# American Institute of Mathematical Sciences

May  2013, 33(5): 1835-1856. doi: 10.3934/dcds.2013.33.1835

## Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance

 1 Department of Mathematics - University of Torino, Via Carlo Alberto, 10 - 10123 Torino, Italy, Italy 2 School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received  January 2012 Revised  July 2012 Published  December 2012

In this paper we deal with the existence of unbounded orbits of the map $$\left\{\begin{array}{l} θ_1= θ+\frac{1}{ρ} [u(θ)-l_1(ρ)]+h_1(ρ, θ), ρ_1=ρ-u'(θ)+l_2(ρ)+h_2(ρ, θ), \end{array} \right.$$ where $\mu$ is continuous and $2\pi$-periodic, $l_1$, $l_2$ are continuous and bounded, $h_1(\rho, \theta)=o(\rho^{-1})$, $h_2(\rho, \theta)=o(1)$, for $\rho\to+\infty$. We prove that every orbit of the map tends to infinity in the future or in the past for $\rho$ large enough provided that $$[\liminf_{\rho\to+\infty}l_1(\rho), \limsup_{\rho\to+\infty}l_1(\rho)]\cap Range(\mu)=\emptyset$$ and other conditions hold. On the basis of this conclusion, we prove that the system $Jz'=\nabla H(z)+f(z)+p(t)$ has unbounded solutions when $H$ is positively homogeneous of degree 2 and positive. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this system.
Citation: Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835
##### References:
 [1] J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator,, J. Differential Equations, 143 (1998), 201. doi: 10.1006/jdeq.1997.3367. Google Scholar [2] D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance,, Differential Integral Equations, 15 (2002), 1139. Google Scholar [3] D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance,, Discrete Contin. Dynam. Systems, 8 (2002), 907. doi: 10.3934/dcds.2002.8.907. Google Scholar [4] A. Capietto, W. Dambrosio and Z. Wang, Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance,, Proc. Edinburgh Math. Soc. A, 138 (2008), 15. doi: 10.1017/S030821050600062X. Google Scholar [5] A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, J. London Math. Soc., 68 (2003), 119. doi: 10.1112/S0024610703004459. Google Scholar [6] N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations,, Bull. Austral. Math. Soc., 15 (1976), 321. Google Scholar [7] C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,, J. Differential Equations, 147 (1998), 58. doi: 10.1006/jdeq.1998.3441. Google Scholar [8] C. Fabry and A. Fonda, Unbounded motions of perturbed isochronous hamiltonian systems at resonance,, Adv. Nonlinear Stud., 5 (2005), 351. Google Scholar [9] C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance,, Nonlinearity, 13 (2000), 493. doi: 10.1088/0951-7715/13/3/302. Google Scholar [10] A. Fonda, Positively homogeneous Hamiltonian systems in the plane,, J. Differential Equations, 200 (2004), 162. doi: 10.1016/j.jde.2004.02.001. Google Scholar [11] A. Fonda and J. Mawhin, Planar differential systems at resonance,, Adv. Differential Equations, 11 (2006), 1111. Google Scholar [12] N. G. Lloyd, "Degree Theory,", Cambridge University Press, (1978). Google Scholar [13] A. I. Luré, Some nonlinear problems of the theory of automatic regulation (Russian),, Gostekhizdat, (1951). Google Scholar [14] D. B. Qian, On resonance phenomena for asymmetric weakly nonlinear oscillator,, Sci. China Ser. A, 45 (2002), 214. Google Scholar [15] Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives,, Discrete Contin. Dynam. Systems, 9 (2003), 751. doi: 10.3934/dcds.2003.9.751. Google Scholar [16] Z. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, Sci. China Ser. A, 8 (2007), 1205. doi: 10.1007/s11425-007-0070-z. Google Scholar [17] X. Yang, Unboundedness of the large solutions of someasymmetric oscillators at resonance,, Math. Nachr., 276 (2004), 89. doi: 10.1002/mana.200310215. Google Scholar

show all references

##### References:
 [1] J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator,, J. Differential Equations, 143 (1998), 201. doi: 10.1006/jdeq.1997.3367. Google Scholar [2] D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance,, Differential Integral Equations, 15 (2002), 1139. Google Scholar [3] D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance,, Discrete Contin. Dynam. Systems, 8 (2002), 907. doi: 10.3934/dcds.2002.8.907. Google Scholar [4] A. Capietto, W. Dambrosio and Z. Wang, Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance,, Proc. Edinburgh Math. Soc. A, 138 (2008), 15. doi: 10.1017/S030821050600062X. Google Scholar [5] A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, J. London Math. Soc., 68 (2003), 119. doi: 10.1112/S0024610703004459. Google Scholar [6] N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations,, Bull. Austral. Math. Soc., 15 (1976), 321. Google Scholar [7] C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,, J. Differential Equations, 147 (1998), 58. doi: 10.1006/jdeq.1998.3441. Google Scholar [8] C. Fabry and A. Fonda, Unbounded motions of perturbed isochronous hamiltonian systems at resonance,, Adv. Nonlinear Stud., 5 (2005), 351. Google Scholar [9] C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance,, Nonlinearity, 13 (2000), 493. doi: 10.1088/0951-7715/13/3/302. Google Scholar [10] A. Fonda, Positively homogeneous Hamiltonian systems in the plane,, J. Differential Equations, 200 (2004), 162. doi: 10.1016/j.jde.2004.02.001. Google Scholar [11] A. Fonda and J. Mawhin, Planar differential systems at resonance,, Adv. Differential Equations, 11 (2006), 1111. Google Scholar [12] N. G. Lloyd, "Degree Theory,", Cambridge University Press, (1978). Google Scholar [13] A. I. Luré, Some nonlinear problems of the theory of automatic regulation (Russian),, Gostekhizdat, (1951). Google Scholar [14] D. B. Qian, On resonance phenomena for asymmetric weakly nonlinear oscillator,, Sci. China Ser. A, 45 (2002), 214. Google Scholar [15] Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives,, Discrete Contin. Dynam. Systems, 9 (2003), 751. doi: 10.3934/dcds.2003.9.751. Google Scholar [16] Z. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, Sci. China Ser. A, 8 (2007), 1205. doi: 10.1007/s11425-007-0070-z. Google Scholar [17] X. Yang, Unboundedness of the large solutions of someasymmetric oscillators at resonance,, Math. Nachr., 276 (2004), 89. doi: 10.1002/mana.200310215. Google Scholar
 [1] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [2] Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385 [3] José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653 [4] Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361 [5] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 [6] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [7] Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197 [8] Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023 [9] Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193 [10] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [11] Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487 [12] Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789 [13] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [14] Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121 [15] Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007 [16] Nikolaos S. Papageorgiou, Patrick Winkert. Double resonance for Robin problems with indefinite and unbounded potential. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 323-344. doi: 10.3934/dcdss.2018018 [17] H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127 [18] Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919 [19] Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423 [20] Rinaldo M. Colombo, Mauro Garavello. Comparison among different notions of solution for the $p$-system at a junction. Conference Publications, 2009, 2009 (Special) : 181-190. doi: 10.3934/proc.2009.2009.181

2018 Impact Factor: 1.143