Chaos

For deterministic chaos to exist, a dynamical system must have a dense set of periodic orbits, it must be transitive and it also has to be sensitive to initial conditions. Density on periodic orbits implies that any periodic orbit's trajectory visits an arbitrarily small neighborhood of a nonperiodic one. Transitivity relates to the existence of points $a, b$ for which a third point $c$ can be found that is arbitrarily close to $a$ and whose orbit passes arbitrarily close to $b$. Finally, sensitivity to initial conditions is the property of arbitrarily close initial conditions to give rise to orbits that are eventually separated by a finite amount.

As a corollary, no chaotic dynamical system can have real fixed points (as opposed to virtual ones [7][8]) within its attractor. This, however, is a conditioned observation. In general the equations that specify a dynamical system are dependent on a parameter or set of parameters $( \sigma , \rho$ and $\beta$ for the Lorenz system; $\delta , \gamma$ , and $\omega$ in Duffing's oscillator) and chaotic behavior only manifests itself for certain values of these parameters. It is only under the chaotic regime that the system cannot have real fixed points.

This compels us to ask when and how is it that a dynamical system exhibits chaos. One way is through period doubling bifurcations. Take for example the logistic map introduced by Verhulst as a refinement to a growth model proposed by Malthus [9]

$$x_{k+1}=f_{\lambda} (x_k)=\lambda x_k (x_k -1)$$ (10)

If we keep $\lambda$ between 0 and 4, $x \in \mathcal{R} ^1$ will stay in the range 0 to 1. The equilibrium points of this system must satisfy:

$$x=\lambda x (x-1)$$ (11)

with solutions

$$x_- = 0 \cr x_+$$ (12)

For small enough values of ${\lambda}$ the point $x_-$ is stable, whereas $x_+$ is not. As a result the system reaches $x_-$ asymptotically (it effectively dies out). If we increase $\lambda$ we reach a value in which $x_-$ ceases to be stable and $x_+$ becomes attracting. This is called a Hopf bifurcation

What makes a fixed point of a discrete map stable? Mostly, that nearby points tend to it after successive iterations of the map. The notion of nearness is important, we call a metric a function $d(x,y)$ that measures the distance between two points in phase space. We call a norm a function $\mid x \mid$ that defines the magnitude of a vector in phase space. We will adopt as our distance measure the metric induced by the norm:

$$d(x,y)= \mid x-y \mid$$ (13)

A metric must meet the following requirements [10]:

$$(i) d(x,y) = 0 \Leftrightarrow x=y \cr (ii) d(x,y)$$ (14)

Given this, the notion of convergence to a fixed point $x^*$ under a discrete map like (7) can be expressed as:

$$\mid f(y)-f(x^*) \mid \;\;\; = \; \mid f(y) - x^* \mid \;\;\; \leq \;\;\; \mid y-x^* \mid$$ (15)

Now, if we consider $y$ to be sufficiently close to $x^*$, we may linerize $f(\cdot)$ at $x^*$

$$f(y) \approx f(x^*)+ \frac{\partial f}{\partial x} \Big \arrowvert _{x^*} (y-x^*)$$ (16)

so that (16) reduces to (see p-norms properties in [11]):

$$\Big \arrowvert \frac{\partial f }{\partial x} \Big \arrowvert _{x^*} \Big \arrowvert < 1$$ (17)

Which is met if all eigenvalues of the Jacobian in (18) have less than unit magnitude. In the case of continuous-time systems the eigenvalues must have strictly negative real parts.

In the logistic map $x_-$ is stable if $\lambda$ is in between 0 and 1. For $\lambda$ between 1 and 3, $x_+$ turns stable, but just above $\lambda = 3$ both fixed points become unstable. In turn a stable 2-cycle appears, which means that $x$ oscillates between two values. This transformation of a stable fixed point into a stable 2-cycle is called a period doubling bifurcation. Figure 1 shows how these bifurcations keep up showing up in the logistic map for different values of $\lambda$.

Figure 1: Bifurcation diagram for $f_\lambda (\cdot ), 0 \leq \lambda \leq 4$

It is also noteworthy that the distance between bifurcations diminishes in successive bifurcations. It was Mitchell Feigenbaum who first noted that the ratio of consecutive differences approaches a specific limit [12]:

$$\lim_{n \to \infty} \delta_n = \frac{\lambda_{n+1} - \lambda_{n}}{\lambda_{n+2} - \lambda_{n+1}}= \delta = 4.6692016$$ (18)

This has two immediate implications. First, every dynamical system characterized by a function $f(\cdot)$ that has a local quadratic maximum will show period doubling bifurcations and second, the system will eventually reach chaos by period doubling (with every $\lambda_n$ the period doubles and there is a finite value of $\lambda$ for which the period is infinite). This happens in the logistic map at $\lambda = 4$.

Another interesting observation made by Feigenbaum is that in the limit of high iterations $(k \gg 0)$, the specific form of $f(\cdot)$ is unimportant: qualitatively and quantitatively all systems with local quadratic maxima behave the same.

The existence of a period three window in Figure 1 is significative. The period three theorem of Li and Yorke proves that if a real function has a period three solution then it has solutions of any other period [13]. This is a special case of Sarkovskii's Theorem that, having been published in russian in 1964, was not known in the western world until after the work of Li and Yorke [14].