Bringing order to disorder: a method for stabilising a chaotic system around an arbitrary unstable periodic orbit. (English) Zbl 1533.37163
The authors propose a method for creating an arbitrary periodic point in the Poincaré section of a chaotic dynamical system which is then stabilized by the chaos control method of E. Ott et al. [Phys. Rev. Lett. 64, No. 11, 1196–1199 (1990; Zbl 0964.37501)]. In order to find a small control input that changes the dynamics such that a prescribed unstable periodic orbit exists in the Poincaré map an optimal control problem with piecewise constant controls is solved. The method is illustrated by applying it to the driven and damped Josephson junction in a chaotic parameter regime.
Reviewer: Jörg Härterich (Bochum)
MSC:
| 37M25 | Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) |
| 37N35 | Dynamical systems in control |
| 37C27 | Periodic orbits of vector fields and flows |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34H10 | Chaos control for problems involving ordinary differential equations |
| 65P20 | Numerical chaos |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Citations:
Zbl 0964.37501References:
| [1] | Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501 |
| [2] | Boccaletti, S.; Arecchi, F. T., Adaptive control of chaos, Europhys. Lett., 31, 3, 127-132 (1995) |
| [3] | Barone, A.; Paterno, G., Physics and Applications of the Josephson Effect, vol. 1 (1982), Wiley Online Library |
| [4] | Dauxois, T.; Peyrard, M., Physics of Solitons (2006), Cambridge University Press · Zbl 1192.35001 |
| [5] | Online, T., Mathworks MATLAB (2023), Matlab homepage. https://se.mathworks.com |
| [6] | Kautz, R. L., Chaotic states of rf-biased Josephson junctions, J. Appl. Phys., 52, 10, 6241-6246 (1981) |
| [7] | Kautz, R., Chaos in Josephson circuits, IEEE Trans. Magn., 19, 3, 465-474 (1983) |
| [8] | Kautz, R. L., Global stability of the chaotic state near an interior crisis, (Christiansen, P. L.; Parmentier, R. D., Structure, Coherence and Chaos in Dynamical Systems (1989), Manchester University Press: Manchester University Press Manchester and New York) |
| [9] | Sørensen, M. P.; Davidson, A.; Pedersen, N. F.; Pagano, S., Crises in a driven Josephson junction studied by cell mapping, Phys. Rev. A, 38, 10, 5384-5390 (1988), Copyright (1988) by the American Physical Society. |
| [10] | Grebogi, C.; Ott, E.; Yorke, J. A., Crises, sudden changes in chaotic attractors, and transient chaos, Physica D, 7, 1-3, 181-200 (1983) · Zbl 0561.58029 |
| [11] | Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.-M., Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 1: Theory, Meccanica, 15, 1, 9-20 (1980) · Zbl 0488.70015 |
| [12] | Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica D, 16, 3, 285-317 (1985) · Zbl 0585.58037 |
| [13] | Rawlings, J. B.; Mayne, D. Q.; Diehl, M. M., Model Predictive Control: Theory, Computation, and Design, vol. 1 (2019), Nob Hill Publishing, LCC |
| [14] | C. Thilker, H. Bergsteinsson, P. Bacher, H. Madsen, D. Cali, R. Junker, Non-linear Model Predictive Control for Smart Heating of Buildings, in: Proceedings of Cold Climate HVAC & Energy 2021, Cold Climate HVAC; Energy 2021 ; Conference date: 20-04-2021 Through 21-04-2021, 2021. |
| [15] | Clarke, F. H., The generalized problem of bolza, SIAM J. Control Optim., 14, 4, 682-699 (1976) · Zbl 0333.49023 |
| [16] | SQP-methods for solving optimal control problems with control and state constraints: Adjoint variables, sensitivity analysis and real-time control, J. Comput. Appl. Math. 120, (1), (2000) 85-108, http://dx.doi.org/10.1016/S0377-0427(00)00305-8. · Zbl 0963.65070 |
| [17] | Byrd, R. H.; Hribar, M. E.; Nocedal, J., An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9, 4, 877-900 (1999) · Zbl 0957.65057 |
| [18] | Wächter, A.; Biegler, L. T., On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106, 25-57 (2006) · Zbl 1134.90542 |
| [19] | Andersson, J. A.E.; Gillis, J.; Horn, G.; Rawlings, J. B.; Diehl, M., CasADi – a software framework for nonlinear optimization and optimal control, Math. Program. Comput. (2018) |
| [20] | Dressler, U.; Nitsche, G., Controlling chaos using time delay coordinates, Phys. Rev. Lett., 68, 1-4 (1992) |
| [21] | Arecchi, F.; Boccaletti, S.; Ciofini, M.; Meucci, R.; Grebogi, C., The control of chaos: Theoretical schemes and experimental realizations, Int. J. Bifurcation Chaos, 8, 08, 1643-1655 (1998) · Zbl 0941.93529 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
