November  2019, 12(7): 2019-2034. doi: 10.3934/dcdss.2019130

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

Received  January 2018 Revised  May 2018 Published  December 2018

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.

Citation: Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130
References:
[1]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21, URL https://doi.org/10.1145/513001.513002. doi: 10.1145/513001.513002.

[2]

K. Engelborghs, T. Luzyanina and G. Samaey, Dde-biftool v. 2.00: A matlab package for bifurcation analysis of delay differential equations.

[3]

T. Faria and L. T. Magalhães, Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201–224, URL https://doi.org/10.1006/jdeq.1995.1145. doi: 10.1006/jdeq.1995.1145.

[4]

B. FiedlerV. FlunkertP. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 319-341.  doi: 10.1098/rsta.2009.0232.

[5]

J. Hale, Theory of Functional Differential Equations, 2nd edition, Springer-Verlag, New York-Heidelberg, 1977, Applied Mathematical Sciences, Vol. 3.

[6]

E. Hooton, Z. Balanov, W. Krawcewicz and D. Rachinskii, Noninvasive stabilization of periodic orbits in $O_4$-symmetrically coupled systems near a Hopf bifurcation point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750087, 18pp, URL https://doi.org/10.1142/S0218127417500870. doi: 10.1142/S0218127417500870.

[7]

E. HootonP. Kravetc and D. Rachinskii, Restrictions to the use of time-delayed feedback control in symmetric settings, Continuous Dynamical Systems-B, 23 (2018), 543-556.  doi: 10.3934/dcdsb.2017207.

[8]

E. W. Hooton and A. Amann, Analytical limitation for time-delayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101.

[9]

W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hövel and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Physical Review E, 76 (2007), 026210, 11pp. doi: 10.1103/PhysRevE.76.026210.

[10]

J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, vol. 114 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996, URL https://doi.org/10.1007/978-1-4612-3968-0. doi: 10.1007/978-1-4612-3968-0.

[11]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976, Applied Mathematical Sciences, Vol. 19.

[12]

A. H. Nayfeh, Perturbation Methods, Wiley Classics Library, Wiley-Interscience [John Wiley & Sons], New York, 2000, URL https://doi.org/10.1002/9783527617609, Reprint of the 1973 original. doi: 10.1002/9783527617609.

[13]

K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics letters A, 170 (1992), 421-428. 

[14]

K. Pyragas and A. Tamaševičius, Experimental control of chaos by delayed self-controlling feedback, Physics Letters A, 180 (1993), 99-102. 

[15]

I. Schneider, Delayed feedback control of three diffusively coupled Stuart-Landau oscillators: a case study in equivariant Hopf bifurcation, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472.

[16]

J. Sieber, A. Gonzalez-Buelga, S. Neild, D. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Physical Review Letters, 100 (2008), 244101.

[17]

D. W. SukowM. E. BleichD. J. Gauthier and J. E. Socolar, Controlling chaos in a fast diode resonator using extended time-delay autosynchronization: Experimental observations and theoretical analysis, Chaos: An Interdisciplinary Journal of Nonlinear Science, 7 (1997), 560-576. 

[18]

M. Tlidi, A. Vladimirov, D. Pieroux and D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), 103904.

show all references

References:
[1]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21, URL https://doi.org/10.1145/513001.513002. doi: 10.1145/513001.513002.

[2]

K. Engelborghs, T. Luzyanina and G. Samaey, Dde-biftool v. 2.00: A matlab package for bifurcation analysis of delay differential equations.

[3]

T. Faria and L. T. Magalhães, Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201–224, URL https://doi.org/10.1006/jdeq.1995.1145. doi: 10.1006/jdeq.1995.1145.

[4]

B. FiedlerV. FlunkertP. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 319-341.  doi: 10.1098/rsta.2009.0232.

[5]

J. Hale, Theory of Functional Differential Equations, 2nd edition, Springer-Verlag, New York-Heidelberg, 1977, Applied Mathematical Sciences, Vol. 3.

[6]

E. Hooton, Z. Balanov, W. Krawcewicz and D. Rachinskii, Noninvasive stabilization of periodic orbits in $O_4$-symmetrically coupled systems near a Hopf bifurcation point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750087, 18pp, URL https://doi.org/10.1142/S0218127417500870. doi: 10.1142/S0218127417500870.

[7]

E. HootonP. Kravetc and D. Rachinskii, Restrictions to the use of time-delayed feedback control in symmetric settings, Continuous Dynamical Systems-B, 23 (2018), 543-556.  doi: 10.3934/dcdsb.2017207.

[8]

E. W. Hooton and A. Amann, Analytical limitation for time-delayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101.

[9]

W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hövel and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Physical Review E, 76 (2007), 026210, 11pp. doi: 10.1103/PhysRevE.76.026210.

[10]

J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, vol. 114 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996, URL https://doi.org/10.1007/978-1-4612-3968-0. doi: 10.1007/978-1-4612-3968-0.

[11]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976, Applied Mathematical Sciences, Vol. 19.

[12]

A. H. Nayfeh, Perturbation Methods, Wiley Classics Library, Wiley-Interscience [John Wiley & Sons], New York, 2000, URL https://doi.org/10.1002/9783527617609, Reprint of the 1973 original. doi: 10.1002/9783527617609.

[13]

K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics letters A, 170 (1992), 421-428. 

[14]

K. Pyragas and A. Tamaševičius, Experimental control of chaos by delayed self-controlling feedback, Physics Letters A, 180 (1993), 99-102. 

[15]

I. Schneider, Delayed feedback control of three diffusively coupled Stuart-Landau oscillators: a case study in equivariant Hopf bifurcation, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472.

[16]

J. Sieber, A. Gonzalez-Buelga, S. Neild, D. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Physical Review Letters, 100 (2008), 244101.

[17]

D. W. SukowM. E. BleichD. J. Gauthier and J. E. Socolar, Controlling chaos in a fast diode resonator using extended time-delay autosynchronization: Experimental observations and theoretical analysis, Chaos: An Interdisciplinary Journal of Nonlinear Science, 7 (1997), 560-576. 

[18]

M. Tlidi, A. Vladimirov, D. Pieroux and D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), 103904.

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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