Figure 5.5. (a) Trajectories of two realizations of the periodically driven pendulum for two nearby initial conditions {e0,w0,(t>0) = (0.36564,-7.4964,100) (solid curve) and (6ยป0, u>0, <J>0) = (0.36564,-7.4964,100.1) (dotted curve), (b) How a rectangular block of initial points get deformed in a simulation interval of 4 units, stretching in one direction and squeezing in the other, (c) Same initial rectangular block after a simulation time of 10 units.

Using the length of the ith ellipsoidal principal axis Pi{t), we can define the ith Lyapunov exponent of the system from

when the limit exists. Xt are conventionally ordered from largest to the smallest. Note that, this definition is akin to the definition of eigenvalues for linear systems, but unlike the eigenvalues, there is no unique direction associated with a given Lyapunov exponent. This is understandable, since the eigenvalue is a local definition, and, characterizes a steady state, while the Lyapunov exponent is a time average associated with a principal axis, that continuously changes orientation as it evolves.

As one classifies linear systems using their eigenvalues, Lyapunov spectra can be used to classify the asymptotic behavior of nonlinear systems. For example, for a system to be dissipative, the sum of its Lyapunov exponents should be negative. Likewise, if we have a Hamiltonian system, the sum of its Lyapunov exponents should be zero, due to the volume preserving property of such systems. A continuous dynamical system is chaotic, if it has at least one positive Lyapunov exponent.

In the investigation of chaotic systems, we have mentioned that third-order systems have a special importance. For third order dissipative systems, we can easily classify the possible spectra of attractors in four groups, based on Lyapunov exponents.

Therefore, the last configuration is the only possible third-order chaotic system. However, in a continuous fourth-order dissipative system, there are three possible types of strange attractors with Lyapunov spectra (+, 0, โ, โ), (+, 0,0, โ) and (+, +, 0, -). Note that, all three configurations have at least one vanishing Lyapunov exponent. In fact, it is required by the theorem of Haken [215] that the system should have at least one zero Lyapunov exponent, if the trajectory of its attractor does not have a fixed point. The last case where there are two positive Lyapunov exponents is called the hyper chaos.

The classical Lyapunov exponent computation method of Wolf et al. [669] is based on observing the long time evolution of the axes of an infinitesimal sphere of states. It is implemented by defining the principal

Figure 5.6. Time evolution of the fiducial trajectory and the principal axis (axes), (a) The largest Lyapunov exponent is computed from the growth of length elements, (b) The sum of the largest two Lyapunov exponents is computed from the growth of area elements.

Figure 5.6. Time evolution of the fiducial trajectory and the principal axis (axes), (a) The largest Lyapunov exponent is computed from the growth of length elements, (b) The sum of the largest two Lyapunov exponents is computed from the growth of area elements.

axes, with initial conditions that are separated as small as the computer arithmetic allows, and by evolving these using the nonlinear model equations. The trajectory followed by the center of the sphere is called the fiducial trajectory. The principal axes are defined throughout the flow via the linearized equations of an initially orthonormal vector frame "anchored" to the fiducial trajectory. To implement the procedure, the fiducial trajectory on the attractor is integrated simultaneously with the vector tips defining n arbitrarily oriented orthonormal vectors. Eventually, each vector in the set tends to fall along the local direction of most rapid growth (or a least rapid shrink for a non-chaotic system). On the other hand, the collapse toward a common direction causes the tangent space orientation of all axis vectors to become indistinguishable. Therefore, after a certain interval, the principal axis vectors are corrected into an orthonormal set, using the Gram-Schmidt reorthonormalization. Projection of the evolved vectors onto the new orthonormal frame correctly updates the rates of growth of each of the principal axes, providing estimates of the Lyapunov exponents. Following this procedure, the rate of change of a length element, around the fiducial trajectory, as shown in Figure 5.6.a, would indicate the dominant Lyapunov exponent, with

Similarly, the rate of change of an area element, as shown in Figure 5.6.b would indicate the sum of the largest two Lyapunov exponents, with

The idea can be generalized to higher dimensions, considering volume elements for the largest three Lyapunov exponents, hypervolume elements for the largest four Lyapunov exponents, and so on.

The reorthonormalization procedure can further be implemented in every infinitesimal time step. This continuum limit of the procedure can be expressed by the set of differential equations

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