which has a trace r = —a — 1-/3, that is less than zero. Thus, an initial phase space volume, V(0) shrinks with time as V(t) = V(0)ert.

A similar definition is made for the map of Eq. (5.2), using the magnitude of the determinant of the tangent flow matrix, r =

Eq. (5.7) defines the factor by which the n-dimensional phase space volume changes. If this factor is greater (less) than one, the phase space volume grows (shrinks) at the next iteration.

Example 2 Phase volume change of a map

Consider the two dimensional Henon map [231], given by xi(i + 1) = a - xt(i) + (3x2(i) (5.8)

x2(i + \) = Xi{t), where a and 0 axe constants. The tangent flow matrix for this system

has a constant determinant r — |/3|. The hyper volume defined in this phase space is in fact an area, since the phase space is two dimensional. If the "absolute value" of the parameter ¡3 is less than one, then the area shrinks by a factor of |/?|. □

Systems with phase space contraction, such as the ones presented in the last two examples, are commonly characterized by the presence of attrac-tors. The trajectories of the system originating from a specific region of the phase space are attracted to a bounded subset of the phase space, called the attractor, and that specific region that hosts all such initial conditions is called the basin of attraction for that attractor.

Chaos— Sensitive Dependence on Initial Conditions

In a two dimensional phase space, possible scenarios are limited for a continuous flow. Distinct trajectories cannot intersect because of the existence and uniqueness conditions. They either diverge to infinity, or converge to a limit point1 or a limit cycle2 However, when the phase space dimension increases to three, something fascinating happens. The trajectories can enjoy the freedom of staying confined in the phase space, without converging to a limit point or to a limit cycle. Let us illustrate this with an example.

1 Limit point: a final state, where the trajectory of a dynamics converge.

2Limit cycle: the periodic motion displayed by the trajectory of a dynamics.

Example 3 Driven pendulum

Using Newton's second law, the dynamics of a damped, sinusoidally driven pendulum are expressed as j2 a J a

where 9 is the angular displacement from the vertical, £ is the damping coefficient accounting for friction, and a and ¡3 are the forcing amplitude and forcing frequency, respectively. Note that, this is a non-autonomous second order system. Applying the definitions,

we can transform the system into an autonomous third order system. First, consider a pendulum with no forcing (a = 0), which reduces the phase space dimension to two. The system will have infinitely many steady states, located at 9 = ±kir and w = 0, with k = 0,1,2,____ The steady states for even k (corresponding to the lower vertical position) are stable, and those for odd k (corresponding to the upper vertical position) are unstable. If we also set £ = 0 to eliminate friction, the pendulum will swing back-and-forth, or rotate in an infinite loop, determined solely by its initial conditions, as shown in Figure 5.2.a. Note that, the pendulum with no friction is a Hamiltonian system, hence it conserves the phase space volume due to Liouville theorem. If, however, we consider a finite friction, the energy of the trajectories will eventually be consumed, and the pendulum will come to a halt at one of its steady states (Figure 5.2.b). To have a better understanding of the mechanism of this, take a closer look at the trajectories near the two types of steady states. Near the stable ones, (Figure 5.3.a) the trajectories spiral down to the steady state. Near the unstable steady states, trajectories approach the (saddle) steady state from one direction, and are repelled in another direction. There are some trajectories that seem to violate the uniqueness of the solution; they approach the steady state from opposite directions to meet at the steady state, and diverge from it in opposite direction, starting from the steady state. If we consider the time aspect of the problem, the uniqueness condition is not actually violated, since it takes infinitely long to converge to the steady state, on the trajectories that end in there. Similarly, for the trajectories that emanate dw ~dt d9

~dt from the steady state, it takes infinitely long to diverge from the steady state. Another property of the trajectories that converge to this steady state is the partitioning of the phase space in the sense that trajectories on the right hand side of this trajectory cannot cross over to the left hand side of it, and vice versa. Therefore, they define the boundaries of the basins of attraction.

Next, introducing the forcing back into the system, we have a three dimensional phase space. For certain combinations of driving amplitude and frequency, we observe a rich dynamic behavior, which neither converges to a steady state, nor gets attracted by a limit cycle. Instead, the trajectories explore a finite subset of the phase space, converging to a strange attractor.

When the system converges to a steady state (also called a limit point), the limit set of the system in phase space is an object of zero dimension. When it converges to a limit cycle, the limit set is still an object of integer dimension (one). However, when the system exhibits a rich dynamic behavior, such as the one shown in Figure 5.2.C, the limit set is a fractal object with a non-integer dimension. We will discuss the concept of dimension in the next section in more detail.

