Selective Pyragas control of Hamiltonian systems

Edward Hooton; Pavel Kravetc; Dmitrii Rachinskii; Qingwen Hu

doi:10.3934/dcdss.2019130

Article Contents

2019, Volume 12, Issue 7: 2019-2034. Doi: 10.3934/dcdss.2019130

This issue Previous Article Global symmetry-breaking bifurcations of critical orbits of invariant functionals Next Article Existence of positive ground state solutions for Choquard equation with variable exponent growth

Selective Pyragas control of Hamiltonian systems

1.
Department of Mathematical Sciences, University of Texas at Dallas, 800 W Campbell Road, Richardson, TX 75080, USA
2.
Department of Mathematics and Computer Science, Alcorn State University, 1000 ASU Drive, Lorman, MS 39096, USA

* Corresponding author
* Corresponding author

Received: December 31, 2017

Revised: April 30, 2018

Published: June 2019

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Abstract

We consider a Newton system which has a branch (surface) of neutrally stable periodic orbits. We discuss sufficient conditions which allow arbitrarily small delayed Pyragas control to make one selected cycle asymptotically stable. In the case of small amplitude periodic solutions we give conditions in terms of the asymptotic expansion of the right hand side, while in the case of larger cycles we frame the conditions in terms of the Floquet modes of the target orbit as a solution of the uncontrolled system.

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Mathematics Subject Classification: Primary: 34K20; Secondary: 34K13, 34K35.

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Figure 1. Panel (A): Bifurcation diagram of the controlled system for $ \tilde \kappa = (-0.02, 0.012) $. Points $ A $, $ B $ and $ C $ on the branch correspond to the delays $ \tau_A = 7.1857 $, $ \tau_B = 5.9655 $ and $ \tau_C = 5.1761 $, respectively. Panel (B): The same branch of periodic solutions has different stability properties for $ \hat\kappa = (-0.02, 0.02) $. Stable and unstable solutions are shown by solid and dashed lines, respectively.

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Figure 2. Control parameters plane. Conditions (18) of Theorem 2.1 are satisfied within the sector $ O_1OO_2 $. Sectors $ A_1 O A_2 $, $ B_1 O B_2 $, $ C_1 O C_2 $ are defined by conditions (23) and (24) for the periodic solutions indicated by points $ A $, $ B $ and $ C $, respectively, on Figure 1. Point $ \tilde{\kappa} = \left(-0.02, 0.012\right) $ corresponds to control parameters used in Figure Figure 1(A); parameters $ \hat{\kappa} = \left(-0.02, 0.02\right) $ are used in Figure 1(B)

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