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

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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

Anna Capietto ^1, , Walter Dambrosio ^1, , Tiantian Ma ^2, and Zaihong Wang ^2,

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

Abstract
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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.

Keywords: Hamiltonian system, unbounded solution., periodic solution, resonance.

Mathematics Subject Classification: Primary: 34C25; Secondary: 34B1.

Citation: Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete and Continuous Dynamical Systems, 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), 201-220. doi: 10.1006/jdeq.1997.3367.
[2]	D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral Equations, 15 (2002), 1139-1152.
[3]	D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907-930. doi: 10.3934/dcds.2002.8.907.
[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), 15-32. doi: 10.1017/S030821050600062X.
[5]	A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc., 68 (2003), 119-132. doi: 10.1112/S0024610703004459.
[6]	N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328.
[7]	C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. doi: 10.1006/jdeq.1998.3441.
[8]	C. Fabry and A. Fonda, Unbounded motions of perturbed isochronous hamiltonian systems at resonance, Adv. Nonlinear Stud., 5 (2005), 351-373.
[9]	C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance, Nonlinearity, 13 (2000), 493-505. doi: 10.1088/0951-7715/13/3/302.
[10]	A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differential Equations, 200 (2004), 162-184. doi: 10.1016/j.jde.2004.02.001.
[11]	A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations, 11 (2006), 1111-1133.
[12]	N. G. Lloyd, "Degree Theory," Cambridge University Press, 1978.
[13]	A. I. Luré, Some nonlinear problems of the theory of automatic regulation (Russian), Gostekhizdat, (1951), pp.26.
[14]	D. B. Qian, On resonance phenomena for asymmetric weakly nonlinear oscillator, Sci. China Ser. A, 45 (2002), 214-222.
[15]	Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives, Discrete Contin. Dynam. Systems, 9 (2003), 751-770. doi: 10.3934/dcds.2003.9.751.
[16]	Z. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, Sci. China Ser. A, 8 (2007), 1205-1216. doi: 10.1007/s11425-007-0070-z.
[17]	X. Yang, Unboundedness of the large solutions of someasymmetric oscillators at resonance, Math. Nachr., 276 (2004), 89-102. doi: 10.1002/mana.200310215.

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), 201-220. doi: 10.1006/jdeq.1997.3367.
[2]	D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral Equations, 15 (2002), 1139-1152.
[3]	D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907-930. doi: 10.3934/dcds.2002.8.907.
[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), 15-32. doi: 10.1017/S030821050600062X.
[5]	A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc., 68 (2003), 119-132. doi: 10.1112/S0024610703004459.
[6]	N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328.
[7]	C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. doi: 10.1006/jdeq.1998.3441.
[8]	C. Fabry and A. Fonda, Unbounded motions of perturbed isochronous hamiltonian systems at resonance, Adv. Nonlinear Stud., 5 (2005), 351-373.
[9]	C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance, Nonlinearity, 13 (2000), 493-505. doi: 10.1088/0951-7715/13/3/302.
[10]	A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differential Equations, 200 (2004), 162-184. doi: 10.1016/j.jde.2004.02.001.
[11]	A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations, 11 (2006), 1111-1133.
[12]	N. G. Lloyd, "Degree Theory," Cambridge University Press, 1978.
[13]	A. I. Luré, Some nonlinear problems of the theory of automatic regulation (Russian), Gostekhizdat, (1951), pp.26.
[14]	D. B. Qian, On resonance phenomena for asymmetric weakly nonlinear oscillator, Sci. China Ser. A, 45 (2002), 214-222.
[15]	Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives, Discrete Contin. Dynam. Systems, 9 (2003), 751-770. doi: 10.3934/dcds.2003.9.751.
[16]	Z. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, Sci. China Ser. A, 8 (2007), 1205-1216. doi: 10.1007/s11425-007-0070-z.
[17]	X. Yang, Unboundedness of the large solutions of someasymmetric oscillators at resonance, Math. Nachr., 276 (2004), 89-102. doi: 10.1002/mana.200310215.

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