doi: 10.3934/dcdsb.2021281
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Stabilizing multiple equilibria and cycles with noisy prediction-based control

1. 

Dept. of Math. and Stats., University of Calgary, 2500 University Drive N.W. Calgary, AB, T2N 1N4, Canada

2. 

Department of Mathematics, the University of the West Indies, Mona Campus, Kingston, Jamaica

Received  May 2020 Revised  July 2021 Early access November 2021

Fund Project: E. Braverman is a corresponding author. The first author is supported by NSERC grant RGPIN-2020-03934

Pulse stabilization of cycles with Prediction-Based Control including noise and stochastic stabilization of maps with multiple equilibrium points is analyzed for continuous but, generally, non-smooth maps. Sufficient conditions of global stabilization are obtained. Introduction of noise can relax restrictions on the control intensity. We estimate how the control can be decreased with noise and verify it numerically.

Citation: Elena Braverman, Alexandra Rodkina. Stabilizing multiple equilibria and cycles with noisy prediction-based control. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021281
References:
[1]

M. Benaïm and S. J. Schreiber, Persistence and extinction for stochastic ecological models with internal and external variables, J. Math. Biol., 79 (2019), 393-431.  doi: 10.1007/s00285-019-01361-4.  Google Scholar

[2]

E. BravermanC. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31.  doi: 10.1016/j.physd.2016.02.004.  Google Scholar

[3]

E. Braverman, C. Kelly and A. Rodkina, Stabilization of cycles with stochastic prediction-based and target-oriented control, Chaos, 30 (2020), 093116, 15 pp. doi: 10.1063/1.5145304.  Google Scholar

[4]

E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201.  doi: 10.1016/j.camwa.2012.01.013.  Google Scholar

[5]

E. Braverman and A. Rodkina, Stochastic control stabilizing unstable or chaotic maps, J. Difference Equ. Appl., 25 (2019), 151-178.  doi: 10.1080/10236198.2018.1561882.  Google Scholar

[6]

E. Braverman and A. Rodkina, Stochastic difference equations with the Allee effect, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 5929-5949.  doi: 10.3934/dcds.2016060.  Google Scholar

[7]

P. Cull, Global stability of population models, Bull. Math. Biol., 43 (1981), 47-58.   Google Scholar

[8]

E. Liz and D. Franco, Global stabilization of fixed points using predictive control, Chaos, 20 (2010), 023124, 9 pages. doi: 10.1063/1.3432558.  Google Scholar

[9]

E. Liz and C. Pótzsche, PBC-based pulse stabilization of periodic orbits, Phys. D, 272 (2014), 26-38.  doi: 10.1016/j.physd.2014.01.003.  Google Scholar

[10]

P. Hitczenko and G. Medvedev, Stability of equilibria of randomly perturbed maps, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 369-381.  doi: 10.3934/dcdsb.2017017.  Google Scholar

[11]

M. Nag and S. Poria, Synchronization in a network of delay coupled maps with stochastically switching topologies, Chaos Solitons Fractals, 91 (2016), 9-16.  doi: 10.1016/j.chaos.2016.04.022.  Google Scholar

[12]

M. Porfiri and I. Belykh, Memory matters in synchronization of stochastically coupled maps, SIAM J. Appl. Dyn. Syst., 16 (2017), 1372-1396.  doi: 10.1137/17M111136X.  Google Scholar

[13]

S. J. Schreiber, Coexistence in the face of uncertainty, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, 349–384, Fields Inst. Commun., 79, Springer, New York, 2017. doi: 10.1007/978-1-4939-6969-2_12.  Google Scholar

[14]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

[15]

A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

[16]

T. Ushio and S. Yamamoto, Prediction-based control of chaos, Phys. Lett. A, 264 (1999), 30-35.  doi: 10.1016/S0375-9601(99)00782-3.  Google Scholar

show all references

References:
[1]

M. Benaïm and S. J. Schreiber, Persistence and extinction for stochastic ecological models with internal and external variables, J. Math. Biol., 79 (2019), 393-431.  doi: 10.1007/s00285-019-01361-4.  Google Scholar

[2]

