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Selective Pyragas control of Hamiltonian systems

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  • 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.

    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.

    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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