Continuos-time and discrete-time dynamical systems

We will start by defining what is it we mean by a dynamical system. Georg Cantor (1845-1918), the father of modern set theory, defines a set as ``Any collection of definite, distinguishable objects of our intuition or imagination to be conceived as a whole`` [4]. If we now impose an order on top of a set so that its elements could be represented by a state vector and we have a rule that determines how this vector evolves with time we get a dynamical system.

If time is considered to be a continuous parameter we have a continuos-time dynamical system whose development is governed by a some differential equation. Moreover, by an appropriate selection of state variables (the components of the state vector) we can reduce the original $n$th differential equation to a system of $n$ first-order differential equations.

Lorenz's original equations, for example, define a three dimensional continuous-time dynamical system

$$\dot{x}_{1} = \frac{dx_1}{dt}=\sigma(x_2-x_1) \cr \dot{x}_{2}$$ (1)

If we consider $(x_1, x_2, x_3)^T$ as the components of a vector $X$ and $f(\cdot)$ as a vector function of $X$, then we may re-write 1 in a more compact form:

$$\dot{X}=f(X)$$ (2)

where $f(\cdot)$ denotes the nonlinear function (1).

A particular solution to (1) or (2) is a time dependent vector $X(t)$ that might, as well, be interpreted as a trajectory or orbit in phase space (also called state space). The orbit of a point $X_0$ is the collection of points in phase space visited by $X(t)$ when $X(0)=X_0$.

Interestingly enough, even though in general it is not possible to solve for $X(t)$ explicitly, the region in phase space in which $X(t)$ unwinds is not, in general, the whole phase space. So, $X(t)$ is often confined to a subspace of the phase space. This implies that certain general considerations on the evolution of the system can be made.

If we suppose that, at least in principle, (2) can be integrated, then

$$X(t)=\Phi(t,X_0)$$ (3)

The region of phase space to which the solutions (3) converge (or are eventually drawn to) is called the attractor of the system. It is one of several invariant spaces associated with a dynamical system. More precisely, the attractor $G$ of a dynamical system characterized by a function $f(\cdot)$ over a space $S$ is given by

$$G=\left\{ g \in S \mid g \in G \rightarrow \forall \;\;\; t \geq 0 \;\;\; \Phi (t,g) \in G \right\}$$ (4)

If $A$ is a subset of $S$, the function $f(\cdot)$ applied to $A$ , the set of images of the elements within $A$:

$$f(A)=\left\{ y \mid \forall \;\;\; x \in A, \hspace{0.1cm} y=f(x) \right\}$$ (5)

Thus, condition (4) can be expressed as:

$$G=\Phi (t,G) \forall \;\;\; t \geq 0$$ (6)

For discrete-time dynamical systems, the equivalent of (2) is a difference equation that defines the one time-step evolution of the state vector.

In this case, the attractor $G$ is given by

$$G=f(G)$$ (7)

Technically, $G$ must be a compact set of finite diameter and nearby trajectories should tend to $G$. This is because there are other invariant sets (the unstable manifold, for example) for which this is not the case [5].

For practical purposes, and at least from a computational viewpoint, the formulation of a dynamical system in terms like (6) and (7) is not only easier but also more useful. The reason why is that to simulate a dynamical system with a digital computer one must discretize it to convert the process of integration into a an iterative procedure involving only addition. There is no unique way to affect this discretization [2][6]. However, in this paper we will not need to make any conversion from continuous-time to discrete-time nor vice versa.

Actually the dynamical systems we will discuss are very simple, like the logistic map $f_\lambda (\cdot)$, the quadratic transformation $Q_c(\cdot)$, and others we will introduce in due time. Interestingly, from such simple functions it is possible to obtain a surprisingly complex behavior.