settle up, because from that moment on the curve will repeat itself, giving rise to a periodic state that allows easy predictions of the system dynamics.
The bizarre geometric figure drawn by this trajectory is called strange attractor: the name betrays the surprise of those who saw it for the very first time and did not know how else that rich dynamic could be called. The set of all states that give raise to trajectories evolving with this curious geometry is called the attractor's basin of attraction.
Not all points of the infinite phase space (also known as state space) are possible points for the system: indeed, most of them are impossible points and the system will never reach them. Mathematically this means that those points (combination of status variables) are not a solution of the differential equations governing the dynamics.
If we consider all possible points of the system in a chronological order, we will find that it draws a trajectory line in the phase space, which looks like the flight of a fly. The study of these trajectories tells much about systems.
Generally speaking the development of a trajectory depends on the initial state and thus by the initial conditions.
Left free to evolve a system displays normally a short-term dynamic, temporary, known as transient stage, and a long-term dynamic conditions. The latter can be a null state (think of a spring that is discharged, or the swings of a pendulum at the end of its oscillations, when it doesn't receive additional forces) or a steady state. The current in a transformer is periodic indefinitely, for example.
The state we are interested to is the chaotic one. In this case, different initial conditions cause different trajectories, which never overlap nor cross both a given point. The chaotic trajectories are an infinite development of a curve in the phase space that do not ever repeat twice the same even for a tiny tract.
Should we draw it, we would see that this curve never intersects itself. If you think about it, this fact is closely related to the impredicibility of chaos: should an overlap ever occur periodicity would settle up, because from that moment on the curve will repeat itself, giving rise to a periodic state that allows easy predictions of the system dynamics.
The bizarre geometric figure drawn by this trajectory is called strange attractor: the name betrays the surprise of those who saw it for the very first time and did not know how else that rich dynamic could be called. The set of all states that give raise to trajectories evolving with this curious geometry is called the attractor's basin of attraction.
A chaotic system can produce more than one strange attractor, with basins of attraction hardly distinguishable from each other or even overlapped - such as riddled basins, which are fractal basins. In these cases, a dynamic system might jump from one to another attractor due to external causes and perturbations. My words fail to depict the complexity that can surface from a similar case.
But the question is not purely theoretical, rather it is very real and closely concerns the world we live in. An example might be the climate, which so often deserves the honor of the first columns in these years.
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