F: A Brief Introduction to the Characteristics of Chaos
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 24242
In this appendix we will describe some aspects of the phenomenon of chaos as it arises in ODEs. Chaos is one of those notable topics that crosses disciplinary boundaries in mathematics, science, and engineering and captures the intrigue and curiousity of the general public. Numerous popularizations and histories of the topic, from different points of view, have been written; see, for example the books by Lorenz, Diacu and Holmes, Stewart, and Gleick.
Our goal here is to introduce some of the key characteristics of chaos based on notions that we have already developed so as to frame possible future directions of studies that the student might wish to pursue. Our discussion will be in the setting of a flow generated by an autonomous vector field.
The phrase "chaotic behavior" calls to mind a form of randomness and unpredictability. But keep in mind, we are working in the setting of –our ODE satisfies the criteria for existence and uniqueness of solutions. Therefore specifying the initial condition exactly implies that the future evolution is uniquely determined, i.e. there is no "randomness or unpredictability". The key here is the word "exactly". Chaotic systems have an intrinsic property in their dynamics that can result in slight perturbations of the initial conditions leading to behavior, over time, that is unlike the behavior of the trajectory though the original initial condition. Often it is said that a chaotic system exhibits sensitive dependence on initial conditions. Now this is a lot of words for a mathematics course. Just like when we studied stability, we will give a mathematical definition of sensitive dependence on initial conditions, and then consider the meaning of the definition in the context of specific examples.
As mentioned above, we consider an autonomous, \(C^r, r \ge 1\) vector field on \(\mathbb{R}^n\):
\[\dot{x} = f(x), x \in \mathbb{R}^n, \label{F.1}\]
and we denote the flow generated by the vector field by \(\phi_{t}(\cdot)\), and we assume that it exists for all time. We let \(\Lambda \subset \mathbb{R}^n\) denote an invariant set for the flow. Then we have the following definition.
Definition 22 (Sensitive dependence on initial conditions)
The flow \(\phi_{t}(\cdot)\) is said to have sensitive dependence on initial conditions on \(\Lambda\) if there exists \(\epsilon > 0\) such that, for any \(x \in \Lambda\) and any neighborhood U of x there exists \(y \in U\) and t > 0 such that \(\phi_{t}(x)\phi_{t}(y) > \epsilon\).
Now we consider an example and analyze whether or not sensitive dependence on initial conditions is present in the example.
Example \(\PageIndex{45}\)
Consider the autonomous linear vector field on \(\mathbb{R}^2\):
\(\dot{x} = \lambda x\),
\[\dot{y} = \mu y. (x, y) \in \mathbb{R}^2 \label{F.2}\]
with \(\lambda, \mu > 0\). This is just a standard "saddle point". The origin is a fixed point of saddle type with its stable manifold given by the y axis (i.e. x = 0) and its unstable manifold given by the x axis (i.e. y = 0). The flow generated by this vector field is given by:
\[\phi_{t}(x_{0}, y_{0}) = (x_{0}e^{\lambda t}, y_{0}e^{\mu t}). \label{F.3}\]
Following the definition, sensitive dependence on initial conditions is de fined with respect to invariant sets. Therefore we must identify the invariant sets for which we want to determine whether or not they possess the property of sensitive dependence on initial condition.
The simplest invariant set is the fixed point at the origin. However, that invariant set clearly does not exhibit sensitive dependence on initial conditions.
Then we have the one dimensional stable (y axis) and unstable manifolds (x axis). We can consider the issue of sensitive dependence on initial conditions on these invariant sets. The stable and unstable manifolds divide the plane into four quadrants. Each of these is an invariant set (with a segment of the stable and unstable manifold forming part of their boundary), and the entire plane (i.e. the entire phase space) is also an invariant set.
We consider the unstable manifold, y = 0. The flow restricted to the unstable manifold is given by
\[\phi_{t}(x_{0}, 0) = x_{0}e^{\lambda t}, 0) \label{F.4}\]
It should be clear the the unstable manifold is an invariant set that exhibits sensitive dependence on initial conditions. Choose an arbitrary point on the unstable manifold, \(\bar{x}_{1}\). Consider another point arbitrarily close to \(\bar{x}_{1}\), \(\bar{x}_{2}\). Now consider any \(\epsilon > 0\). We have
\[\phi_{t}(\bar{x}_{1}, 0)  \phi_{t}(\bar{x}_{2}, 0) = \bar{x}_{1}  \bar{x}_{2}e^{\lambda t} \label{F.5}\]
Now since \(\bar{x}_{1}  \bar{x}_{2}\) is a fixed constant, we can clearly find a t > 0 such that
\[\bar{x}_{1}  \bar{x}_{2}e^{\lambda t} > \epsilon. \label{F.6}\]
Therefore the invariant unstable manifold exhibits sensitive dependence on initial conditions. Of course, this is not surprising because of the elt term in the expression for the flow since this term implies exponential growth in time of the x component of the flow.
The stable manifold, x = 0, does not exhibit sensitive dependence on initial conditions since the restriction to the stable manifold is given by:
\[\phi t(0, y_{0}) = (0, y_{0}e^{\mu t}), \label{F.7}\]
which implies that neighboring points actually get closer together as t increases.
Moreover, the term \(e^{\lambda t}\) implies that the four quadrants separated by the stable and unstable manifolds of the origin also exhibit sensitive dependence on initial conditions.
Of course, we would not consider a linear autonomous ODE on the plane having a hyperbolic saddle point to be a chaotic dynamical system, even though it exhibits sensitive dependence on initial conditions. Therefore there must be something more to "chaos", and we will explore this through more examples.
Before we consider the next example we point out two features of this example that we will consider in the context of other examples.

