- Previous Article
- DCDS Home
- This Issue
-
Next Article
Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations
The generalized Liénard systems
1. | Department of Mathematics, University of Turku, FIN-20014 Turku, Finland, Finland |
$\frac{dx}{dt} = \frac{1}{a(x)}[h(y)-F(x)],$
$\frac{dy}{dt}= -a(x)g(x),\qquad\qquad\qquad\qquad\qquad$ (0.1)
where $a$ is a positive and continuous function on $R=(-\infty, \infty)$, and $F$, $g$ and $h$ are continuous functions on $R$. Under the assumption that the origin is a unique equilibrium, we obtain necessary and sufficient conditions for the origin of system (0.1) to be globally asymptotically stable by using a nonlinear integral inequality. Our results substantially extend and improve several known results in the literature.
[1] |
Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121 |
[2] |
N. Kamran, K. Tenenblat. Periodic systems for the higher-dimensional Laplace transformation. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 359-378. doi: 10.3934/dcds.1998.4.359 |
[3] |
Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883 |
[4] |
Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485 |
[5] |
Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293 |
[6] |
W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 |
[7] |
Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218 |
[8] |
Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911 |
[9] |
Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579 |
[10] |
Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, 2021, 29 (5) : 2987-3015. doi: 10.3934/era.2021023 |
[11] |
Peng Chen, Linfeng Mei, Xianhua Tang. Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021279 |
[12] |
Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367 |
[13] |
Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080 |
[14] |
Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042 |
[15] |
Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010 |
[16] |
Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315 |
[17] |
Peter Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4839-4865. doi: 10.3934/dcdsb.2020315 |
[18] |
Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589 |
[19] |
Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 |
[20] |
Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]