## Deterministic Systems and Chaos

Assuming that we are living in a causal universe, the cause of the events in the future are attributed to the events of the present and the past. Considering an isolated deterministic system that is characterized by an n-dimensional real phase space (denoted by x £ to describe this dependence, we can write a set of n differential equations

or a set of n difference equations x(i + l) = f(x(i)) (5.2)

with f (•) representing an n-dimensional vector function of x. Under general conditions, the existence and uniqueness properties of solutions hold, and Eq (5.1) or (5.2) determines the trajectory (or orbit) of the dynamical system, given the initial conditions. Eq (5.1) defines a continuous flow for a continuous system, and Eq (5.2) defines a discrete map for a discrete system. Note that both definitions are for autonomous systems, meaning there is no explicit time dependence in the system equations. Even if there were explicit terms in t or i in the system equations, we could augment the system order by one, to have xn+\ = t with fn+i = 1 for the continuous case, and xn+i = i with fn+i = i + 1. Thus, without loss of generality, we will confine our interest to autonomous systems.

### Poincare Map

Although most physical systems manifest continuous dynamics, maps arise naturally in many applications. Furthermore, even when the natural statement of a problem is in continuous time, it is often possible and sometimes desirable to transform the continuous dynamics to a map. Note, however

Figure 5.1. The Poincare surface of section, S, defined by =constant, for a third order system.

Figure 5.1. The Poincare surface of section, S, defined by =constant, for a third order system.

that, this transformation is not necessarily a mere time discretization, but can as well be performed by the Poincare surface of section technique, shown in Figure 5.1 for a third order system. The trajectory of the system can be visualized by a parametric plot of the states xk, in phase space (or state space) (xi,x2,x3). Thus, the trace of this parametric plot is an instance of the system dynamics, and the arrows show the direction of time.

If we choose a surface, S, in this space, say, defined by X3 =constant, and label the points that the system trajectory crosses S, as a,b,c,..., we can collect a two dimensional information about those crossings, given by the coordinates (x\,x2). If the point a corresponds to the ith crossing of the surface of section at t — U, we can define a two-dimensional vector, y(i) = (xi(ti),x2(ti)). Given y(i), we can reconstruct an initial condition for the system dynamics, and solve the model equations —analytically or numerically— to find the next crossing point, b. In other words, point a uniquely determines point b for a given system dynamics. Therefore, there exists a two-dimensional map which can be iterated to find all subsequent crossings of S. Using this Poincare surface of section technique, we can, in general, discretize an nth order continuous flow into an n — 1st order map, called the Poincare map. Note that, the common time discretization via strobing is a special Poincare map, where a periodic function of the system time, e.g., sin(wt), is considered as a state variable, and the surface of section is selected, such as sin(wt) = 0.

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