1 t where ei € lin with i = 1,..., n stand for the orthonormal basis vectors of the tangent flow around a fiducial trajectory[645]. The initial conditions of the state variables are set by the original system. Although the augmenting variables can be initialized arbitrarily, it would be a good practice to select an orthonormal set for e,, such as {e,} = In, and a neutral guess for Aj, such as Aj = 0. In the limit t —> oo, the set {A,} will give the Lyapunov spectrum of the system.

The concept of Lyapunov exponents is illustrated for continuous flows in this presentation, but it can be carried to discrete maps as well. For instance, if the map (or a time series data for that matter) represents samplings of a continuous flow, the amount of growth or contraction associated with the ¿th Lyapunov exponent will be <Ji = eXiAt, and is called the Lyapunov number. Therefore, when ith Lyapunov exponent is positive (negative) the ¿th Lyapunov number will be greater (less )than unity.

Unlike the continuous flows, discrete maps need not have a minimum phase space dimension to exhibit chaotic behavior. Since the values are attained at discrete instances, orbit crossings in data representations are mostly superfluous, hence do not pose the existence-uniqueness problems of continuous flows. Even a first order discrete map can produce chaotic behavior, as shown in the following example.

Example 4 Logistic map

The logistic map is a one dimensional nonlinear system, given by the difference equation x{i + 1) = fix(i)(1 - «(*)) (5.16)

which was originally proposed to model population dynamics in a limited resource environment [375]. The population size, x(i), at instant i is a normalized quantity. It can be easily shown that a choice of /x in the range [0,4] guarantees that, if we start with a physically meaningful population size, i.e., x(0) e [0,1], the population size stays in [0,1].

If we simulate the system with an initial condition x(0) = 0.1, we will obtain the results shown in Figure 5.7 for various ¡j, values. The system goes to a steady state for /j, = 1 and [x = 2, but as n is further increased to 2.9, the behavior of the convergence to a steady state is qualitatively different than the previous cases. It reaches the steady state in an oscillatory manner. For fi = 3.3, the oscillations are not damped anymore, and we have periodic behavior, every other value of Xi being equal for large i. The system is said to have a two-period oscillation in this regime. For fi = 3.5, the asymptotic behavior of the system is similar to the previous case. This time, we have a four-period oscillation though. The demonstrated increase in the period is actually common to many nonlinear systems, and is called period, doubling. The period-doubling mechanism is a route to chaos, that has been studied extensively, since it is encountered in many dynamical systems. One interesting finding is that, period doubling may be characterized by a universal number independent of the underlying dynamics. In our example, if we label the kth. period doubling value of /i with ¡xk, then

where S ~ 4.669 is the Feigenbaum number [150]. Naturally, the period doubling values of the parameter, . depend on the system dynamics. However, the number S is universal (i.e., the same) for all one-dimensional maps. A detailed study of the link between the Feigenbaum number and the period doublings is beyond the scope of this text. (Interested reader can check the rigorous works of Lanford [313] and Collet and Eckmann [109, 110]) However, its implication for our case is important, which suggests that as /x = 3.569 is approached, an infinite number of period doublings will occur.

Although different types of dynamics can be visualized by plotting the time evolution of the system for a specific parameter set (Figure 5.7), or by plotting orbits or Poincare surface of sections in the phase space, they axe far from representing the global behavior of the system for a range of parameter values. The bifurcation diagram provides a summary of the dynamics by plotting the essential dynamics for a range of parameter values. For example, if we plot the steady states of the system versus the system parameter, fi, we obtain the bifurcation diagram of Figure 5.8.a. The solid curve corresponds to the stable steady states, and the dashed curve to the unstable ones. The steady state loses its stability at /i = 3, which is where the period doubling occurs.

Alternatively, we can plot the limiting values of x(i) for the logistic map versus the system parameter, p, to obtain the bifurcation diagram of Figure 5.8.b. This bifurcation diagram summarizes the period doubling cascade that leads to chaos. It shows that, there is a period doubling at fj. = 3, and another around // — 3.45, and indeed the system goes chaotic around ¡j, = 3.569, as suggested by the Feigenbaum number. Looking at this bifurcation diagram, we not only have a complete picture of how the period doubling cascade results in chaotic behavior, but we may also disclose some types of behavior that we did not know have existed, such as the window around fi = 3.83, where the system exhibits periodic behavior. The logistic map is attracted by a three-period orbit when operated with

Figure 5.7. Simulation results of the logistic map for (a) pt = 1, (b) pt = 2, (c) n = 2.9, (d) pi = 3.3, (e) fx = 3.5, and (f) fi = 3.9.

Figure 5.7. Simulation results of the logistic map for (a) pt = 1, (b) pt = 2, (c) n = 2.9, (d) pi = 3.3, (e) fx = 3.5, and (f) fi = 3.9.

¡1 — 3.83 (Figure 5.9), which is an evidence that this map has a region in the parameter space {(x for this example) that it experiences chaotic behavior.

After this periodic window, we again observe a chaotic regime, but this time the chaos is reached via the intermittency route. There are three documented intermittency routes [540]

1. a real eigenvalue crosses the unit circle at +1;

2. a complex conjugate pair of eigenvalues cross the unit circle;

3. a real eigenvalue crosses the unit circle at -1.

The logistic map exhibits a type 1 intermittency after the three-period window. □

Apart from the period doubling and intermittency routes, there is a third route to chaos, called the quasiperiodicity route, originally suggested to understand turbulence [419], In this mechanism, the stable steady state of a dynamical system becomes unstable at a certain value of the bifurcation parameter, in a special manner. In a continuous system, with the changing bifurcation parameter, a complex conjugate pair of eigenvalues cross the imaginary axis, making the steady state unstable and creating a stable limit cycle around it. This transition of the complex conjugate eigenvalues through the imaginary axis with changing bifurcation parameter is called the Hopf bifurcation. After a Hopf bifurcation that makes the steady state unstable, the dynamics can have another Hopf bifurcation, this time making the stable limit cycle unstable. Right after this second bifurcation, the trajectory is confined on a torus, and the states of the system show a quasiperiodic behavior. Some further change in the bifurcation parameter may result in a third Hopf bifurcation, taking the trajectory on a three torus, which decays to a strange attractor after certain infinitesimal perturbations.

Example 5 Autocatalytic reactions Consider the cubic autocatalysis

as a paradigm for population dynamics of sexually reproducing species [64, 65], with kp and dp representing the birth and death rates of the species P, respectively. If we let these reactions occur in two coupled identical with decay

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