The invariant sets that we considered (with the exception of the fixed point at the origin) were unbounded. This was a feature of the linear nature of the vector field.

The "separation of trajectories" occurred at an exponential rate.Therewas no requirement on the "rate of separation" in the definition of sensitive dependence on initial conditions.

Related to these two points is the fact that trajectories continue to separate for all time, i.e. they never again "get closer" to each other.
Example \(\PageIndex{46}\)
Consider the autonomous vector field on the cylinder:
\(\dot{r} = 0\),
\[\dot{\theta} = r, (r, q) \in \mathbb{R}^{+} \times S^1, \label{F.8}\]
The flow generated by this vector field is given by:
\[\phi_{t}(r_{0}, q_{0}) = (r_{0}, r_{0} t + \theta_{0}), \label{F.9}\]
Note that r is constant in time. This implies that any annulus is an invariant set. In particular, choose any \(r_{1} < r_{2}\). Then the annulus
\[\mathcal{A} \equiv \{(r, q) \in \mathbb{R}^{+} \times S^1  r_{1} \le r \le r_{2}, \theta \in S^1\} \label{F.10}\]
is a bounded invariant set.
Now choose initial conditions in \(\mathcal{A}\), \((r_{1}', \theta_{1})\), \((r_{2}', \theta_{2})\), with \(r_{1} \le r_{1}' < r_{2}' \le r_{2}\). Then we have that:
\(\phi_{t}(r_{1}', \theta_{1}) \phi_{t}(r_{2}', \theta_{2}) = (r_{1}', r_{1}' t+ \theta_{1})  (r_{2}', r_{2}' t+\theta_{2})\),
\(= (r_{1}'  r_{2}', (r_{1}'  r_{2}')t+(\theta_{1}  \theta_{2}))\).
Hence we see that the distance between trajectories will grow linearly in time, and therefore trajectories exhibit sensitive dependence on initial conditions. However, the distance between trajectories will not grow unboundedly (as in the previous example). This is because \(\theta\) is on the circle. Trajectories will move apart (in \(\theta\), but their r values will remain constant) and then come close, then move apart, etc. Nevertheless, this is not an example of a chaotic dynamical system.
Example \(\PageIndex{47}\)
Consider the following autonomous vector field defined on the two dimensional torus (i.e. each variable is an angular variable):
\(\dot{q}_{1} = \omega_{1}\),
\[\dot{q}_{2} = \omega_{2}, (\theta_{1}, \theta_{2}) \in S^1 \times S^1 \label{F.11}\]
This vector field is an example that is considered in many dynamical systems courses where it is shown that if \(\frac{\omega_{1}}{\omega_{2}}\) is an irrational number, then the trajectory through any initial condition "densely fills out the torus". This means that given any point on the torus any trajectory will get arbitrarily close to that point at some time of its evolution, and this "close approach" will happen infinitely often. This is the classic example of an ergodic system, and this fact is proven in many textbooks, e.g. Arnold or Wiggins. This behavior is very different from the previous examples. For the case \(\frac{\omega_{1}}{\omega_{2}}\) an irrational number, the natural invariant set to consider is the entire phase space (which is bounded).
Next we consider the issue of sensitive dependence on initial conditions. The flow generated by this vector field is given by:
\[\phi_{t}(\theta_{1}, \theta_{2}) = (\omega_{1} t+\theta_{1}, \omega_{2} t+\theta_{2}). \label{F.12}\]
We choose two initial conditions, \((\theta_{1}, \theta_{2})\), \((\theta_{1}' , \theta_{2}')\). Then we have
\(\phi_{t}(\theta_{1}, \theta_{2})  \phi_{t}(\theta_{1}', \theta_{2}') = (\omega_{1} t+\theta_{1}, \omega_{2} t+\theta_{2})  (\omega_{1} t+\theta_{1}', \omega_{2} t+\theta_{2}'\),
\(= (\theta_{1}  \theta_{1}', \theta_{2}  \theta_{2}')\),
and therefore trajectories always maintain the same distance from each other during the course of their evolution.
Sometimes it is said that chaotic systems contain an infinite number of unstable periodic orbits. We consider an example.
Example \(\PageIndex{48}\)
Consider the following two dimensional autonomous vector field on the cylinder:
\(\dot{r} = sin(\frac{\pi}{r})\),
\(\dot{q} = r, (r, q) \in \mathbb{R}^{+} \times S^1\).
Equilibrium points of the \(\dot{r}\) component of this vector field correspond to periodic orbits. These equilibrium points are given by
\[r = \frac{1}{n}, n = 0,1,2,3,.... \label{F.13}\]
Stability of the periodic orbits can be determined by computing the Jacobian of the r ̇ component of the equation and evaluating it on the periodic orbit. This is given by:
\(\frac{\pi}{r^2} cos(\frac{\pi}{r}\),
and evaluating this on the periodic orbits gives;
\(\frac{\pi}{n^2} (1)^n\)
Therefore all of these periodic orbits are hyperbolic and stable for n even and unstable for n odd. This is an example of a two dimensional autonomous vector field that contains an infinite number of unstable hyperbolic periodic orbits in a bounded region, yet it is not chaotic.
Now we consider what we have learned from these four examples. In example 45 we identified invariant sets on which the trajectories exhibited sensitive dependence on initial conditions (i.e. trajectories separated at an exponential rate), but those invariant sets were unbounded, and the trajectories also became unbounded. This illustrates why boundedness is part of the definition of invariant set in the context of chaotic systems.
In example 46 we identified an invariant set, \(\mathcal{A}\), on which all trajectories were bounded and they exhibited sensitive dependence on initial conditions, although they only separated linearly in time. However, the r coordinates of all trajectories remained constant, indicating that trajectories were constrained to lie on circles ("invariant circles") within \(\mathcal{A}\).
In example 47, for \(\frac{\omega_{1}}{\omega_{2}}\) an irrational number, every trajectory densely fills out the entire phase space, the torus (which is bounded). However, the trajectories did not exhibit sensitive dependence on initial conditions.
Finally, in example 48 we gave an example having an infinite number of unstable hyperbolic orbits in a bounded region of the phase space. We did not explicitly examine the issue of sensitive dependence on initial conditions for this example.
So what characteristics would we require of a chaotic invariant set? A combination of examples 45 and 47 would capture many manifestations of "chaotic invariant sets":

