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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 |
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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