The four Lyapunov exponents
([LE.sub.i]) of the system in (1) are obtained by singular value decomposition as [LE.sub.1] = 0.1354, [LE.sub.2] = 0.0076, [LE.sub.3] = -0.2936, and [LE.sub.4] = -25.6091.
Caption: Figure 2: Box plot representing distribution of (a) correlation dimension (CD) and (b) largest Lyapunov exponent
(LLE) in EEG of encephalopathic patients and EEG of normal subjects.
Table 1: Comparative analysis of some dynamical systems by using the largest Lyapunov exponent
Caption: FIGURE 1: Bifurcation diagram of forced van der Pol oscillator for w = 0.45 and [eta] = 1 using (a) Poincare map and (b) largest nonzero Lyapunov exponent
Considering the limitations of the Lyapunov's stability theory for complicated nonlinear systems, in this section, Lyapunov exponent
method is applied to analyze the dynamic stability of the MUAV system in the case of manipulator movements.
Secondly, for a = 3.3, sample results showing the bifurcation diagram versus b and the related plot of Largest Lyapunov exponent
are provided in Figure 5.
The highly explored nonlinear signal analysis methods include reconstructed phase space analysis, Lyapunov exponents
, correlation dimension, detrended fluctuation analysis (DFA), recurrence plot, Poincare plot, approximate entropy, and sample entropy.
Entropy usually is quite difficult to obtain numerically compared to Lyapunov exponent
. Since every expansive map has positive Lyapunov exponent
, then by confirming the expansiveness of a map, chaos is possible if not guaranteed.
Figure 15 plots the evolution of the largest Lyapunov exponent
in a suspension system.
Baleanu, "Jacobian matrix algorithm for Lyapunov exponents
of the discrete fractional maps," Communications in Nonlinear Science and Numerical Simulation, vol.
In this case, the Lyapunov exponents
of the system are [L.sub.1] = 0.2566, [L.sub.2] = 0.0674, and [L.sub.3] = 0, [L.sub.4] = -19.2935, and the Lyapunov dimension is [D.sub.L] = 3.017.