August  2019, 24(8): 4457-4473. doi: 10.3934/dcdsb.2019127

On the eventual stability of asymptotically autonomous systems with constraints

1. 

School of Mathematics, Tianjin University, Tianjin 300072, China

2. 

Department of Mathematics, Civil Aviation University of China, Tianjin 300300, China

3. 

School of Mathematics, Tianjin Normal University, Tianjin 300387, China

* Corresponding author: Desheng Li

Dedicated to Professor Peter E. Kloeden on the occasion of his 70th birthday

Received  October 2018 Revised  January 2019 Published  June 2019

In this paper we first give a criterion on stability of equilibrium solutions for autonomous systems with constraints. Then we discuss the relationship between asymptotic behaviors of an asymptotically autonomous system with constraint and its limit system. Finally as an example, we revisit an extreme ideology model proposed in the literature and give a more detailed description on the dynamics of the system.

Citation: Jinlong Bai, Xuewei Ju, Desheng Li, Xiulian Wang. On the eventual stability of asymptotically autonomous systems with constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4457-4473. doi: 10.3934/dcdsb.2019127
References:
[1]

D. AldilaN. Nuraini and E. Soewono, Mathematical model for the spread of extreme ideology, AIP Conference Proceedings, 1651 (2015), 33-39.  doi: 10.1063/1.4914429.  Google Scholar

[2]

Z. Artstein, Limiting Equations and Stability of Nonautonomous Ordinary Differential Equations, in: J. P. LaSalle (ed.), The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.  Google Scholar

[3]

A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, Journal of Differential Equations, 233 (2007), 622-653.  doi: 10.1016/j.jde.2006.08.009.  Google Scholar

[4]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity Vol. 1: Theory of Epidemics, Wuerz, Winnipeg, Canada, 1995. Google Scholar

[5]

S. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, 251, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[6]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, Journal of Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

[7]

J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $ \mathbb{R}^{N}$, Journal of Differential Equations, 251 (2011), 1903-1922.  doi: 10.1016/j.jde.2011.04.002.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[9]

S. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Analysis, 46 (2001), 675-698.  doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[10]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[11]

D. Li and Z. Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J., 67 (2018), 583-621.  doi: 10.1512/iumj.2018.67.7292.  Google Scholar

[12]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[13]

K. MischaikowH. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[14]

R. Schnaubelt, Asymptotically autonomous parabolic evolution equations, Journal of Evolution Equations, 1 (2001), 19-37.  doi: 10.1007/PL00001363.  Google Scholar

[15]

A. Strauss and J. A. Yorke, Perturbing uniform asymptotically stable nonlinear systems, Journal of Differential Equations, 6 (1969), 452-483.  doi: 10.1016/0022-0396(69)90004-7.  Google Scholar

[16]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[17]

X. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonlinear Anal., 3 (1996), 43-66.   Google Scholar

show all references

References:
[1]

D. AldilaN. Nuraini and E. Soewono, Mathematical model for the spread of extreme ideology, AIP Conference Proceedings, 1651 (2015), 33-39.  doi: 10.1063/1.4914429.  Google Scholar

[2]

Z. Artstein, Limiting Equations and Stability of Nonautonomous Ordinary Differential Equations, in: J. P. LaSalle (ed.), The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.  Google Scholar

[3]

A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, Journal of Differential Equations, 233 (2007), 622-653.  doi: 10.1016/j.jde.2006.08.009.  Google Scholar

[4]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity Vol. 1: Theory of Epidemics, Wuerz, Winnipeg, Canada, 1995. Google Scholar

[5]

S. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, 251, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4613-8159-4.  Google Scholar

[6]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, Journal of Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

[7]

J. Földes and P. Poláčik, Convergence to a steady state for asymptotically autonomous semilinear heat equations on $ \mathbb{R}^{N}$, Journal of Differential Equations, 251 (2011), 1903-1922.  doi: 10.1016/j.jde.2011.04.002.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[9]

S. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Analysis, 46 (2001), 675-698.  doi: 10.1016/S0362-546X(00)00145-0.  Google Scholar

[10]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[11]

D. Li and Z. Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J., 67 (2018), 583-621.  doi: 10.1512/iumj.2018.67.7292.  Google Scholar

[12]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.  Google Scholar

[13]

K. MischaikowH. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[14]

R. Schnaubelt, Asymptotically autonomous parabolic evolution equations, Journal of Evolution Equations, 1 (2001), 19-37.  doi: 10.1007/PL00001363.  Google Scholar

[15]

A. Strauss and J. A. Yorke, Perturbing uniform asymptotically stable nonlinear systems, Journal of Differential Equations, 6 (1969), 452-483.  doi: 10.1016/0022-0396(69)90004-7.  Google Scholar

[16]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[17]

X. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonlinear Anal., 3 (1996), 43-66.   Google Scholar

[1]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315

[2]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[3]

Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002

[4]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[6]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[7]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017

[8]

Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031

[9]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[10]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[11]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[12]

Manuel Friedrich, Martin Kružík, Ulisse Stefanelli. Equilibrium of immersed hyperelastic solids. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021003

[13]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400

[14]

Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363

[15]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[16]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391

[17]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160

[18]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010

[19]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176

[20]

Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021013

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (135)
  • HTML views (156)
  • Cited by (1)

Other articles
by authors

[Back to Top]