# American Institute of Mathematical Sciences

• Previous Article
The optimal distribution of resources and rate of migration maximizing the population size in logistic model with identical migration
• DCDS-B Home
• This Issue
• Next Article
On the nearest stable $2\times 2$ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas
April  2021, 26(4): 2037-2053. doi: 10.3934/dcdsb.2020365

## Asymptotic dynamics of hermitian Riccati difference equations

 1 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan 2 Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan

* Corresponding author: Huey-Er Lin

(dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas)

Received  October 2020 Revised  November 2020 Published  April 2021 Early access  December 2020

In this paper, we consider the hermitian Riccati difference equations. Analogous to a Riccati differential equation, there is a connection between a Riccati difference equation and its associated linear difference equation. Based on the linear difference equation, we can obtain an explicit representation for the solution of the Riccati difference equation and define the extended solution. Further, we can characterize the asymptotic state of the extended solution and the rate of convergence. Constant equilibrium solutions, periodic solutions and closed limit cycles are exhibited in the investigation of asymptotic behavior of the hermitian Riccati difference equations.

Citation: Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2037-2053. doi: 10.3934/dcdsb.2020365
##### References:

show all references

##### References:
 [1] Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485 [2] José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1 [3] Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Mean--field control and Riccati equations. Networks & Heterogeneous Media, 2015, 10 (3) : 699-715. doi: 10.3934/nhm.2015.10.699 [4] Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034 [5] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [6] Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074 [7] Wei-guo Wang, Wei-chao Wang, Ren-cang Li. Deflating irreducible singular M-matrix algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 491-518. doi: 10.3934/naco.2013.3.491 [8] Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169 [9] John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 [10] Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017 [11] Ewa Schmeidel, Robert Jankowski. Asymptotically zero solution of a class of higher nonlinear neutral difference equations with quasidifferences. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2691-2696. doi: 10.3934/dcdsb.2014.19.2691 [12] Amira Khelifa, Yacine Halim. Global behavior of P-dimensional difference equations system. Electronic Research Archive, 2021, 29 (5) : 3121-3139. doi: 10.3934/era.2021029 [13] Gennadi M. Henkin, Victor M. Polterovich. A difference-differential analogue of the Burgers equations and some models of economic development. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 697-728. doi: 10.3934/dcds.1999.5.697 [14] Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577 [15] Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017 [16] Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575 [17] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 [18] Eugenia N. Petropoulou. On some difference equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2587-2594. doi: 10.3934/dcdsb.2017098 [19] Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055 [20] Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861

2020 Impact Factor: 1.327