E. BravermanC. Kelly and A. Rodkina, Stabilisation of difference equations with noisy prediction-based control, Physica D, 326 (2016), 21-31.  doi: 10.1016/j.physd.2016.02.004.  Google Scholar

[3]

E. Braverman, C. Kelly and A. Rodkina, Stabilization of cycles with stochastic prediction-based and target-oriented control, Chaos, 30 (2020), 093116, 15 pp. doi: 10.1063/1.5145304.  Google Scholar

[4]

E. Braverman and E. Liz, On stabilization of equilibria using predictive control with and without pulses, Comput. Math. Appl., 64 (2012), 2192-2201.  doi: 10.1016/j.camwa.2012.01.013.  Google Scholar

[5]

E. Braverman and A. Rodkina, Stochastic control stabilizing unstable or chaotic maps, J. Difference Equ. Appl., 25 (2019), 151-178.  doi: 10.1080/10236198.2018.1561882.  Google Scholar

[6]

E. Braverman and A. Rodkina, Stochastic difference equations with the Allee effect, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 5929-5949.  doi: 10.3934/dcds.2016060.  Google Scholar

[7]

P. Cull, Global stability of population models, Bull. Math. Biol., 43 (1981), 47-58.   Google Scholar

[8]

E. Liz and D. Franco, Global stabilization of fixed points using predictive control, Chaos, 20 (2010), 023124, 9 pages. doi: 10.1063/1.3432558.  Google Scholar

[9]

E. Liz and C. Pótzsche, PBC-based pulse stabilization of periodic orbits, Phys. D, 272 (2014), 26-38.  doi: 10.1016/j.physd.2014.01.003.  Google Scholar

[10]

P. Hitczenko and G. Medvedev, Stability of equilibria of randomly perturbed maps, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 369-381.  doi: 10.3934/dcdsb.2017017.  Google Scholar

[11]

M. Nag and S. Poria, Synchronization in a network of delay coupled maps with stochastically switching topologies, Chaos Solitons Fractals, 91 (2016), 9-16.  doi: 10.1016/j.chaos.2016.04.022.  Google Scholar

[12]

M. Porfiri and I. Belykh, Memory matters in synchronization of stochastically coupled maps, SIAM J. Appl. Dyn. Syst., 16 (2017), 1372-1396.  doi: 10.1137/17M111136X.  Google Scholar

[13]

S. J. Schreiber, Coexistence in the face of uncertainty, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, 349–384, Fields Inst. Commun., 79, Springer, New York, 2017. doi: 10.1007/978-1-4939-6969-2_12.  Google Scholar

[14]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020.  Google Scholar

[15]

A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

[16]

T. Ushio and S. Yamamoto, Prediction-based control of chaos, Phys. Lett. A, 264 (1999), 30-35.  doi: 10.1016/S0375-9601(99)00782-3.  Google Scholar

Figure 1.  The second iteration of the Ricker map for $ r = 2.7 $
Figure 2.  The fourth iteration of the Ricker map for $ r = 2.6 $
Figure 3.  A bifurcation diagram for the second iterate of the Ricker map with $ r = 2.7 $, $ \alpha \in (0.1, 0.3) $ and (left) no noise, (right) $ \ell = 0.15 $
Figure 4.  A bifurcation diagram for the third iterate of the Ricker map with $ r = 3.5 $ for $ \alpha\in (0.75, 0.9) $ and (left) without noise, (right) $ \ell = 0.06 $. The last bifurcation leading to two stable equilibrium points occurs for $ \alpha \approx 0.88 $ in the deterministic case and $ \alpha<0.86 $ in the stochastic case
Figure 5.  The graph of the map defined in (4.1), together with $ y = x $
Figure 6.  A bifurcation diagram for the map defined in \protect{(4.1)} with $ \alpha \in (0.45, 0.65) $ and (left) no noise, we have two stable equilibrium points starting from $ \alpha\approx 0.605 $, (right) for Bernoulli noise with $ \ell = 0.04 $, the last bifurcation happens for smaller $ \alpha\approx 0.535 $. Here the two attractors correspond to two stable equilibrium points with separate basins of attraction, not to a cycle
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