American Institute of Mathematical Sciences

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

On the eventual stability of asymptotically autonomous systems with constraints

Jinlong Bai 1, , Xuewei Ju 2, , Desheng Li 1,, and  Xiulian Wang 3,

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.

Keywords: Asymptotically autonomous system, constraint, equilibrium, eventual asymptotic stability, eventual instability.
Mathematics Subject Classification: 34D05, 34D20, 37B55, 93D20.
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. Aldila, N. 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. Li, L. 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. Mischaikow, H. 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. Aldila, N. 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. Li, L. 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. Mischaikow, H. 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

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