the invariant set is bounded,

every trajectory comes arbitrarily close to every point in the invariant set during the course of its evolution in time, and

every trajectory has sensitive dependence on initial condition.
While simple to state, developing a unique mathematical framework that makes these three criteria mathematically rigorous, and provides a way to verify them in particular examples, is not so straightforward.
Property 1 is fairly straightforward, once we have identified a candidate invariant set (which can be very difficult in explicit ODEs). If the phase space is equipped with a norm, then we have a way of verifying whether or not the invariant set is bounded.
Property 2 is very difficult to verify, as well as to develop a universally accepted definition amongst mathematicians as to what it means for "every trajectory to come arbitrarily close to every point in phase space during the course of its evolution". Its definition is understood within the context of recurrence properties of trajectories. Those can be studied from either the topological point of view (see Akin). or from the point of view of ergodic theory (see Katok and Hasselblatt or Brin and Stuck). The settings for both of these points of view utilize different mathematical structures (topology in the former case, measure theory in the latter case). A book that describes how both of these points of view are used in the application of mixing is Sturman et al.
Verifying that Property 3 holds for all trajectories is also not straightforward. What "all" means is different in the topological setting ("open set", Baire category) and the ergodic theoretic setting (sets of "full measure"). What "sensitive dependence on initial conditions" means is also different in each setting. The definition we gave above was more in the spirit of the topological point of view (no specific "rate of separation" was given) and the ergodic theoretic framework focuses
on Lyapunov exponents ("Lyapunov’s first method") and exponential rate of separation of trajectories.
Therefore we have not succeeded in giving a specific example of an ODE whose trajectories behave in a chaotic fashion. We have been able to describe some of the issues, but the details will be left for other courses (which could be either courses in dynamical systems theory or ergodic theory or, ideally, a bit of both). But we have illustrated just how difficult it can be to formulate mathematically precise definitions that can be verified in specific examples.
All of our examples above were two dimensional, autonomous vector fields. The type of dynamics that can be exhibited by such systems is very limited, according to the PoincaréBendixson theorem (see Hirsch et al. or Wiggins). There are a number of variations of this theorem, so we will leave the exploration of this theorem to the interested student.