Secondly, the conventional dynamic analyses are carried out by Lyapunov exponents
, bifurcation diagrams, Poincaree maps, phase diagrams, and so forth.
Swinney, "Determining Lyapunov exponents
from a time series," Physica D: Nonlinear Phenomena, vol.
Pham, "Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents
and its SPICE implementation," Archives of Control Sciences, vol.
are computed for each a evidencing the bifurcations and chaotic attractors.
Then, the calculated Lyapunov exponents
will also be different based on Figure 2 and after n iteration, the trajectory distance between the disturbed MUAV system and the original system is as follows:
Caption: Figure 3: The bifurcation diagrams depicting the local maxima (black dots) and local minima (gray dots) of x(t) (a) and the largest Lyapunov exponents
(b) versus the parameter a for b = 3.
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.
This implies that Lyapunov exponents
measure the rate of divergence of orbits away from each other.
We employed bifurcation diagrams, phase portraits, Poincare maps, frequency spectra, and Lyapunov exponents
to explain periodic and chaotic motions in a vehicle suspension system that exerts hysteretic nonlinear damping forces.
Baleanu, "Jacobian matrix algorithm for Lyapunov exponents
of the discrete fractional maps," Communications in Nonlinear Science and Numerical Simulation, vol.