One way of identifying chaotic behavior is using the Poincare surface of section technique. For example, let us consider the periodically driven pendulum again, and use a surface of section on the angle of the forcing term <f>. If we operate the system with £ = 0.4, a — 1 and ¡3 — 2/3, it converges to a periodic trajectory which gives a single point in the Poincare surface of section of Figure 5.4.b. If we operate it with £ = 0.4, a = 1.4 and /3 = 2/3, the dynamics would be richer, and we observe a fractal object resembling the shape in the projection of the attractor on the (<?, w)-plane (Figure 5.4.d). This kind of an attractor is called a strange attractor.

The manifestation of chaos in the dynamics of a system is often associated with a sensitive dependence on its initial conditions. If we initialize our driven pendulum with slightly different initial conditions around its strange attractor, initially nearby trajectories diverge exponentially in time as shown by solid and dotted curves in Figure 5.5.a.

If we observe a block of initial conditions, shown in Figure 5.5.b, for 4 units of simulation time, the volume element that we started with shrinks in one direction, and is stretched in another. If we keep on observing the system, since the trajectories stay confined in a certain region of the phase space, the volume element cannot perform this shrinking and stretching without eventually folding on itself (Figure 5.5.c). This stretch-and-fold routine repeats itself as the dynamics further evolves. Hence, we will eventually find points that were arbitrarily close initially, separated in the phase space by a finite distance. In fact, the stretch-and-fold is the very mechanism that generates the fractal set of the strange attractor. □

Figure 5.2. Phase space of pendulum, projected on (0, w) plane, (a) Trajectories for no friction case with several initial conditions either oscillating or rotating around the stable steady state, (b) Trajectories for several initial conditions converge to the stable steady state, when there is friction, (c) The chaotic trajectory of the driven pendulum with friction.

Figure 5.2. Phase space of pendulum, projected on (0, w) plane, (a) Trajectories for no friction case with several initial conditions either oscillating or rotating around the stable steady state, (b) Trajectories for several initial conditions converge to the stable steady state, when there is friction, (c) The chaotic trajectory of the driven pendulum with friction.

Figure 5.3. Trajectories near steady states of the pendulum, 0 = iir (a) for i — even, and (b) for i — odd.

Figure 5.3. Trajectories near steady states of the pendulum, 0 = iir (a) for i — even, and (b) for i — odd.

There are several different measures that quantify this fingerprint of chaos. For example, the exponential divergence and convergence of nearby trajectories in different axial directions is measured by the Lyapunov exponents of the system. An nth order system has n Lyapunov exponents associated with the exponential rate of growth or shrinkage along its principal axes, and the set of all n Lyapunov exponents of a system is called its Lyapunov spectrum.

The notion of Lyapunov exponents can best be visualized by considering the experiment of putting a droplet of ink in a glass of water. The sphere described by the ink at the instant it is dropped, represents a family of nearby trajectories. In the course of time, the droplet gets deformed slowly, first into an ellipsoid, and then diffuses in the liquid. If we watch the droplet in slow motion, we can see that it is stretched in some directions, and squeezed in others. After some time, we see a folding occurring, as if to keep the droplet in the glass. In this analogy, the stretch of the droplet corresponds to a positive Lyapunov exponent, and the squeeze to a negative one. Since the phase volume is conserved in this system, i.e., total amount of ink in the glass is constant, the amount of squeezes is equal to the amount of stretches. Therefore, the sum of the Lyapunov exponents is equal to zero.

To put the idea in mathematical context, consider the autonomous continuous flow of Eq. (5.1), and observe the long-term evolution of an infinitesimal n-sphere of initial conditions. As in the ink analogy, the sphere will become an n-ellipsoid due to the locally deforming nature of the flow.

Figure 5.4. Periodically driven pendulum (a) goes to a limit cycle for £ = 0.4, a — 1 and (3 = 2/3. (b) Strobing it with a frequency that is a multiple of the oscillation frequency results in a single point in the Poincare section, (c) If we operate the system with £ = 0.4, a = 1.4 and (3 = 2/3, it leads to this chaotic orbit, and (d) strobing this motion results in a fractal object in the Poincare section.

Figure 5.4. Periodically driven pendulum (a) goes to a limit cycle for £ = 0.4, a — 1 and (3 = 2/3. (b) Strobing it with a frequency that is a multiple of the oscillation frequency results in a single point in the Poincare section, (c) If we operate the system with £ = 0.4, a = 1.4 and (3 = 2/3, it leads to this chaotic orbit, and (d) strobing this motion results in a fractal object in the Poincare section.

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