Dynamical Systems

Why

We want to model natural phenomena.¹

Definition

Let $X_0, X_1, \dots , X_T$ be a sequence of sets and let $f_t: X_t \to X_{t+1}$ for $t = 0, \dots , T-1$. We call $((X_0, \dots , X_T), (f_1, \dots , f_{T-1}))$ a deterministic discrete-time dynamical system.

We call the index $t$ the epoch, the stage or the period. We call $X_t$ the state space at period $t$. We call $f_t$ the transition function or dynamics function.

Let $x_0 \in \mathcal{X} _0$. Define a state sequence $x_1 \in \mathcal{X} _1, \dots , x_T \in \mathcal{X} _T$ by

\[ x_{t+1} = f_t(x_t, u_t). \]

In this case we call $x_0$ the initial state. We call the $x_t$ the trajectory associatd with initial state $x_0$.

We call $T$ the horizon. In the case that we have an infinite sequence of state sets, input sets, and dynamics, then we refer to a infinite-horizon dynamical system. To use language in contrast with this case, we refer to the dynamical system when $T$ is finite as a finite-horizon dynamical system.

Future editions will modify, and may develop dynamic systems via the genetic approach by appealing to their classical use in Newtonian physics for modeling celestial mechanics. ↩︎