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Partial control of chaos: How to avoid undesirable behaviors with small controls in presence of noise
1.  Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain 
2.  Institute for New Economic Thinking at the Oxford Martin School, Mathematical Institute, University of Oxford, Walton Well Road, Eagle House OX2 6ED, Oxford, UK 
3.  Department of Applied Informatics, Kaunas University of Technology, Studentu 50415, Kaunas LT51368, Lithuania 
4.  Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA 
The presence of a nonattractive chaotic set, also called chaotic saddle, in phase space implies the appearance of a finite time kind of chaos that is known as transient chaos. For a given dynamical system in a certain region of phase space with transient chaos, trajectories eventually abandon the chaotic region escaping to an external attractor, if no external intervention is done on the system. In some situations, this attractor may involve an undesirable behavior, so the application of a control in the system is necessary to avoid it. Both, the nonattractive nature of transient chaos and eventually the presence of noise may hinder this task. Recently, a new method to control chaos called partial control has been developed. The method is based on the existence of a set, called the safe set, that allows to sustain transient chaos by only using a small amount of control. The surprising result is that the trajectories can be controlled by using an amount of control smaller than the amount of noise affecting it. We present here a broad survey of results of this control method applied to a wide variety of dynamical systems. We also review here all the variations of the partial control method that have been developed so far. In all the cases various systems of different dimensionality are treated in order to see the potential of this method. We believe that this method is a step forward in controlling chaos in presence of disturbances.
References:
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R. Capeáns, J. Sabuco and M. A. F. Sanjuán, When less is more: Partial control to avoid extinction of predators in an ecological model, Ecol. Complex., 19 (2014), 18. 
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R. Capeáns, J. Sabuco, M. A. F. Sanjuán and J. A. Yorke, Partially controlling transient chaos in the Lorenz equations, Phil. Trans. R. Soc. A, 375 (2017), 20160211, 18pp. doi: 10.1098/rsta.2016.0211. 
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R. Capeáns, J. Sabuco and M. A. F. Sanjuán, Parametric partial control of chaotic systems, Nonlinear Dyn., 2 (2016), 869876. doi: 10.1007/s1107101629294. 
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R. Capeáns, J. Sabuco, M. A. F. Sanjuán and J. A. Yorke, Partial control of delaycoordinate maps, Nonlinear Dyn, (2018), 111. 
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J. Duarte, C. Januário, N. Martins and J. Sardanyés, Chaos and crises in a model for cooperative hunting:a symbolic dynamics approach, Chaos, 58 (2009), 863883. 
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P. Holmes, A Nonlinear Oscillator with a Strange Attractor, Phil. Trans. R. Soc. Soc. Lond. Ser. A, 292 (1979), 419448. doi: 10.1098/rsta.1979.0068. 
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V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), 171. doi: 10.1016/00255564(92)90078B. 
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V. In, M. L. Spano, J. D. Neff, W. L. Ditto, C. S. Daw, K. D. Edwards and K. Nguyen, Maintenance of chaos in a computational model of thermal pulse combustor, Chaos, 7 (1997), 605613. doi: 10.1063/1.166260. 
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J. Jacobs, E. Ott, T. Antonsen and J. A. Yorke, Modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps, Physica D, 110 (1997), 117. doi: 10.1016/S01672789(97)00122X. 
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I. V. Kolmanovski and E. G. Gilbert, Multimode regulators for systems with state and control constraints and disturbance inputs, Control Using Logicbased Switching (Block Island, RI, 1995), 104117, Lect. Notes Control Inf. Sci., 222, Springer, London, 1997. doi: 10.1007/BFb0036088. 
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K. McCann and P. Yodzis, Bifurcation structure of a threespecies food chain model, Theo. Pop. Biol., 48 (1995), 93125. doi: 10.1006/tpbi.1995.1023. 
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M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.Math. Soc. Japan, 24 (1942), 551559. 
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Z. Neufeld and T. Tél, Advection in chaotically timedependent open flows, Phys.Rev.E, 57 (1995), 28322842. doi: 10.1103/PhysRevE.57.2832. 
[23] 
E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 11961199. doi: 10.1103/PhysRevLett.64.1196. 
[24] 
M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Soliton Fract., 27 (2006), 395403. doi: 10.1016/j.chaos.2005.03.045. 
