-
Previous Article
Entire solutions with merging fronts to a bistable periodic lattice dynamical system
- DCDS Home
- This Issue
-
Next Article
A classification of volume preserving generating forms in $\mathbb{R}^3$
Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics
1. | School of Mathematic Science, Yancheng Teacher's University, Yancheng 224001, China |
2. | School of Mathematical Sciences, Soochow University, Suzhou 215006 |
3. | School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin, 541003, China |
References:
[1] |
D. Bonheune and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bouncing, Nonlinearity, 15 (2002), 1281-1297.
doi: 10.1088/0951-7715/15/4/314. |
[2] |
T. Burton and R. Grimmer, On the continuability of solutions of second-order differential equations, Proc. Amer. Math. Soc., 29 (1971), 277-283. |
[3] |
G. J. Butler, Rapid oscillation, nonextendability and the existence of periodic solutions to second-order nonlinear differential equations, J. Differential Equations, 22 (1976), 467-477.
doi: 10.1016/0022-0396(76)90041-3. |
[4] |
A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438.
doi: 10.1006/jdeq.2001.4080. |
[5] |
T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.
doi: 10.1016/0022-0396(92)90076-Y. |
[6] |
J. Hale, Ordinary Differential Equation, Robert E. Krieger Publishing Co., Inc., Huntington, 1980. |
[7] |
M. Jiang, Periodic solutions of second order differential equations with an obstacle, Nonlinearity, 19 (2006), 1165-1183.
doi: 10.1088/0951-7715/19/5/007. |
[8] |
M. Kunze, Non-Smooth Dynamical Systems, Spring-Verlag, New York, 2000.
doi: 10.1007/BFb0103843. |
[9] |
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator, Physica D, 82 (1995), 117-135.
doi: 10.1016/0167-2789(94)00222-C. |
[10] |
A. Lazer and P. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differential and Integral Equations, 5 (1992), 165-172.
doi: 10.2307/2152750. |
[11] |
Q. Liu and Z. Wang, Periodic impact behavior of a class of Hamiltonian oscillators with obstacles, J. Math. Anal. Appl., 365 (2010), 67-74.
doi: 10.1016/j.jmaa.2009.09.054. |
[12] |
G. Luo and J. Xie, Bifurcations and chaos in a system with impacts, Physica D, 148 (2001), 183-200.
doi: 10.1016/S0167-2789(00)00170-6. |
[13] |
D. Papini, Infinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign, J. Math. Anal. Appl, 247 (2000), 217-235.
doi: 10.1006/jmaa.2000.6849. |
[14] |
D. Papini, Prescribing the nodal behaviour of periodic solutions of a superlinear equation with indefinite weight, Atti. Sem. Mat. Fis. Univ. Modena., 51 (2003), 43-63. |
[15] |
D. Papini and F. Zanolin, A topological approach to superlinear indefinite boundary-value problem, Topol. Methods Nonlinear Anal., 15 (2000), 203-233. |
[16] |
M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, Cambridge Univercity Press, Cambridge, UK, 1998.
doi: 10.1017/CBO9781139173049. |
[17] |
D. Qian, Large amplitude periodic bouncing in impact oscillators with damping, Proc. Amer. Math. Soc., 133 (2005), 1797-1804.
doi: 10.1090/S0002-9939-04-07759-7. |
[18] |
D. Qian and P. Torres, Periodic motions of linear impact oscillators via successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.
doi: 10.1137/S003614100343771X. |
[19] |
D. Qian and P. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edingburgh Sect. A, 134 (2004), 201-213.
doi: 10.1017/S0308210500003164. |
[20] |
A. Ruiz-Herrera and P. Torres, Periodic solutions and chaotic dynamics in forced impact oscillators, SIAM J. Appl. Dyn. Syst., 12 (2013), 383-414.
doi: 10.1137/120880902. |
[21] |
S. W. Shaw and P. Holmes, Periodically forced linear oscillator with impact: Chaos and long-period motions, Phy. Rev. Lett., 51 (1983), 623-626.
doi: 10.1103/PhysRevLett.51.623. |
[22] |
R. Srzednicki, On geometric detection of periodic solutions and chaos, Non-linear analysis and boundary value problems for ordinary differential equations (Udine), CISM Courses and Lectures, Springer, Vienna, 371 (1996), 197-209. |
[23] |
C. Wang, The periodic motions of a class of symmetric superlinear Hill's impact equations, (in Chinese) Sci. Sin. Math., 44 (2014), 235-248. |
[24] |
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts, Comm. Math. Phys., 211 (2000), 289-302.
doi: 10.1007/s002200050813. |
show all references
References:
[1] |
D. Bonheune and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bouncing, Nonlinearity, 15 (2002), 1281-1297.
doi: 10.1088/0951-7715/15/4/314. |
[2] |
T. Burton and R. Grimmer, On the continuability of solutions of second-order differential equations, Proc. Amer. Math. Soc., 29 (1971), 277-283. |
[3] |
G. J. Butler, Rapid oscillation, nonextendability and the existence of periodic solutions to second-order nonlinear differential equations, J. Differential Equations, 22 (1976), 467-477.
doi: 10.1016/0022-0396(76)90041-3. |
[4] |
A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438.
doi: 10.1006/jdeq.2001.4080. |
[5] |
T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.
