Abraham and Shaw: Dynamics: Basic Concepts of Dynamics

by Carson Reynolds

After a brief introduction of the major players in the development of dynamics, Abraham and Shaw launch into a discussion of state spaces. They argue that physical systems can be modeled with a few observable parameters, which under a conventional interpretation we agree describe the behavior of a system.

The real number line, and the plane are presented as state spaces for one and two dimensional systems respectively. Trajectories are introduced as is the time-series representation.

Vectorfields are introduced next. Instantaneous velocities can be obtained by differentiating the trajectories. The state space populated with trajectories is the phase portrait, while the differentiation of the phase portrait is the vectorfield, aka dynamical system.

A set of assumptions on which Dynamic systems are built are articulated next:

(1) a velocity vector is in a real system is the same as the velocity system in our dynamical model.

(2) the vectorfield of the model is smooth.

Dynamical systems do not seem to be able to handle discontinuities, or signals that violate the lipschitz condition. This is my conjecture though.

By integrating vector fields we can recover trajectories, or the phase portrait.

The vectorfield / phase portrait model extends to manifolds, and higher dimensional systems.

In the subsection “special trajectories” critical (equilibrium) points are discussed along with constant trajectories. A special set of equilibriums: closed orbits, closed trajectory, periodic trajectory, cycle, or oscillation. Lastly solenoidal or almost periodic trajectories are shown as extensions of periodic trajectories.

We then move on to asymptotic approach to set limits. Certain trajectories approach equilibrium or closed orbits asymptotically, another set asymptotically diverge from equilibria. Thee equilibria can be discussed in terms of a limit point or limit cycle (in the orbit case). A system can have a constant or dynamic equilibrium.

By dividing the phase portrait into alpha-limit sets and omega-limit sets we can divide it into regions that are attractors, basins, or separatricies. An attractor is a limit set with an open inset: a limit point with an open inset is a static attractor, a limit cycle with an open inset is a periodic attractor. The inset of an attractor is its basin, dividing boundaries are separaticies.

Non-attractors have thin insets (probability of zero). These are also called exceptional limit sets, improbable limit sets. Separatix consist of insets of exceptional limit sets. They are also called saddle points.

A dynamical system with a potential function can be represented as a gradient vectorfield. From this model we can easily compute a potential surface or level curves.