# American Institute of Mathematical Sciences

April  2016, 36(4): 2305-2328. doi: 10.3934/dcds.2016.36.2305

## 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

Received  November 2014 Revised  July 2015 Published  September 2015

In this paper, we are concerned with superlinear impact oscillators of Hill's type with indefinite weight $$\left\{\begin{array}{lll} x''+f(x)x'+q(t)g(x)=0, ~\text{for}~ x(t)>0;\\ x(t)\geq0;\\ x'(t_0+)=-x'(t_0-),~\text{if}~x(t_0)=0,\end{array}\right.$$ where the indefinite weight $q(t)$, defined in $(a,b)$ with $-\infty\leq a< b \leq+\infty,$ has infinitely many zeros in $(a,b),$ $g$ is superlinear and $f$ is bounded. We prove the existence of globally defined bouncing solutions with prescribed number of impacts in the intervals of negativity and positivity of $q$. Furthermore, we show that when $q$ is periodic, the equation under consideration exhibits an interesting phenomenon of chaotic-like dynamics. Finally, in case that $q$ is even and periodic, we prove the existence and multiplicity of the even and periodic bouncing solutions for the Hill's type equation in case of $f\equiv0.$
Citation: Chao Wang, Dingbian Qian, Qihuai Liu. Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2305-2328. doi: 10.3934/dcds.2016.36.2305
##### 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.

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##### 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.
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