[25] 
J. Sabuco, S. Zambrano, M. A. F. Sanjuán and J. A. Yorke, Finding safety in partially controllable chaotic systems, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 42744280. doi: 10.1016/j.cnsns.2012.02.033. 
[26] 
J. Sabuco, S. Zambrano, M. A. F. Sanjuán and J. A. Yorke, Dynamics of partial control, Chaos, 22 (2012), 047507, 9pp. doi: 10.1063/1.4754874. 
[27] 
M. A. F. Sanjuán, J. Kennedy, C. Grebogi and J. A. Yorke, Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders, Chaos, 7 (1997), 125138. doi: 10.1063/1.166244. 
[28] 
M. A. F. Sanjuán, J. Kennedy, E. Ott and J. A. Yorke, Indecomposable Continua and the characterization of strange sets in Nonlinear Dynamics, Phys. Rev. Lett., 78 (1997), 18921895. 
[29] 
I. B. Schwartz and I. Triandaf, The slow invariant manifold of a conservative pendulumoscillator system, Int. J. Bifurcation Chaos, 6 (1996), 673692. 
[30] 
I. B. Schwartz and I. Triandaf, Sustainning chaos by using basin boundary saddles, Phys. Rev. Lett., 77 (1996), 47404743. 
[31] 
S.F. Wang, X.C. Li, F. Xia and Z.S. Xie, The novel control method of three dimensional discrete hyperchaotic Hénon map, Appl.Math.Comput, 247 (2014), 487493. doi: 10.1016/j.amc.2014.09.011. 
[32] 
W. Yang, M. Ding, A. J. Mandell and E. Ott, Preserving chaos: Control strategies to preserve complex dynamics with potential relevance to biological disorders, Phys. Rev. E, 51 (1995), 102110. doi: 10.1103/PhysRevE.51.102. 
[33] 
J. A. Yorke and E. D. Yorke, Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys., 21 (1979), 263277. doi: 10.1007/BF01011469. 
show all references
References:
[1] 
J. Aguirre and M. A. F. Sanjuán, Unpredictable behavior in the duffing oscillator: Wada Basins, Physica D, 171 (2002), 4151. doi: 10.1016/S01672789(02)005651. 
[2] 
D. P. Bertsekas, Infinitetime reachability of statespace regions by using feedback control, IEEE Trans. Autom. Control, 17 (1972), 604613. 
[3] 
D. P. Bertsekas and I. B. Rhodes, On the minimax reachability of target set and target tubes, Automatica, 7 (1971), 233247. doi: 10.1016/00051098(71)900665. 
[4] 
F. Blanchini, Set invariance in control, Automatica, 35 (1999), 17471767. doi: 10.1016/S00051098(99)001132. 
[5] 
R. Capeáns, J. Sabuco and M. A. F. Sanjuán, Parametric partial control of chaotic systems, Nonlinear Dyn., 86 (2016), 869876. doi: 10.1007/s1107101629294. 
[6] 
R. Capeáns, J. Sabuco and M. A. F. Sanjuán, When less is more: Partial control to avoid extinction of predators in an ecological model, Ecol. Complex., 19 (2014), 18. 
[7] 
R. Capeáns, J. Sabuco, M. A. F. Sanjuán and J. A. Yorke, Partially controlling transient chaos in the Lorenz equations, Phil. Trans. R. Soc. A, 375 (2017), 20160211, 18pp. doi: 10.1098/rsta.2016.0211. 
[8] 
R. Capeáns, J. Sabuco and M. A. F. Sanjuán, Parametric partial control of chaotic systems, Nonlinear Dyn., 2 (2016), 869876. doi: 10.1007/s1107101629294. 
[9] 
R. Capeáns, J. Sabuco, M. A. F. Sanjuán and J. A. Yorke, Partial control of delaycoordinate maps, Nonlinear Dyn, (2018), 111. 
[10] 
D. Dangoisse, P. Glorieux and D. Hannequin, Laser chaotic attractors in crisis, Phys. Rev. Lett., 57 (1986), 26572660. doi: 10.1103/PhysRevLett.57.2657. 
[11] 
M. Dhamala and Y. C. Lai, Controlling transient chaos in deterministic flows with applications. to electrical power systems and ecology, Phys. Rev. E, 59 (1999), 16461655. doi: 10.1103/PhysRevE.59.1646. 
[12] 
J. Duarte, C. Januário, N. Martins and J. Sardanyés, Chaos and crises in a model for cooperative hunting:a symbolic dynamics approach, Chaos, 58 (2009), 863883. 
