Calculating the Lyapunov exponents of a piecewise-smooth soft impacting system with a time-delayed feedback controller

•Calculation of Lyapunov exponents of piecewise-smooth DDEs is studied.•We focus on the grazing event at where trajectory approaches discontinuity surface.•A grazing estimation algorithm is proposed to improve computational accuracy.•The method is validated on an impacting system under the delayed f...

Full description

Saved in:
Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2020-12, Vol.91, p.105451, Article 105451
Main Authors: Zhang, Zhi, Liu, Yang, Sieber, Jan
Format: Article
Language:English
Subjects:
Algorithms
Control systems
Delay differential equation
Dynamical systems
Eigenvalues
Feedback control
Grazing
Impact oscillator
Jacobi matrix method
Jacobian matrix
Jacobians
Liapunov exponents
Lyapunov exponents
Mathematical analysis
Nonlinear equations
Numerical integration
Numerical methods
Piecewise-smooth dynamical system
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:•Calculation of Lyapunov exponents of piecewise-smooth DDEs is studied.•We focus on the grazing event at where trajectory approaches discontinuity surface.•A grazing estimation algorithm is proposed to improve computational accuracy.•The method is validated on an impacting system under the delayed feedback control. Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. This paper studies how to build a variational equation for the efficient construction of Jacobians along trajectories of a delayed nonsmooth system. Trajectories of a piecewise-smooth system may encounter the so-called grazing event where the trajectory approaches a discontinuity surface in the state space in a non-transversal manner. For this event we develop a grazing point estimation algorithm to ensure the accuracy of trajectories for the nonlinear and the variational equations. We show that the eigenvalues of the Jacobian matrix computed by the algorithm converge with an order consistent with the order of the numerical integration method, therefore guaranteeing the reliability of the proposed numerical method. Finally, the method is demonstrated on a periodically forced impacting oscillator under the time-delayed feedback control.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2020.105451