-
Previous Article
Existence of solutions for the $p-$Laplacian with crossing nonlinearity
- DCDS Home
- This Issue
-
Next Article
A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$
Higher order Melnikov function for a quartic hamiltonian with cuspidal loop
| 1. | Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China |
| 2. | Department of Mathematics, Zhongshan (Sun Yat-sen) University, 510275, Guangzhou, China |
$X_\epsilon=(H_y+\epsilon f(x,y,\epsilon))\frac{\partial}{\partial x}+ (-H_x+\epsilon g(x,y,\epsilon))\frac{\partial}{\partial y},$
where the Hamiltonian $H(x,y)=\frac{1}{2}y^2+U(x)$ has one center and one cuspidal loop, $deg U(x)=4$. In present paper we find an upper bound for the number of zeros of the $k$th order Melnikov function $M_k(h)$ for arbitrary polynomials $f(x,y,\epsilon)$ and $g(x,y,\epsilon)$.
| [1] |
Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5661-5679. doi: 10.3934/dcdsb.2020375 |
| [2] |
Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367 |
| [3] |
Hang Zheng, Yonghui Xia. Chaotic threshold of a class of hybrid piecewise-smooth system by an impulsive effect via Melnikov-type function. Discrete & Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021319 |
| [4] |
Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021, 11 (3) : 653-679. doi: 10.3934/mcrf.2021017 |
| [5] |
Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387 |
| [6] |
Guy Joseph Eyebe, Betchewe Gambo, Alidou Mohamadou, Timoleon Crepin Kofane. Melnikov analysis of the nonlocal nanobeam resting on fractional-order softening nonlinear viscoelastic foundations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2213-2228. doi: 10.3934/dcdss.2020252 |
| [7] |
R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435 |
| [8] |
Dmitry Strunin, Mayada Mohammed. Validity and dynamics in the nonlinearly excited 6th-order phase equation. Conference Publications, 2013, 2013 (special) : 719-728. doi: 10.3934/proc.2013.2013.719 |
| [9] |
Daniele Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 537-549. doi: 10.3934/dcds.2008.21.537 |
| [10] |
Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3803-3819. doi: 10.3934/dcdss.2021019 |
| [11] |
Monica Marras, Stella Vernier-Piro. A note on a class of 4th order hyperbolic problems with weak and strong damping and superlinear source term. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2047-2055. doi: 10.3934/dcdss.2020157 |
| [12] |
Ravi P. Agarwal, Abdullah Özbekler. Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2281-2300. doi: 10.3934/cpaa.2016037 |
| [13] |
Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 |
| [14] |
Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293 |
| [15] |
Guillaume Duval, Andrzej J. Maciejewski. Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1969-2009. doi: 10.3934/dcds.2015.35.1969 |
| [16] |
Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure & Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164 |
| [17] |
Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134 |
| [18] |
V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511 |
| [19] |
John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443 |
| [20] |
Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]