# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021281
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 and 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. [2] E. Braverman, C. 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. [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. [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. [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. [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. [7] P. Cull, Global stability of population models, Bull. Math. Biol., 43 (1981), 47-58. [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. [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. [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. [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. [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. [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. [14] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020. [15] A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1. [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.

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. [2] E. Braverman, C. 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. [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. [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. [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. [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. [7] P. Cull, Global stability of population models, Bull. Math. Biol., 43 (1981), 47-58. [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. [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. [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. [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. [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. [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. [14] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.  doi: 10.1137/0135020. [15] A. N. Shiryaev, Probability, (2nd edition), Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1. [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.
The second iteration of the Ricker map for $r = 2.7$
The fourth iteration of the Ricker map for $r = 2.6$
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) $\ell = 0.15$
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
The graph of the map defined in (4.1), together with $y = x$
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
 [1] Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 [2] Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091 [3] Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014 [4] Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545 [5] Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052 [6] Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control and Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191 [7] John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 [8] Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459 [9] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 [10] Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623 [11] B. E. Ainseba, W. E. Fitzgibbon, M. Langlais, J. J. Morgan. An application of homogenization techniques to population dynamics models. Communications on Pure and Applied Analysis, 2002, 1 (1) : 19-33. doi: 10.3934/cpaa.2002.1.19 [12] Robert Carlson. Myopic models of population dynamics on infinite networks. Networks and Heterogeneous Media, 2014, 9 (3) : 477-499. doi: 10.3934/nhm.2014.9.477 [13] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [14] Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107 [15] MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 [16] G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517 [17] Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640-649. doi: 10.3934/proc.2009.2009.640 [18] Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365 [19] Xiujuan Wang, Mingshu Peng. Rich dynamics in some generalized difference equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3205-3212. doi: 10.3934/dcdss.2020191 [20] Eric Ruggieri, Sebastian J. Schreiber. The Dynamics of the Schoener-Polis-Holt model of Intra-Guild Predation. Mathematical Biosciences & Engineering, 2005, 2 (2) : 279-288. doi: 10.3934/mbe.2005.2.279

2021 Impact Factor: 1.497

## Metrics

• HTML views (280)
• Cited by (0)

• on AIMS