American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[2]

D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral Equations, 15 (2002), 1139-1152.  Google Scholar

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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.  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), 15-32. 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), 119-132. doi: 10.1112/S0024610703004459.  Google Scholar

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N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328.  Google Scholar

[7]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. 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), 351-373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations, 11 (2006), 1111-1133.  Google Scholar

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N. G. Lloyd, "Degree Theory," Cambridge University Press, 1978.  Google Scholar

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A. I. Luré, Some nonlinear problems of the theory of automatic regulation (Russian), Gostekhizdat, (1951), pp.26.  Google Scholar

[14]

D. B. Qian, On resonance phenomena for asymmetric weakly nonlinear oscillator, Sci. China Ser. A, 45 (2002), 214-222.  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), 751-770. 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), 1205-1216. doi: 10.1007/s11425-007-0070-z.  Google Scholar

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X. Yang, Unboundedness of the large solutions of someasymmetric oscillators at resonance, Math. Nachr., 276 (2004), 89-102. 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), 201-220. 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), 1139-1152.  Google Scholar

[3]

D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907-930. 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), 15-32. 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), 119-132. doi: 10.1112/S0024610703004459.  Google Scholar

[6]

N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328.  Google Scholar

[7]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. 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), 351-373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations, 11 (2006), 1111-1133.  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), pp.26.  Google Scholar

[14]

D. B. Qian, On resonance phenomena for asymmetric weakly nonlinear oscillator, Sci. China Ser. A, 45 (2002), 214-222.  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), 751-770. 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), 1205-1216. 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), 89-102. doi: 10.1002/mana.200310215.  Google Scholar

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