[13] 
R. Genesio, M. Tartaglia and A. Vicino, On the estimate of asymptotic stability regions: Stateof art and new proposal, IEEE Trans. Autom. Control, 30 (1985), 747755. doi: 10.1109/TAC.1985.1104057. 
[14] 
P. Holmes, A Nonlinear Oscillator with a Strange Attractor, Phil. Trans. R. Soc. Soc. Lond. Ser. A, 292 (1979), 419448. doi: 10.1098/rsta.1979.0068. 
[15] 
V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci., 111 (1992), 171. doi: 10.1016/00255564(92)90078B. 
[16] 
V. In, M. L. Spano, J. D. Neff, W. L. Ditto, C. S. Daw, K. D. Edwards and K. Nguyen, Maintenance of chaos in a computational model of thermal pulse combustor, Chaos, 7 (1997), 605613. doi: 10.1063/1.166260. 
[17] 
J. Jacobs, E. Ott, T. Antonsen and J. A. Yorke, Modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps, Physica D, 110 (1997), 117. doi: 10.1016/S01672789(97)00122X. 
[18] 
J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67 (1979), 93108. doi: 10.1007/BF01221359. 
[19] 
I. V. Kolmanovski and E. G. Gilbert, Multimode regulators for systems with state and control constraints and disturbance inputs, Control Using Logicbased Switching (Block Island, RI, 1995), 104117, Lect. Notes Control Inf. Sci., 222, Springer, London, 1997. doi: 10.1007/BFb0036088. 
[20] 
K. McCann and P. Yodzis, Bifurcation structure of a threespecies food chain model, Theo. Pop. Biol., 48 (1995), 93125. doi: 10.1006/tpbi.1995.1023. 
[21] 
M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.Math. Soc. Japan, 24 (1942), 551559. 
[22] 
Z. Neufeld and T. Tél, Advection in chaotically timedependent open flows, Phys.Rev.E, 57 (1995), 28322842. doi: 10.1103/PhysRevE.57.2832. 
[23] 
E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 11961199. doi: 10.1103/PhysRevLett.64.1196. 
[24] 
M. Perc and M. Marhl, Chaos in temporarily destabilized regular systems with the slow passage effect, Chaos Soliton Fract., 27 (2006), 395403. doi: 10.1016/j.chaos.2005.03.045. 
[25] 
J. Sabuco, S. Zambrano, M. A. F. Sanjuán and J. A. Yorke, Finding safety in partially controllable chaotic systems, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 42744280. doi: 10.1016/j.cnsns.2012.02.033. 
[26] 
J. Sabuco, S. Zambrano, M. A. F. Sanjuán and J. A. Yorke, Dynamics of partial control, Chaos, 22 (2012), 047507, 9pp. doi: 10.1063/1.4754874. 
[27] 
M. A. F. Sanjuán, J. Kennedy, C. Grebogi and J. A. Yorke, Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders, Chaos, 7 (1997), 125138. doi: 10.1063/1.166244. 
[28] 
M. A. F. Sanjuán, J. Kennedy, E. Ott and J. A. Yorke, Indecomposable Continua and the characterization of strange sets in Nonlinear Dynamics, Phys. Rev. Lett., 78 (1997), 18921895. 
[29] 
I. B. Schwartz and I. Triandaf, The slow invariant manifold of a conservative pendulumoscillator system, Int. J. Bifurcation Chaos, 6 (1996), 673692. 
[30] 
I. B. Schwartz and I. Triandaf, Sustainning chaos by using basin boundary saddles, Phys. Rev. Lett., 77 (1996), 47404743. 
[31] 
S.F. Wang, X.C. Li, F. Xia and Z.S. Xie, The novel control method of three dimensional discrete hyperchaotic Hénon map, Appl.Math.Comput, 247 (2014), 487493. doi: 10.1016/j.amc.2014.09.011. 
[32] 
W. Yang, M. Ding, A. J. Mandell and E. Ott, Preserving chaos: Control strategies to preserve complex dynamics with potential relevance to biological disorders, Phys. Rev. E, 51 (1995), 102110. doi: 10.1103/PhysRevE.51.102. 
[33] 
J. A. Yorke and E. D. Yorke, Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys., 21 (1979), 263277. doi: 10.1007/BF01011469. 
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