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Research Papers

# Implementation of Periodicity Ratio in Analyzing Nonlinear Dynamic Systems: A Comparison With Lyapunov Exponent

[+] Author and Article Information
Liming Dai1

Mathematics and Physics, North China Electric Power University, Beijing 102206, P.R.C.; Industrial Systems Engineering, University of Regina, Regina, SK, S4S 0A2, Canadaliming̱dai@hotmail.com

Guoqing Wang

Industrial Systems Engineering, University of Regina, Regina, SK, S4S 0A2, Canada

1

Corresponding author.

J. Comput. Nonlinear Dynam 3(1), 011006 (Nov 12, 2007) (9 pages) doi:10.1115/1.2802581 History: Received February 22, 2007; Revised June 20, 2007; Published November 12, 2007

## Abstract

The present research intends to investigate the characteristics of the periodicity ratio and its implementation in analyzing the nonlinear behavior of dynamic systems governed by second-order differential equations. Numerical analyses on the nonlinear dynamic systems with employment of the periodicity ratio for diagnosing chaotic, regular, and irregular behaviors of dynamic systems are performed. To characterize the approach with periodicity ratio in distinguishing different behaviors of the nonlinear dynamic systems, a comparison of periodicity ratio with the widely used Lyapunov exponent in numerically assessing the responses of nonlinear dynamic systems is presented.

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## Figures

Figure 1

Lyapunov exponent corresponding to a chaotic response of a driven pendulum. Q=2 and ωD=2∕3.

Figure 2

State of motion determined with periodicity ratio and Lyapunov exponent for a driven pendulum. Q=2 and ωD=2∕3.

Figure 3

Trajectory of driven pendulum in phase space. Q=2 and ωD=2∕3.

Figure 4

Trajectory of driven pendulum in phase space. Q=2, ωD=2∕3, and F=1.4243.

Figure 5

Trajectory of driven pendulum in phase space. Q=2, ωD=2∕3, and F=1.093.

Figure 6

Trajectory of driven pendulum in phase space. Q=2, ωD=2∕3, F=1.032.

Figure 7

Bifurcation diagram of the driven pendulum with Q=2 and ωD=2∕3.

Figure 8

Phase diagram of the driven pendulum with F=1.2635, Q=2, and ωD=2∕3

Figure 9

Poincaré map of the driven pendulum with F=1.2635, Q=2, and ωD=2∕3

Figure 10

Phase diagram of the driven pendulum with F=1.2625, Q=2, and ωD=2∕3

Figure 11

Poincaré map of the driven pendulum with F=1.2625, Q=2, and ωD=2∕3

Figure 12

Phase diagram of the driven pendulum with F=1.2615, Q=2, and ωD=2∕3

Figure 13

Poincaré map of the driven pendulum with F=1.2615, Q=2, and ωD=2∕3

Figure 14

Poincaré map of the driven pendulum in chaos with Q=2 and ωD=2∕3

Figure 15

Periodic-chaotic region diagram of Duffing’s equation

Figure 16

Periodicity-ratio diagram corresponding to varying values of amplitude and frequency of the external excitation acting on the pendulum system

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