Application of average positive lyapunov in estimation of chaotic response peak excitation frequency of harmonically excited pendulum (2013)
"The fact that the drive parameters space of harmonically excited pendulum consist of mix parameters combination leading to different dynamics phenomena including chaotic and periodic responses is a strong motivation for this study aim at estimating the peak frequency that favour chaotic response. Simulation of pendulum and estimation of the average Lyapunov exponents by Grahm Schmidt Orthogonal rules at parameter nodal points selected from damp quality (2.0≤q≤ 4.0). excitation amplitude (0.9≤g ≤1.5) and drive frequency (0.5 ≤ ωD≤1.0) were effected using popular constant time step Runge-Kutta schemes (RK4, RK5 and RK5B) from two initial conditions through transient and steady periods. The impact of resolution on the measure of percentage of parameters combination leading to chaotic response (PPCLCR) was examined at resolution levels (RI to R5) for increasing drive frequency. The validation cases were from those reported by Gregory and Jerry (1990) for (ώᶹ,q,g≡ 2/3,4,1.5) and (ωυq,g≡ 2/3,4,1.5) simulated from (0. 0) initial conditions. Corresponding validation results compare well with reported results of Gregory and Jerry (1990). The estimated peak frequency (0.6 radian /s) is the same across studied resolutions initial conditions and Runge-Kutta schemes. The peak value of PPCLCR is 69.5. 69.4 and 69.4 respectively for RK4. RK5 and RK5B at initial conditions (0. 0). When initial conditions is (I. 0) the corresponding PPCLR value changes in significantly to 69.6. 69.7 and 69.6 for RK4, RK5 and RK5B. Therefore affirms the utility and reliability of Lyapunov exponent as chaotic response identification tool. "