doi: 10.1016/0022-0396(92)90076-Y. |
[6] |
J. Hale, Ordinary Differential Equation, Robert E. Krieger Publishing Co., Inc., Huntington, 1980. |
[7] |
M. Jiang, Periodic solutions of second order differential equations with an obstacle, Nonlinearity, 19 (2006), 1165-1183.
doi: 10.1088/0951-7715/19/5/007. |
[8] |
M. Kunze, Non-Smooth Dynamical Systems, Spring-Verlag, New York, 2000.
doi: 10.1007/BFb0103843. |
[9] |
H. Lamba, Chaotic, regular and unbounded behaviour in the elastic impact oscillator, Physica D, 82 (1995), 117-135.
doi: 10.1016/0167-2789(94)00222-C. |
[10] |
A. Lazer and P. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differential and Integral Equations, 5 (1992), 165-172.
doi: 10.2307/2152750. |
[11] |
Q. Liu and Z. Wang, Periodic impact behavior of a class of Hamiltonian oscillators with obstacles, J. Math. Anal. Appl., 365 (2010), 67-74.
doi: 10.1016/j.jmaa.2009.09.054. |
[12] |
G. Luo and J. Xie, Bifurcations and chaos in a system with impacts, Physica D, 148 (2001), 183-200.
doi: 10.1016/S0167-2789(00)00170-6. |
[13] |
D. Papini, Infinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign, J. Math. Anal. Appl, 247 (2000), 217-235.
doi: 10.1006/jmaa.2000.6849. |
[14] |
D. Papini, Prescribing the nodal behaviour of periodic solutions of a superlinear equation with indefinite weight, Atti. Sem. Mat. Fis. Univ. Modena., 51 (2003), 43-63. |
[15] |
D. Papini and F. Zanolin, A topological approach to superlinear indefinite boundary-value problem, Topol. Methods Nonlinear Anal., 15 (2000), 203-233. |
[16] |
M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, Cambridge Univercity Press, Cambridge, UK, 1998.
doi: 10.1017/CBO9781139173049. |
[17] |
D. Qian, Large amplitude periodic bouncing in impact oscillators with damping, Proc. Amer. Math. Soc., 133 (2005), 1797-1804.
doi: 10.1090/S0002-9939-04-07759-7. |
[18] |
D. Qian and P. Torres, Periodic motions of linear impact oscillators via successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.
doi: 10.1137/S003614100343771X. |
[19] |
D. Qian and P. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edingburgh Sect. A, 134 (2004), 201-213.
doi: 10.1017/S0308210500003164. |
[20] |
A. Ruiz-Herrera and P. Torres, Periodic solutions and chaotic dynamics in forced impact oscillators, SIAM J. Appl. Dyn. Syst., 12 (2013), 383-414.
doi: 10.1137/120880902. |
[21] |
S. W. Shaw and P. Holmes, Periodically forced linear oscillator with impact: Chaos and long-period motions, Phy. Rev. Lett., 51 (1983), 623-626.
doi: 10.1103/PhysRevLett.51.623. |
[22] |
R. Srzednicki, On geometric detection of periodic solutions and chaos, Non-linear analysis and boundary value problems for ordinary differential equations (Udine), CISM Courses and Lectures, Springer, Vienna, 371 (1996), 197-209. |
[23] |
C. Wang, The periodic motions of a class of symmetric superlinear Hill's impact equations, (in Chinese) Sci. Sin. Math., 44 (2014), 235-248. |
[24] |
V. Zharnitsky, Invariant tori in Hamiltonian systems with impacts, Comm. Math. Phys., 211 (2000), 289-302.
doi: 10.1007/s002200050813. |
[1] |
Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167 |
[2] |
Weijun Zhan, Qian Guo, Yuhao Cong. The truncated Milstein method for super-linear stochastic differential equations with Markovian switching. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3663-3682. doi: 10.3934/dcdsb.2021201 |
[3] |
E. N. Dancer. On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3861-3869. doi: 10.3934/dcds.2012.32.3861 |
[4] |
Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049 |
[5] |
Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014 |
[6] |
Zuzana Došlá, Mauro Marini, Serena Matucci. Global Kneser solutions to nonlinear equations with indefinite weight. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3297-3308. doi: 10.3934/dcdsb.2018252 |
[7] |
Daxiong Piao, Xiang Sun. Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials. Communications on Pure and Applied Analysis, 2014, 13 (2) : 645-655. doi: 10.3934/cpaa.2014.13.645 |
[8] |
Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025 |
[9] |
Alberto Boscaggin, Maurizio Garrione. Positive solutions to indefinite Neumann problems when the weight has positive average. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5231-5244. doi: 10.3934/dcds.2016028 |
[10] |
M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure and Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411 |
[11] |
Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315 |
[12] |
Wenxian Shen, Xiaoxia Xie. Spectraltheory for nonlocal dispersal operators with time periodic indefinite weight functions and applications. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1023-1047. doi: 10.3934/dcdsb.2017051 |
[13] |
Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436 |
[14] |
Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195 |
[15] |
Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283 |
[16] |
Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363 |
[17] |
Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013 |
[18] |
Nikolaos S. Papageorgiou, Vicenšiu D. Rădulescu, Dušan D. Repovš. Robin problems with indefinite linear part and competition phenomena. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1293-1314. doi: 10.3934/cpaa.2017063 |
[19] |
Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044 |
[20] |
Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]