TY - JOUR

T1 - Numerical inclusion of exact periodic solutions for time delay Duffing equation

AU - Oishi, Shin'ichi

N1 - Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2020/7

Y1 - 2020/7

N2 - Numerical inclusion results for exact periodic solutions are presented for the time delay autonomous Duffing equations. Constructive implicit function theorem is used for including one dimensional solution manifolds consisting of exact periodic solutions. A conjecture of a lower bound for a number of periodic solutions is given as a function of the time delay. If the delay time is less than 30, we have proved this conjecture using verified numerical computations. Theory for proving the existence of periodic solutions of the forced delay Duffing equation is proposed based on the verified numerical computations. The forced term is sinusoidal waves. Stress is on a study of the bifurcation of periodic solutions synchronizing to the external forces. A rich bifurcation phenomena of periodic solutions are reported taking the delay time as parameters. Especially, a kind of fractal structure is observed concerning resonance peaks.

AB - Numerical inclusion results for exact periodic solutions are presented for the time delay autonomous Duffing equations. Constructive implicit function theorem is used for including one dimensional solution manifolds consisting of exact periodic solutions. A conjecture of a lower bound for a number of periodic solutions is given as a function of the time delay. If the delay time is less than 30, we have proved this conjecture using verified numerical computations. Theory for proving the existence of periodic solutions of the forced delay Duffing equation is proposed based on the verified numerical computations. The forced term is sinusoidal waves. Stress is on a study of the bifurcation of periodic solutions synchronizing to the external forces. A rich bifurcation phenomena of periodic solutions are reported taking the delay time as parameters. Especially, a kind of fractal structure is observed concerning resonance peaks.

KW - Bifurcation of periodic solutions

KW - Constructive implicit function theorem

KW - Delay differential equation

KW - Fractal structure

KW - Inclusion of periodic solution

UR - http://www.scopus.com/inward/record.url?scp=85076200908&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85076200908&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2019.112620

DO - 10.1016/j.cam.2019.112620

M3 - Article

AN - SCOPUS:85076200908

SN - 0377-0427

VL - 372

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

M1 - 112620

ER -